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Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models

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Andrew McCulloch, Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models, Journal of the Royal Statistical Society Series A: Statistics in Society , Volume 168, Issue 2, March 2005, Page 466, https://doi.org/10.1111/j.1467-985X.2005.358_16.x

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Sensitivity analysis is defined as

‘the study of how the uncertainty in the output of a model (numerical or otherwise) can be apportioned to different sources of uncertainty in the model input’

(page 45). This book aims to guide a non-expert user through the most widely used methods in the area. It includes discussion of implementation of the methods by using SIMLAB, a Windows software package for sensitivity analysis that has been written by the authors. The main difficulty in sensitivity analysis is ensuring that the range of variation in parameters and/or input variables has been examined in a combined way. The general approach that is introduced in Chapter 1 is to use sampling-based sensitivity analysis in which the model is executed repeatedly for combinations of values sampled from the distribution of the input factors. Chapter 2 sets out the five general steps that are required for a sensitivity analysis: design the experiment, assign density functions or ranges of variation to input factors, generate the input vectors through the design, create the corresponding output distribution and assess the relative importance of inputs. The remainder of the book is an elaboration of how to perform these five steps in different circumstances and particularly how to use quantitative indicators in the assessment stage. Throughout the book the approach is that a complete sensitivity analysis cannot be performed where all combinations of parameter values are examined and so a sampling procedure must be used. The methods are presented via worked examples and case-studies. I did not find this successful, perhaps because I found the SIMLAB software difficult to use. The statistical methods are perhaps more clearly presented in the authors’ published papers and I would have preferred a more traditional lay-out which set out the methods before presenting applications. Most of the worked examples are from the engineering and physical sciences. The methods that are presented may have less application in the social sciences where what causes the model output is usually the focus of investigation, rather than what the sources of variation in the output are.

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9.7   Sensitivity analyses

The process of undertaking a systematic review involves a sequence of decisions. Whilst many of these decisions are clearly objective and non-contentious, some will be somewhat arbitrary or unclear. For instance, if inclusion criteria involve a numerical value, the choice of value is usually arbitrary: for example, defining groups of older people may reasonably have lower limits of 60, 65, 70 or 75 years, or any value in between. Other decisions may be unclear because a study report fails to include the required information. Some decisions are unclear because the included studies themselves never obtained the information required: for example, the outcomes of those who unfortunately were lost to follow-up. Further decisions are unclear because there is no consensus on the best statistical method to use for a particular problem.

It is desirable to prove that the findings from a systematic review are not dependent on such arbitrary or unclear decisions. A sensitivity analysis is a repeat of the primary analysis or meta-analysis, substituting alternative decisions or ranges of values for decisions that were arbitrary or unclear. For example, if the eligibility of some studies in the meta-analysis is dubious because they do not contain full details, sensitivity analysis may involve undertaking the meta-analysis twice: first, including all studies and second, only including those that are definitely known to be eligible. A sensitivity analysis asks the question, “Are the findings robust to the decisions made in the process of obtaining them?”.

There are many decision nodes within the systematic review process which can generate a need for a sensitivity analysis. Examples include:

Searching for studies:

Should abstracts whose results cannot be confirmed in subsequent publications be included in the review?

Eligibility criteria:

Characteristics of participants: where a majority but not all people in a study meet an age range, should the study be included?

Characteristics of the intervention: what range of doses should be included in the meta-analysis?

Characteristics of the comparator: what criteria are required to define usual care to be used as a comparator group?

Characteristics of the outcome: what time-point or range of time-points are eligible for inclusion?

Study design: should blinded and unblinded outcome assessment be included, or should study inclusion be restricted by other aspects of methodological criteria?

What data should be analysed?

Time-to-event data: what assumptions of the distribution of censored data should be made?

Continuous data: where standard deviations are missing, when and how should they be imputed? Should analyses be based on change scores or on final values?

Ordinal scales: what cut-point should be used to dichotomize short ordinal scales into two groups?

Cluster-randomized trials: what values of the intraclass correlation coefficient should be used when trial analyses have not been adjusted for clustering?

Cross-over trials: what values of the within-subject correlation coefficient should be used when this is not available in primary reports?

All analyses: what assumptions should be made about missing outcomes to facilitate intention-to-treat analyses? Should adjusted or unadjusted estimates of treatment effects used?

Analysis methods:

Should fixed-effect or random-effects methods be used for the analysis?

For dichotomous outcomes, should odds ratios, risk ratios or risk differences be used?

And for continuous outcomes, where several scales have assessed the same dimension, should results be analysed as a standardized mean difference across all scales or as mean differences individually for each scale?

Some sensitivity analyses can be pre-specified in the study protocol, but many issues suitable for sensitivity analysis are only identified during the review process where the individual peculiarities of the studies under investigation are identified. When sensitivity analyses show that the overall result and conclusions are not affected by the different decisions that could be made during the review process, the results of the review can be regarded with a higher degree of certainty. Where sensitivity analyses identify particular decisions or missing information that greatly influence the findings of the review, greater resources can be deployed to try and resolve uncertainties and obtain extra information, possibly through contacting trial authors and obtained individual patient data. If this cannot be achieved, the results must be interpreted with an appropriate degree of caution. Such findings may generate proposals for further investigations and future research.

Reporting of sensitivity analyses in a systematic review may best be done by producing a summary table. Rarely is it informative to produce individual forest plots for each sensitivity analysis undertaken.

Sensitivity analyses are sometimes confused with subgroup analysis. Although some sensitivity analyses involve restricting the analysis to a subset of the totality of studies, the two methods differ in two ways. First, sensitivity analyses do not attempt to estimate the effect of the intervention in the group of studies removed from the analysis, whereas in subgroup analyses, estimates are produced for each subgroup. Second, in sensitivity analyses, informal comparisons are made between different ways of estimating the same thing, whereas in subgroup analyses, formal statistical comparisons are made across the subgroups.

Introduction to Sensitivity Analysis

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• Bertrand Iooss 4 , 5 &
• Andrea Saltelli 6 , 7  

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Sensitivity analysis provides users of mathematical and simulation models with tools to appreciate the dependency of the model output from model input and to investigate how important is each model input in determining its output. All application areas are concerned, from theoretical physics to engineering and socio-economics. This introductory paper provides the sensitivity analysis aims and objectives in order to explain the composition of the overall “Sensitivity Analysis” chapter of the Springer Handbook. It also describes the basic principles of sensitivity analysis, some classification grids to understand the application ranges of each method, a useful software package, and the notations used in the chapter papers. This section also offers a succinct description of sensitivity auditing, a new discipline that tests the entire inferential chain including model development, implicit assumptions, and normative issues and which is recommended when the inference provided by the model needs to feed into a regulatory or policy process. For the “Sensitivity Analysis” chapter, in addition to this introduction, eight papers have been written by around twenty practitioners from different fields of application. They cover the most widely used methods for this subject: the deterministic methods as the local sensitivity analysis, the experimental design strategies, the sampling-based and variance-based methods developed from the 1980s, and the new importance measures and metamodel-based techniques established and studied since the 2000s. In each paper, toy examples or industrial applications illustrate their relevance and usefulness.

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Iooss, B., Saltelli, A. (2015). Introduction to Sensitivity Analysis. In: Ghanem, R., Higdon, D., Owhadi, H. (eds) Handbook of Uncertainty Quantification. Springer, Cham. https://doi.org/10.1007/978-3-319-11259-6_31-1

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March 6th, 2024

What Is Sensitivity Analysis in Statistics & How Is It Used?

By Connor Martin · 9 min read

Numbers and data don’t lie, but, when you carry out analysis of that data, a range of variables and factors may influence the outcomes of that analysis. Sensitivity analysis statistics help you understand and even visualize the impact of those variables.

Thus, sensitivity analysis is a crucial tool for assessing the overall robustness and reliability of any study’s findings. This guide breaks down what sensitivity analysis statistics are and the benefits they bring.

What Is Sensitivity Analysis?

First, a definition. Sensitivity analysis is, in simple terms, the process of working out how different variables impact the end result of a study or test. It involves analyzing those variables closely, and assessing how a change in one or more variables may affect the overall outcome.

In essence, sensitivity analyses are about asking “What if?” – “What if variable X was a bit bigger?” or “What if we cut variable Y in half, what would happen?” They’re even sometimes referred to as “What if?” tests by those who use them.

In short, it’s about exploring possibilities, and it’s beneficial in a vast array of scenarios, from scientific research to the world of finance (more on that below).

How Does Sensitivity Analysis Work?

How sensitivity analysis statistics are gathered and used will vary from industry to industry and field to field.

However, the general principles remain the same:

- Researchers must first identify variables that impact their outcomes.

- Then, they explore the effects of those variables.

This may involve processes like repeating a study numerous times or using other, pre-existing data to estimate the impact of altered variables.

Let’s look at an example to see the concept in action more clearly.

Imagine medical researchers are testing the effectiveness of a new medication. Their clinical trials up to that point suggest that the drug has a positive impact on recovery times and symptom reduction. However, they wish to carry out a sensitivity analysis to deepen their understanding.

The primary analysis would begin by identifying variables from their study that could have influenced the end results. Then, they explore those variables.

As an example, they might have initially run clinical trials on people within a certain age bracket. They may like to explore how the presence of older or younger participants may impact the results, so they could run the test again with a broader selection of age groups.

Alternatively, perhaps their initial study used the drug in a set dose every time. They may want to see how slightly smaller or larger doses affect its effectiveness.

By asking these kinds of “What if?” questions and finding sensitivity analysis statistics to answer them, the researchers gain additional insight and information about their subject of study.

How & Where Is Sensitivity Analysis Used?

Sensitivity analyses aren’t just relevant for scientific studies. They have far-reaching, widespread applications across multiple industries.

Scientific Research

Scientific researchers regularly use sensitivity analysis statistics to expand their understanding of study subjects. As the previous example illustrated, they might expand clinical trials of a drug to account for an array of variables, like dosage or the age, gender, and health state of the participants.

Climate Models

Sensitivity analysis may apply to climate models and projections of how climate could change in the future. Naturally, at a time when climate change is under such close scrutiny, this is particularly important.

For example, it’s widely understood that carbon emissions cause gradual increases in global temperatures. But we can dig deeper into this concept through sensitivity analysis, exploring how changes in variables, like the amount of emissions or the expansion of industry, might affect the rate of temperature rise.

Business ROI

Sensitivity analysis statistics are also highly prized by those in business and finance. They can help companies make informed and intelligent investment decisions to maximize their ROI (return on investment).

As an example, a company prepares a marketing drive for a new product. They estimate a 20% ROI. To determine if there’s a better way to boost their ROI, they conduct sensitivity analyses, looking at how conversion rates and ad costs could impact the outcome.

Engineering

In the field of engineering and infrastructure, sensitivity analysis often helps to make small but crucial changes or improvements to a design or concept.

For example, imagine engineers are planning to build a bridge that needs to withstand a certain amount of weight. They conduct sensitivity analysis to see how other variable factors, like wind speed and changes in weather conditions, impact the bridge’s structural integrity, and how different materials might make the final product more or less stable.

Tips for Sensitivity Analysis

Sensitivity analysis statistics are invaluable for discovering and deepening our understanding of any study or piece of research. But, to maximize the value of statistical analyses , they have to be carried out correctly. Here are some tips to help:

- Focus on Key Variables: It’s easy to get overwhelmed when commencing a sensitivity analysis, considering the dozens of potential variables to examine. Ignore those that are less influential and focus on a few key factors that you deem to have the biggest impact on the outcome.

- Be Reasonable with Your Ranges: When you’re looking at how a variable might impact the result if it were bigger, smaller, broader, or narrower, be sensible and realistic with how far you push it.

- Harness Emerging Technologies: Emerging tech tools, like artificial intelligence (AI), make sensitivity analysis and other forms of stats analysis dramatically simpler. Identify and use these tools to help with your variable study and results visualization.

Make the Most of Your Analysis with Julius AI

Whether you’re seeking to make the most of an investment or digging into the nitty-gritty of a scientific study, sensitivity analysis will help you. But it can be tricky and time-consuming to carry out correctly. Julius AI is here to help.

As the most advanced AI data analyst, Julius AI can assist with all major aspects of stats analysis, including identifying variables for your sensitivity analyses, generating visuals to display your findings, and much more. Give it a try today and take your analysis to the next level.

Frequently Asked Questions (FAQs)

What is an example of a sensitivity analysis?

An example of a sensitivity analysis is examining the impact of different dosage levels of a new medication on patient recovery times in a clinical trial. Researchers might vary the dose and observe changes in outcomes to determine the optimal dosage for effectiveness.

What is the sensitivity analysis method?

The sensitivity analysis method involves identifying key variables that affect the outcome of a study, systematically altering these variables, and analyzing the resulting changes in the outcome. This helps researchers understand the influence of each variable and the robustness of their findings.

How is sensitivity measured in statistics?

In statistics, sensitivity is often measured by calculating how changes in input variables lead to changes in the output of a model. This can be quantified using metrics like partial derivatives, scenario analysis, or through simulations that assess the variability of outcomes under different conditions.

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• Published: 16 July 2013

A tutorial on sensitivity analyses in clinical trials: the what, why, when and how

• Lehana Thabane 1 , 2 , 3 , 4 , 5 ,
• Lawrence Mbuagbaw 1 , 4 ,
• Shiyuan Zhang 1 , 4 ,
• Zainab Samaan 1 , 6 , 7 ,
• Maura Marcucci 1 , 4 ,
• Chenglin Ye 1 , 4 ,
• Marroon Thabane 1 , 8 ,
• Lora Giangregorio 9 ,
• Brittany Dennis 1 , 4 ,
• Daisy Kosa 1 , 4 , 10 ,
• Victoria Borg Debono 1 , 4 ,
• Rejane Dillenburg 11 ,
• Vincent Fruci 12 ,
• Monica Bawor 13 ,
• Juneyoung Lee 14 ,
• George Wells 15 &
• Charles H Goldsmith 1 , 4 , 16  

BMC Medical Research Methodology volume  13 , Article number:  92 ( 2013 ) Cite this article

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Sensitivity analyses play a crucial role in assessing the robustness of the findings or conclusions based on primary analyses of data in clinical trials. They are a critical way to assess the impact, effect or influence of key assumptions or variations—such as different methods of analysis, definitions of outcomes, protocol deviations, missing data, and outliers—on the overall conclusions of a study.

The current paper is the second in a series of tutorial-type manuscripts intended to discuss and clarify aspects related to key methodological issues in the design and analysis of clinical trials.

In this paper we will provide a detailed exploration of the key aspects of sensitivity analyses including: 1) what sensitivity analyses are, why they are needed, and how often they are used in practice; 2) the different types of sensitivity analyses that one can do, with examples from the literature; 3) some frequently asked questions about sensitivity analyses; and 4) some suggestions on how to report the results of sensitivity analyses in clinical trials.

When reporting on a clinical trial, we recommend including planned or posthoc sensitivity analyses, the corresponding rationale and results along with the discussion of the consequences of these analyses on the overall findings of the study.

Peer Review reports

The credibility or interpretation of the results of clinical trials relies on the validity of the methods of analysis or models used and their corresponding assumptions. An astute researcher or reader may be less confident in the findings of a study if they believe that the analysis or assumptions made were not appropriate. For a primary analysis of data from a prospective randomized controlled trial (RCT), the key questions for investigators (and for readers) include:

How confident can I be about the results?

Will the results change if I change the definition of the outcome (e.g., using different cut-off points)?

Will the results change if I change the method of analysis?

Will the results change if we take missing data into account? Will the method of handling missing data lead to different conclusions?

How much influence will minor protocol deviations have on the conclusions?

How will ignoring the serial correlation of measurements within a patient impact the results?

What if the data were assumed to have a non-Normal distribution or there were outliers?

Will the results change if one looks at subgroups of patients?

Will the results change if the full intervention is received (i.e. degree of compliance)?

The above questions can be addressed by performing sensitivity analyses—testing the effect of these “changes” on the observed results. If, after performing sensitivity analyses the findings are consistent with those from the primary analysis and would lead to similar conclusions about treatment effect, the researcher is reassured that the underlying factor(s) had little or no influence or impact on the primary conclusions. In this situation, the results or the conclusions are said to be “robust”.

The objectives of this paper are to provide an overview of how to approach sensitivity analyses in clinical trials. This is the second in a series of tutorial-type manuscripts intended to discuss and clarify aspects related to some key methodological issues in the design and analysis of clinical trials. The first was on pilot studies [ 1 ]. We start by describing what sensitivity analysis is, why it is needed and how often it is done in practice. We then describe the different types of sensitivity analyses that one can do, with examples from the literature. We also address some of the commonly asked questions about sensitivity analysis and provide some guidance on how to report sensitivity analyses.

Sensitivity Analysis

What is a sensitivity analysis in clinical research.

Sensitivity Analysis (SA) is defined as “a method to determine the robustness of an assessment by examining the extent to which results are affected by changes in methods, models, values of unmeasured variables, or assumptions” with the aim of identifying “results that are most dependent on questionable or unsupported assumptions” [ 2 ]. It has also been defined as “a series of analyses of a data set to assess whether altering any of the assumptions made leads to different final interpretations or conclusions” [ 3 ]. Essentially, SA addresses the “what-if-the-key-inputs-or-assumptions-changed”-type of question. If we want to know whether the results change when something about the way we approach the data analysis changes, we can make the change in our analysis approach and document the changes in the results or conclusions. For more detailed coverage of SA, we refer the reader to these references [ 4 – 7 ].

Why is sensitivity analysis necessary?

The design and analysis of clinical trials often rely on assumptions that may have some effect, influence or impact on the conclusions if they are not met. It is important to assess these effects through sensitivity analyses. Consistency between the results of primary analysis and the results of sensitivity analysis may strengthen the conclusions or credibility of the findings. However, it is important to note that the definition of consistency may depend in part on the area of investigation, the outcome of interest or even the implications of the findings or results.

It is equally important to assess the robustness to ensure appropriate interpretation of the results taking into account the things that may have an impact on them. Thus, it imperative for every analytic plan to have some sensitivity analyses built into it.

The United States (US) Food and Drug Administration (FDA) and the European Medicines Association (EMEA), which offer guidance on Statistical Principles for Clinical Trials, state that “it is important to evaluate the robustness of the results and primary conclusions of the trial.” Robustness refers to “the sensitivity of the overall conclusions to various limitations of the data, assumptions, and analytic approaches to data analysis” [ 8 ]. The United Kingdom (UK) National Institute of Health and Clinical Excellence (NICE) also recommends the use of sensitivity analysis in “exploring alternative scenarios and the uncertainty in cost-effectiveness results” [ 9 ].

How often is sensitivity analysis reported in practice?

To evaluate how often sensitivity analyses are used in medical and health research, we surveyed the January 2012 editions of major medical journals (British Medical Journal, New England Journal of Medicine, the Lancet, Journal of the American Medical Association and the Canadian Medical Association Journal) and major health economics journals (Pharmaco-economics, Medical Decision making, European Journal of Health Economics, Health Economics and the Journal of Health Economics). From every article that included some form of statistical analyses, we evaluated: i) the percentage of published articles that reported results of some sensitivity analyses; and ii) the types of sensitivity analyses that were performed. Table  1 provides a summary of the findings. Overall, the point prevalent use of sensitivity analyses is about 26.7% (36/135) —which seems very low. A higher percentage of papers published in health economics than in medical journals (30.8% vs. 20.3%) reported some sensitivity analyses. Among the papers in medical journals, 18 (28.1%) were RCTs, of which only 3 (16.6%) reported sensitivity analyses. Assessing robustness of the findings to different methods of analysis was the most common type of sensitivity analysis reported in both types of journals. Therefore despite their importance, sensitivity analyses are under-used in practice. Further, sensitivity analyses are more common in health economics research—for example in conducting cost-effectiveness analyses, cost-utility analyses or budget-impact analyses—than in other areas of health or medical research.

Types of sensitivity analyses

In this section, we describe scenarios that may require sensitivity analyses, and how one could use sensitivity analyses to assess the robustness of the statistical analyses or findings of RCTs. These are not meant to be exhaustive, but rather to illustrate common situations where sensitivity analyses might be useful to consider (Table  2 ). In each case, we provide examples of actual studies where sensitivity analyses were performed, and the implications of these sensitivity analyses.

Impact of outliers

An outlier is an observation that is numerically distant from the rest of the data. It deviates markedly from the rest of the sample from which it comes [ 14 , 15 ]. Outliers are usually exceptional cases in a sample. The problem with outliers is that they can deflate or inflate the mean of a sample and therefore influence any estimates of treatment effect or association that are derived from the mean. To assess the potential impact of outliers, one would first assess whether or not any observations meet the definition of an outlier—using either a boxplot or z-scores [ 16 ]. Second, one could perform a sensitivity analysis with and without the outliers.

In a cost–utility analysis of a practice-based osteopathy clinic for subacute spinal pain, Williams et al. reported lower costs per quality of life year ratios when they excluded outliers [ 17 ]. In other words, there were certain participants in the trial whose costs were very high, and were making the average costs look higher than they probably were in reality. The observed cost per quality of life year was not robust to the exclusion of outliers, and changed when they were excluded.

A primary analysis based on the intention-to-treat principle showed no statistically significant differences in reducing depression between a nurse-led cognitive self-help intervention program compared to standard care among 218 patients hospitalized with angina over 6 months. Some sensitivity analyses in this trial were performed by excluding participants with high baseline levels of depression (outliers) and showed a statistically significant reduction in depression in the intervention group compared to the control. This implies that the results of the primary analysis were affected by the presence of patients with baseline high depression [ 18 ].

Impact of non-compliance or protocol deviations

In clinical trials some participants may not adhere to the intervention they were allocated to receive or comply with the scheduled treatment visits. Non-adherence or non-compliance is a form of protocol deviation. Other types of protocol deviations include switching between intervention and control arms (i.e. treatment switching or crossovers) [ 19 , 20 ], or not implementing the intervention as prescribed (i.e. intervention fidelity) [ 21 , 22 ].

Protocol deviations are very common in interventional research [ 23 – 25 ]. The potential impact of protocol deviations is the dilution of the treatment effect [ 26 , 27 ]. Therefore, it is crucial to determine the robustness of the results to the inclusion of data from participants who deviate from the protocol. Typically, for RCTs the primary analysis is based on an intention-to-treat (ITT) principle—in which participants are analyzed according to the arm to which they were randomized, irrespective of whether they actually received the treatment or completed the prescribed regimen [ 28 , 29 ]. Two common types of sensitivity analyses can be performed to assess the robustness of the results to protocol deviations: 1) per-protocol (PP) analysis—in which participants who violate the protocol are excluded from the analysis [ 30 ]; and 2) as-treated (AT) analysis—in which participants are analyzed according to the treatment they actually received [ 30 ]. The PP analysis provides the ideal scenario in which all the participants comply, and is more likely to show an effect; whereas the ITT analysis provides a “real life” scenario, in which some participants do not comply. It is more conservative, and less likely to show that the intervention is effective. For trials with repeated measures, some protocol violations which lead to missing data can be dealt with alternatively. This is covered in more detail in the next section.

A trial was designed to investigate the effects of an electronic screening and brief intervention to change risky drinking behaviour in university students. The results of the ITT analysis (on all 2336 participants who answered the follow-up survey) showed that the intervention had no significant effect. However, a sensitivity analysis based on the PP analysis (including only those with risky drinking at baseline and who answered the follow-up survey; n = 408) suggested a small beneficial effect on weekly alcohol consumption [ 31 ]. A reader might be less confident in the findings of the trial because of the inconsistency between the ITT and PP analyses—the ITT was not robust to sensitivity analyses. A researcher might choose to explore differences in the characteristics of the participants who were included in the ITT versus the PP analyses.

A study compared the long-term effects of surgical versus non-surgical management of chronic back pain. Both the ITT and AT analyses showed no significant difference between the two management strategies [ 32 ]. A reader would be more confident in the findings because the ITT and AT analyses were consistent—the ITT was robust to sensitivity analyses.

Impact of missing data

Missing data are common in every research study. This is a problem that can be broadly defined as “missing some information on the phenomena in which we are interested” [ 33 ]. Data can be missing for different reasons including (1) non-response in surveys due to lack of interest, lack of time, nonsensical responses, and coding errors in data entry/transfer; (2) incompleteness of data in large data registries due to missing appointments, not everyone is captured in the database, and incomplete data; and (3) missingness in prospective studies as a result of loss to follow up, dropouts, non-adherence, missing doses, and data entry errors.

The choice of how to deal with missing data would depend on the mechanisms of missingness. In this regard, data can be missing at random (MAR), missing not at random (MNAR), or missing completely at random (MCAR). When data are MAR, the missing data are dependent on some other observed variables rather than any unobserved one. For example, consider a trial to investigate the effect of pre-pregnancy calcium supplementation on hypertensive disorders in pregnancy. Missing data on the hypertensive disorders is dependent (conditional) on being pregnant in the first place. When data are MCAR, the cases with missing data may be considered a random sample drawn from all the cases. In other words, there is no “cause” of missingness. Consider the example of a trial comparing a new cancer treatment to standard treatment in which participants are followed at 4, 8, 12 and 16 months. If a participant misses the follow up at the 8th and 16th months and these are unrelated to the outcome of interest, in this case mortality, then this missing data is MCAR. Reasons such as a clinic staff being ill or equipment failure are often unrelated to the outcome of interest. However, the MCAR assumption is often challenging to prove because the reason data is missing may not be known and therefore it is difficult to determine if it is related to the outcome of interest. When data are MNAR, missingness is dependent on some unobserved data. For example, in the case above, if the participant missed the 8th month appointment because he was feeling worse or the 16th month appointment because he was dead, the missingness is dependent on the data not observed because the participant was absent. When data are MAR or MCAR, they are often referred to as ignorable (provided the cause of MAR is taken into account). MNAR on the other hand, is nonignorable missingness. Ignoring the missingness in such data leads to biased parameter estimates [ 34 ]. Ignoring missing data in analyses can have implications on the reliability, validity and generalizability of research findings.

The best way to deal with missing data is prevention, by steps taken in the design and data collection stages, some of which have been described by Little et al. [ 35 ]. But this is difficult to achieve in most cases. There are two main approaches to handling missing data: i) ignore them—and use complete case analysis; and ii) impute them—using either single or multiple imputation techniques. Imputation is one of the most commonly used approaches to handling missing data. Examples of single imputation methods include hot deck, cold deck method, mean imputation, regression technique, last observation carried forward (LOCF) and composite methods—which uses a combination of the above methods to impute missing values. Single imputation methods often lead to biased estimates and under-estimation of the true variability in the data. Multiple imputation (MI) technique is currently the best available method of dealing with missing data under the assumption that data are missing at random (MAR) [ 33 , 36 – 38 ]. MI addresses the limitations of single imputation by using multiple imputed datasets which yield unbiased estimates, and also accounts for the within- and between-dataset variability. Bayesian methods using statistical models that assume a prior distribution for the missing data can also be used to impute data [ 35 ].

It is important to note that ignoring missing data in the analysis would be implicitly assuming that the data are MCAR, an assumption that is often hard to verify in reality.

There are some statistical approaches to dealing with missing data that do not necessarily require formal imputation methods. For example, in studies using continuous outcomes, linear mixed models for repeated measures are used for analyzing outcomes measured repeatedly over time [ 39 , 40 ]. For categorical responses or count data, generalized estimating equations [GEE] and random-effects generalized linear mixed models [GLMM] methods may be used [ 41 , 42 ]. In these models it is assumed that missing data are MAR. If this assumption is valid, then the complete-case analysis by including predictors of missing observations will provide consistent estimates of the parameter.

The choice of whether to ignore or impute missing data, and how to impute it, may affect the findings of the trial. Although one approach (ignore or impute, and if the latter, how to impute) should be made a priori, a sensitivity analysis can be done with a different approach to see how “robust” the primary analysis is to the chosen method for handling missing data.

A 2011 paper reported the sensitivity analyses of different strategies for imputing missing data in cluster RCTs with a binary outcome using the community hypertension assessment trial (CHAT) as an example. They found that variance in the treatment effect was underestimated when the amount of missing data was large and the imputation strategy did not take into account the intra-cluster correlation. However, the effects of the intervention under various methods of imputation were similar. The CHAT intervention was not superior to usual care [ 43 ].

In a trial comparing methotrexate with to placebo in the treatment of psoriatic arthritis, the authors reported both an intention-to-treat analysis (using multiple imputation techniques to account for missing data) and a complete case analysis (ignoring the missing data). The complete case analysis, which is less conservative, showed some borderline improvement in the primary outcome (psoriatic arthritis response criteria), while the intention-to-treat analysis did not [ 44 ]. A reader would be less confident about the effects of methotrexate on psoriatic arthritis, due to the discrepancy between the results with imputed data (ITT) and the complete case analysis.

Impact of different definitions of outcomes (e.g. different cut-off points for binary outcomes)

Often, an outcome is defined by achieving or not achieving a certain level or threshold of a measure. For example in a study measuring adherence rates to medication, levels of adherence can be dichotomized as achieving or not achieving at least 80%, 85% or 90% of pills taken. The choice of the level a participant has to achieve can affect the outcome—it might be harder to achieve 90% adherence than 80%. Therefore, a sensitivity analysis could be performed to see how redefining the threshold changes the observed effect of a given intervention.

In a trial comparing caspofungin to amphotericin B for febrile neutropoenic patients, a sensitivity analysis was conducted to investigate the impact of different definitions of fever resolution as part of a composite endpoint which included: resolution of any baseline invasive fungal infection, no breakthrough invasive fungal infection, survival, no premature discontinuation of study drug, and fever resolution for 48 hours during the period of neutropenia. They found that response rates were higher when less stringent fever resolution definitions were used, especially in low-risk patients. The modified definitions of fever resolution were: no fever for 24 hours before the resolution of neutropenia; no fever at the 7-day post-therapy follow-up visit; and removal of fever resolution completely from the composite endpoint. This implies that the efficacy of both medications depends somewhat on the definition of the outcomes [ 45 ].

In a phase II trial comparing minocycline and creatinine to placebo for Parkinson’s disease, a sensitivity analysis was conducted based on another definition (threshold) for futility. In the primary analysis a predetermined futility threshold was set at 30% reduction in mean change in Unified Parkinson’s Disease Rating Scale (UPDRS) score, derived from historical control data. If minocycline or creatinine did not bring about at least a 30% reduction in UPDRS score, they would be considered as futile and no further testing will be conducted. Based on the data derived from the current control (placebo) group, a new threshold of 32.4% (more stringent) was used for the sensitivity analysis. The findings from the primary analysis and the sensitivity analysis both confirmed that that neither creatine nor minocycline could be rejected as futile and should both be tested in Phase III trials [ 46 ]. A reader would be more confident of these robust findings.

Impact of different methods of analysis to account for clustering or correlation

Interventions can be administered to individuals, but they can also be administered to clusters of individuals, or naturally occurring groups. For example, one might give an intervention to students in one class, and compare their outcomes to students in another class – the class is the cluster. Clusters can also be patients treated by the same physician, physicians in the same practice center or hospital, or participants living in the same community. Likewise, in the same trial, participants may be recruited from multiple sites or centers. Each of these centers will represent a cluster. Patients or elements within a cluster often have some appreciable degree of homogeneity as compared to patients between clusters. In other words, members of the same cluster are more likely to be similar to each other than they are to members of another cluster, and this similarity may then be reflected in the similarity or correlation measure, on the outcome of interest.

There are several methods of accounting or adjusting for similarities within clusters, or “clustering” in studies where this phenomenon is expected or exists as part of the design (e.g., in cluster randomization trials). Therefore, in assessing the impact of clustering one can build into the analytic plans two forms of sensitivity analyses: i) analysis with and without taking clustering into account—comparing the analysis that ignores clustering (i.e. assumes that the data are independent) to one primary method chosen to account for clustering; ii) analysis that compares several methods of accounting for clustering.

Correlated data may also occur in longitudinal studies through repeat or multiple measurements from the same patient, taken over time or based on multiple responses in a single survey. Ignoring the potential correlation between several measurements from an individual can lead to inaccurate conclusions [ 47 ].

Here are a few references to studies that compared the outcomes that resulted when different methods were/were not used to account for clustering. Noteworthy, is the fact that the analytical approaches for cluster-RCTs and multi-site RCTs are similar.

Ma et al. performed sensitivity analyses of different methods of analysing cluster RCTs [ 48 ]. In this paper they compared three cluster-level methods (un-weighted linear regression, weighted linear regression and random-effects meta-regression) to six individual level analysis methods (standard logistic regression, robust standard errors approach, GEE, random effects meta-analytic approach, random-effects logistic regression and Bayesian random-effects regression). Using data from the CHAT trial, in this analysis, all nine methods provided similar results, re-enforcing the hypothesis that the CHAT intervention was not superior to usual care.

Peters et al. conducted sensitivity analyses to compare different methods—three cluster-level (un-weighted regression of practice log odds, regression of log odds weighted by their inverse variance and random-effects meta-regression of log odds with cluster as a random effect) and five individual-level methods (standard logistic regression ignoring clustering, robust standard errors, GEE, random-effects logistic regression and Bayesian random-effects logistic regression.)—for analyzing cluster randomized trials using an example involving a factorial design [ 13 ]. In this analysis, they demonstrated that the methods used in the analysis of cluster randomized trials could give varying results, with standard logistic regression ignoring clustering being the least conservative.

Cheng et al. used sensitivity analyses to compare different methods (six models for clustered binary outcomes and three models for clustered nominal outcomes) of analysing correlated data in discrete choice surveys [ 49 ]. The results were robust to various statistical models, but showed more variability in the presence of a larger cluster effect (higher within-patient correlation).

A trial evaluated the effects of lansoprazole on gastro-esophageal reflux disease in children from 19 clinics with asthma. The primary analysis was based on GEE to determine the effect of lansoprazole in reducing asthma symptoms. Subsequently they performed a sensitivity analysis by including the study site as a covariate. Their finding that lansoprazole did not significantly improve symptoms was robust to this sensitivity analysis [ 50 ].

In addition to comparing the performance of different methods to estimate treatment effects on a continuous outcome in simulated multicenter randomized controlled trials [ 12 ], the authors used data from the Computerization of Medical Practices for the Enhancement of Therapeutic Effectiveness (COMPETE) II [ 51 ] to assess the robustness of the primary results (based on GEE to adjust for clustering by provider of care) under different methods of adjusting for clustering. The results, which showed that a shared electronic decision support system improved care and outcomes in diabetic patients, were robust under different methods of analysis.

Impact of competing risks in analysis of trials with composite outcomes

A competing risk event happens in situations where multiple events are likely to occur in a way that the occurrence of one event may prevent other events from being observed [ 48 ]. For example, in a trial using a composite of death, myocardial infarction or stroke, if someone dies, they cannot experience a subsequent event, or stroke or myocardial infarction—death can be a competing risk event. Similarly, death can be a competing risk in trials of patients with malignant diseases where thrombotic events are important. There are several options for dealing with competing risks in survival analyses: (1) to perform a survival analysis for each event separately, where the other competing event(s) is/are treated as censored; the common representation of survival curves using the Kaplan-Meier estimator is in this context replaced by the cumulative incidence function (CIF) which offers a better interpretation of the incidence curve for one risk, regardless of whether the competing risks are independent; (2) to use a proportional sub-distribution hazard model (Fine & Grey approach) in which subjects that experience other competing events are kept in the risk set for the event of interest (i.e. as if they could later experience the event); (3) to fit one model, rather than separate models, taking into account all the competing risks together (Lunn-McNeill approach) [ 13 ]. Therefore, the best approach to assessing the influence of a competing risk would be to plan for sensitivity analysis that adjusts for the competing risk event.

A previously-reported trial compared low molecular weight heparin (LMWH) with oral anticoagulant therapy for the prevention of recurrent venous thromboembolism (VTE) in patients with advanced cancer, and a subsequent study presented sensitivity analyses comparing the results from standard survival analysis (Kaplan-Meier method) with those from competing risk methods—namely, the cumulative incidence function (CIF) and Gray's test [ 52 ]. The results using both methods were similar. This strengthened their confidence in the conclusion that LMWH reduced the risk of recurrent VTE.

For patients at increased risk of end stage renal disease (ESRD) but also of premature death not related to ESRD, such as patients with diabetes or with vascular disease, analyses considering the two events as different outcomes may be misleading if the possibility of dying before the development of ESRD is not taken into account [ 49 ]. Different studies performing sensitivity analyses demonstrated that the results on predictors of ESRD and death for any cause were dependent on whether the competing risks were taken into account or not [ 53 , 54 ], and on which competing risk method was used [ 55 ]. These studies further highlight the need for a sensitivity analysis of competing risks when they are present in trials.

Impact of baseline imbalance in RCTs

In RCTs, randomization is used to balance the expected distribution of the baseline or prognostic characteristics of the patients in all treatment arms. Therefore the primary analysis is typically based on ITT approach unadjusted for baseline characteristics. However, some residual imbalance can still occur by chance. One can perform a sensitivity analysis by using a multivariable analysis to adjust for hypothesized residual baseline imbalances to assess their impact on effect estimates.

A paper presented a simulation study where the risk of the outcome, effect of the treatment, power and prevalence of the prognostic factors, and sample size were all varied to evaluate their effects on the treatment estimates. Logistic regression models were compared with and without adjustment for the prognostic factors. The study concluded that the probability of prognostic imbalance in small trials could be substantial. Also, covariate adjustment improved estimation accuracy and statistical power [ 56 ].

In a trial testing the effectiveness of enhanced communication therapy for aphasia and dysarthria after stroke, the authors conducted a sensitivity analysis to adjust for baseline imbalances. Both primary and sensitivity analysis showed that enhanced communication therapy had no additional benefit [ 57 ].

Impact of distributional assumptions

Most statistical analyses rely on distributional assumptions for observed data (e.g. Normal distribution for continuous outcomes, Poisson distribution for count data, or binomial distribution for binary outcome data). It is important not only to test for goodness-of-fit for these distributions, but to also plan for sensitivity analyses using other suitable distributions. For example, for continuous data, one can redo the analysis assuming a Student-T distribution—which is symmetric, bell-shaped distribution like the Normal distribution, but with thicker tails; for count data, once can use the Negative-binomial distribution—which would be useful to assess the robustness of the results if over-dispersion is accounted for [ 52 ]. Bayesian analyses routinely include sensitivity analyses to assess the robustness of findings under different models for the data and prior distributions [ 58 ]. Analyses based on parametric methods—which often rely on strong distributional assumptions—may also need to be evaluated for robustness using non-parametric methods. The latter often make less stringent distributional assumptions. However, it is essential to note that in general non-parametric methods are less efficient (i.e. have less statistical power) than their parametric counter-parts if the data are Normally distributed.

Ma et al. performed sensitivity analyses based on Bayesian and classical methods for analysing cluster RCTs with a binary outcome in the CHAT trial. The similarities in the results after using the different methods confirmed the results of the primary analysis: the CHAT intervention was not superior to usual care [ 10 ].

A negative binomial regression model was used [ 52 ] to analyze discrete outcome data from a clinical trial designed to evaluate the effectiveness of a pre-habilitation program in preventing functional decline among physically frail, community-living older persons. The negative binomial model provided an improved fit to the data than the Poisson regression model. The negative binomial model provides an alternative approach for analyzing discrete data where over-dispersion is a problem [ 59 ].

Commonly asked questions about sensitivity analyses

Q: Do I need to adjust the overall level of significance for performing sensitivity analyses?

A: No. Sensitivity analysis is typically a re-analysis of either the same outcome using different approaches, or different definitions of the outcome—with the primary goal of assessing how these changes impact the conclusions. Essentially everything else including the criterion for statistical significance needs to be kept constant so that we can assess whether any impact is attributable to underlying sensitivity analyses.

Q: Do I have to report all the results of the sensitivity analyses?

A: Yes, especially if the results are different or lead to different a conclusion from the original results—whose sensitivity was being assessed. However, if the results remain robust (i.e. unchanged), then a brief statement to this effect may suffice.

Q: Can I perform sensitivity analyses posthoc?

A: It is desirable to document all planned analyses including sensitivity analyses in the protocol a priori . Sometimes, one cannot anticipate all the challenges that can occur during the conduct of a study that may require additional sensitivity analyses. In that case, one needs to incorporate the anticipated sensitivity analyses in the statistical analysis plan (SAP), which needs to be completed before analyzing the data. Clear rationale is needed for every sensitivity analysis. This may also occur posthoc .

Q: How do I choose between the results of different sensitivity analyses? (i.e. which results are the best?)

A: The goal of sensitivity analyses is not to select the “best” results. Rather, the aim is to assess the robustness or consistency of the results under different methods, subgroups, definitions, assumptions and so on. The assessment of robustness is often based on the magnitude, direction or statistical significance of the estimates. You cannot use the sensitivity analysis to choose an alternate conclusion to your study. Rather, you can state the conclusion based on your primary analysis, and present your sensitivity analysis as an example of how confident you are that it represents the truth. If the sensitivity analysis suggests that the primary analysis is not robust, it may point to the need for future research that might address the source of the inconsistency. Your study cannot answer the question which results are best? To answer the question of which method is best and under what conditions, simulation studies comparing the different approaches on the basis of bias, precision, coverage or efficiency may be necessary.

Q: When should one perform sensitivity analysis?

A: The default position should be to plan for sensitivity analysis in every clinical trial. Thus, all studies need to include some sensitivity analysis to check the robustness of the primary findings. All statistical methods used to analyze data from clinical trials rely on assumptions—which need to either be tested whenever possible, with the results assessed for robustness through some sensitivity analyses. Similarly, missing data or protocol deviations are common occurrences in many trials and their impact on inferences needs to be assessed.

Q: How many sensitivity analyses can one perform for a single primary analysis?

A: The number is not an important factor in determining what sensitivity analyses to perform. The most important factor is the rationale for doing any sensitivity analysis. Understanding the nature of the data, and having some content expertise are useful in determining which and how many sensitivity analyses to perform. For example, varying the ways of dealing with missing data is unlikely to change the results if 1% of data are missing. Likewise, understanding the distribution of certain variables can help to determine which cut points would be relevant. Typically, it is advisable to limit sensitivity analyses to the primary outcome. Conducting multiple sensitivity analysis on all outcomes is often neither practical, nor necessary.

Q: How many factors can I vary in performing sensitivity analyses?

A: Ideally, one can study the impact of all key elements using a factorial design—which would allow the assessment of the impact of individual and joint factors. Alternatively, one can vary one factor at a time to be able to assess whether the factor is responsible for the resulting impact (if any). For example, in a sensitivity analysis to assess the impact of the Normality assumption (analysis assuming Normality e.g. T-test vs. analysis without assuming Normality e.g. Based on a sign test) and outlier (analysis with and without outlier), this can be achieved through 2x2 factorial design.

Q: What is the difference between secondary analyses and sensitivity analyses?

A: Secondary analyses are typically analyses of secondary outcomes. Like primary analyses which deal with primary outcome(s), such analyses need to be documented in the protocol or SAP. In most studies such analyses are exploratory—because most studies are not powered for secondary outcomes. They serve to provide support that the effects reported in the primary outcome are consistent with underlying biology. They are different from sensitivity analyses as described above.

Q: What is the difference between subgroup analyses and sensitivity analyses?

A: Subgroup analyses are intended to assess whether the effect is similar across specified groups of patients or modified by certain patient characteristics [ 60 ]. If the primary results are statistically significant, subgroup analyses are intended to assess whether the observed effect is consistent across the underlying patient subgroups—which may be viewed as some form of sensitivity analysis. In general, for subgroup analyses one is interested in the results for each subgroup, whereas in subgroup “sensitivity” analyses, one is interested in the similarity of results across subgroups (ie. robustness across subgroups). Typically subgroup analyses require specification of the subgroup hypothesis and rationale, and performed through inclusion of an interaction term (i.e. of the subgroup variable x main exposure variable) in the regression model. They may also require adjustment for alpha—the overall level of significance. Furthermore, most studies are not usually powered for subgroup analyses.

Reporting of sensitivity analyses

There has been considerable attention paid to enhancing the transparency of reporting of clinical trials. This has led to several reporting guidelines, starting with the CONSORT Statement [ 61 ] in 1996 and its extensions [ http://www.equator-network.org ]. Not one of these guidelines specifically addresses how sensitivity analyses need to be reported. On the other hand, there is some guidance on how sensitivity analyses need to be reported in economic analyses [ 62 ]—which may partly explain the differential rates of reporting of sensitivity analyses shown in Table  1 . We strongly encourage some modifications of all reporting guidelines to include items on sensitivity analyses—as a way to enhance their use and reporting. The proposed reporting changes can be as follows:

In Methods Section: Report the planned or posthoc sensitivity analyses and rationale for each.

In Results Section: Report whether or not the results of the sensitivity analyses or conclusions are similar to those based on primary analysis. If similar, just state that the results or conclusions remain robust. If different, report the results of the sensitivity analyses along with the primary results.

In Discussion Section: Discuss the key limitations and implications of the results of the sensitivity analyses on the conclusions or findings. This can be done by describing what changes the sensitivity analyses bring to the interpretation of the data, and whether the sensitivity analyses are more stringent or more relaxed than the primary analysis.

Some concluding remarks

Sensitivity analyses play an important role is checking the robustness of the conclusions from clinical trials. They are important in interpreting or establishing the credibility of the findings. If the results remain robust under different assumptions, methods or scenarios, this can strengthen their credibility. The results of our brief survey of January 2012 editions of major medical and health economics journals that show that their use is very low. We recommend that some sensitivity analysis should be the default plan in statistical or economic analyses of any clinical trial. Investigators need to identify any key assumptions, variations, or methods that may impact or influence the findings, and plan to conduct some sensitivity analyses as part of their analytic strategy. The final report must include the documentation of the planned or posthoc sensitivity analyses, rationale, corresponding results and a discussion of their consequences or repercussions on the overall findings.

Abbreviations

• Sensitivity analysis

United States

Food and Drug Administration

European Medicines Association

United Kingdom

National Institute of Health and Clinical Excellence

Randomized controlled trial

Intention-to-treat

Per-protocol

Last observation carried forward

Multiple imputation

Missing at random

Generalized estimating equations

Generalized linear mixed models

Community hypertension assessment trial

Prostate specific antigen

Cumulative incidence function

End stage renal disease

Instrumental variable

Analysis of covariance

Statistical analysis plan

Consolidated Standards of Reporting Trials.

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LT conceived the idea and drafted the outline and paper. GW, CHG and MT commented on the idea and draft outline. LM and SZ performed literature search and data abstraction. ZS, LG and CY edited and formatted the manuscript. MM, BD, DK, VBD, RD, VF, MB, JL reviewed and revised draft versions of the manuscript. All authors reviewed several draft versions of the manuscript and approved the final manuscript.

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Thabane, L., Mbuagbaw, L., Zhang, S. et al. A tutorial on sensitivity analyses in clinical trials: the what, why, when and how. BMC Med Res Methodol 13 , 92 (2013). https://doi.org/10.1186/1471-2288-13-92

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What Is Sensitivity Analysis?

How it works, how businesses use sensitivity analysis.

• Pros and Cons

The Bottom Line

• Corporate Finance
• Financial Analysis

Sensitivity analysis shows how different values of an independent variable affect a dependent variable under a given set of assumptions. Companies use sensitivity analysis to identify opportunities, mitigate risk, and communicate decisions to upper management.

Sensitivity analysis is deployed in business and economics by financial analysts and economists and is also known as a "what-if" analysis.

Key Takeaways

• Sensitivity analysis shows how different values of an independent variable affect a dependent variable under a given set of assumptions.
• This model is called "what-if" or simulation analysis.
• Sensitivity analysis helps predict share prices of publicly traded companies or how interest rates affect bond prices.

Investopedia / Lara Antal

Sensitivity analysis is a financial model that determines how target variables are affected based on changes in input variables. By creating a given set of variables, an analyst can determine how changes in one variable affect the outcome.

The independent and dependent variables are fully analyzed when sensitivity analysis is conducted. Sensitivity analysis allows for forecasting using historical, true data. By studying all the variables and the possible outcomes, important decisions can be made about businesses, the economy, and making investments. Sensitivity analysis can be used to:

• Make predictions about the share prices of public companies . Some variables that affect stock prices include company earnings, the number of shares outstanding, the debt-to-equity ratios (D/E), and the number of competitors in the industry.
• Determine the effect that changes in interest rates have on bond prices. In this case, the interest rates are the independent variable, while bond prices are the dependent variable.

Sensitivity analysis can provide management feedback useful in many different scenarios including:

• Understand influencing factors. What and how external factors interact with a specific project or undertaking. This allows management to see what input variables may impact output variables.
• Reduce uncertainty. Complex sensitivity analysis models inform project members what to be alert for or what to plan for.
• Catch errors. The original assumptions for the baseline analysis may have had some uncaught errors. By performing analytical iterations, management may catch mistakes in the original analysis.
• Simplify the model. By performing sensitivity analysis, users can better understand what factors don't matter and can be removed from the model due to its lack of materiality.
• Communicate results. Upper management may already be defensive. Compiling analysis on different situations helps inform decision-makers of other outcomes they may be interested in knowing about.
• Achieve goals. Management may lay long-term strategic plans that must meet specific benchmarks. By performing sensitivity analysis, a company can better understand how a project may change and what conditions must be present for the team to meet its metric targets.

Because sensitivity analysis answers questions such as "What if XYZ happens?", this type of analysis is also called what-if analysis.

A sales manager wants to understand the impact of customer traffic on total sales. The company determines that sales are a function of price and transaction volume. The price of a widget is $1,000, and the company sold 100 last year for a total sales of $100,000.

The manager determines that a 10% increase in customer traffic increases transaction volume by 5%. This allows the company to build a financial model and sensitivity analysis based on what-if statements. It can tell the manager what happens to sales if customer traffic increases by 10%, 50%, or 100%.

Based on 100 transactions, a 10%, 50%, or 100% increase in customer traffic equates to an increase in transactions by 5%, 25%, or 50% respectively. The sensitivity analysis demonstrates that sales are sensitive to changes in customer traffic.

Advantages and Disadvantages

Sensitivity analysis provides several benefits for decision-makers. It acts as an in-depth study of all the variables so the predictions may be more reliable. It allows decision-makers to identify where they can make improvements in the future.

However, the outcomes are based on assumptions because the variables are based on historical data only. Complex models may be system-intensive, and models with too many variables may distort a user's ability to analyze influential variables.

May help management target specific inputs to achieve more specific results

May easily communicate areas to focus on or greatest risks to control

May identify mistakes in the original benchmark

Generally reduces the uncertainty and unpredictability of a given undertaking

Heavily relies on assumptions that may not become true in the future

May burden computer systems with complex, intensive models

May become overly complicated which distorts an analysts ability to

May not accurately integrate independent variables as one variable may not accurately the impact of another variable

What Is Sensitivity Analysis in NPV?

Sensitivity analysis in NPV analysis is a technique to evaluate how the profitability of a specific project will change based on changes to underlying input variables. Though a company may have calculated the Net Present Value (NPV) , it may want to understand how better or worse conditions will impact the return the company receives. 

How Do Businesses Calculate Sensitivity Analysis?

Sensitivity analysis is often performed in analysis software, and Excel has functions to perform the analysis. In general, sensitivity analysis is calculated using formulas with different input cells. For example, a company may perform NPV analysis using a discount rate of 6%. Sensitivity analysis can be performed by analyzing scenarios of 5%, 8%, and 10% discount rates and maintaining the formula but referencing the different variable values. 

What Is the Difference Between Sensitivity Analysis and Scenario Analysis?

A sensitivity analysis is not the same as a scenario analysis . Assume an equity analyst wants to do a sensitivity analysis and a scenario analysis around the impact of earnings per share (EPS) on a company's relative valuation by using the price-to-earnings (P/E) multiple. The sensitivity analysis is based on the variables that affect valuation, which a financial model can depict using the variables' price and EPS. For a scenario analysis, an analyst determines a certain scenario such as a stock market crash or change in industry regulation that would affect the company valuation.

When a company wants to determine different potential outcomes for a given project, it may consider performing a sensitivity analysis. Sensitivity analysis entails manipulating independent variables to see the resulting financial impacts. Companies employ it to identify opportunities, mitigate risk, and communicate decisions to upper management.

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The Sense and Sensibility of Sensitivity Analyses

Affiliation.

• 1 From the Department of Biostatistics, Boston University School of Public Health, Boston (D.M.C.); and the Department of Biostatistics, Brown University School of Public Health, Providence, RI (J.W.H.).
• PMID: 39282939
• DOI: 10.1056/NEJMp2403318

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• Open access
• Published: 16 September 2024

Sensitivity analysis and global stability of epidemic between Thais and tourists for Covid -19

• Rattiya Sungchasit 1 ,
• I.-Ming Tang 2 &
• Puntani Pongsumpun 3  

Scientific Reports volume  14 , Article number:  21569 ( 2024 ) Cite this article

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• Applied mathematics
• Computational science

This study employs a mathematical model to analyze and forecast the severe outbreak of SARS-CoV-2 (Severe Acute Respiratory Syndrome Coronavirus 2), focusing on the socio-economic ramifications within the Thai population and among foreign tourists. Specifically, the model examines the impact of the disease on various population groups, including susceptible (S), exposed (E), infected (I), quarantined (Q), and recovered (R) individuals among tourists visiting the country. The stability theory of differential equations is utilized to validate the mathematical model. This involves assessing the stability of both the disease-free equilibrium and the endemic equilibrium using the basic reproduction number. Emphasis is placed on local stability, the positivity of solutions, and the invariant regions of solutions. Additionally, a sensitivity analysis of the model is conducted. The computation of the basic reproduction number (R0) reveals that the disease-free equilibrium is locally asymptotically stable when R0 is less than 1, whereas the endemic equilibrium is locally asymptotically stable when R0 exceeds 1. Notably, both equilibriums are globally asymptotically stable under the same conditions. Through numerical simulations, the study concludes that the outcome of COVID-19 is most sensitive to reductions in transmission rates. Furthermore, the sensitivity of the model to all parameters is thoroughly considered, informing strategies for disease control through various intervention measures.

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Introduction.

Coronavirus disease (COVID-19) is an infectious disease caused by the SARS-COV-2 virus which has spread throughout the world. The World Health Organization (WHO) has declared it a serious epidemic 1 , 2 , 3 , 4 , 5 . The World Health Organization (WHO) has coordinated and asked for international cooperation to stop the spread of the coronavirus -19, which the epidemic is continuously spreading. From reports around the world starting with H1N1 influenza infection, there was a clear outbreak in 2009, with a new outbreak starting on December 31, 2019. A group of cases of pneumonia of unknown ethology in Wuhan, Hubei Province in China. The outbreak was later reported to WHO in January 2022. An outbreak of a new virus 1 , 2 , 3 , 4 and 6 , 7 , 8 , 9 was identified, and the new virus was later named the 2019 novel coronavirus. By analyzing the genetics of viruses from personal illnesses, including Coronavirus Disease 2019 by WHO in February 2020 on behalf of the virus. This virus is called SARS-CoV-2 and a disease in the same family is COVID-19 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 .

Coronaviruses are a set of viruses that cause sicknesses such as respiratory diseases or gastrointestinal diseases. Respiratory diseases can extend from the common cold to the more serious diseases e.g. Middle East Respiratory Syndrome (MERS-COV), Severe Acute Respiratory Syndrome (SARS-COV). The novel coronavirus (nCOV) is a new strain that has not been identified in humans. New diseases caused by viruses are named according to where they were first discovered, such as the Spanish flu and the Hong Kong flu. West Nile Flu, etc. The official name of the disease in this article is COVID-19, not Wuhan Flu (or Chinese flu) Coronaviruses are zoonotic 1 , 2 , 3 and 6 , 7 , 8 , 9 and 14 , 15 , 16 , 17 , 18 , which means they are transmitted between animals and humans meaning that they are transmitted between animals and humans. It has been definite that MERS-COV was transmitted from dromedary camels to humans and SARS-COV from civet cats to humans 5 , 6 , 7 , 8 . While the original source of the COVID-19 virus has not been precisely determined, ongoing investigations point it to be zoonotic 15 , 16 .

In a person infected with the COVID-19 virus, respiratory symptoms can appear almost immediately. In most cases, the person can exhibit no symptoms or mild symptoms. The symptoms of this disease are very similar to those of seasonal flu 17 and 20 , 21 , 22 , 23 . Laboratory and clinical signs of the COVID-19 infection can appear 2–14 days after exposure. The period between the initial exposure to the disease and the time when the symptoms first appear is called the incubation period. During the incubation of the disease, there is a probability of transmitting or spreading the COVID-19 virus. The clinical signs of COVID-19 infections are a fever, a cough, and a general tiredness. Other early symptoms of COVID-19 of the slight loss of taste or smell, shortness of breath or difficult breathing, muscle aches, chills, sore throat, runny nose, headache, chest pain, pink eye (conjunctivitis), nausea. While many of the other illnesses are caused by other viruses, the main cause at the early stages of the current pandemic was the COVID-19 and it seems to have targeted older people 21 , 22 , 23 , 24 , 25 . Older people (people over 70 years of age) often suffer from other serious chronic illnesses, such as diabetes, cardiovascular disease, chronic respiratory disease, cancer, hypertension, chronic liver disease and people who are physically inactive 1 , 2 , 3 , 4 and 13 , 14 , 15 , 16 have weaker immune symptoms and may succumb to the disease (COVID-19). The WHO reported cases of COVID-19 from January 2020 to the present. The number of cases and the number of deaths in Thailand are shown in the following in Fig.  1 . The reasons for separating the populations into Thais and foreigners are that there is shortage of season labor (needed for the farming industry) and tourism is one of the top industries in Thailand. The spread of this disease to become a pandemic is due to the ease of moving from one country to another. The slowness of the great Spanish Flu was the difficulty of traveling from country to country or continent to continent.

Number of patients and deaths of COVID-19 cases per month around the world and in Thailand 1 , 2 , 3 , 4 .

WHO has issued guidelines for the treatment of COVID-19 in the high-risk groups (older people and people with serious chronic illness). These are the people who are the most susceptible to infection by the virus and who are in most danger of dying. The World Health Organization has issued guidelines for preventing COVID-19 infection. Not separated from each other, but able to live together with groups of people at risk. They will take care of their treatment and social care. In Thailand from January 2020 to October 2022, there were 4,689,897 confirmed cases of COVID-19 and 32,922 deaths, according to the WHO 2022 September report. A total of 142,635,014 vaccine doses have been administered.

To understand the nature and dynamics of the COVID-19 of epidemic (a pandemic in the larger scheme), mathematical modeling is used to forecast the transmission dynamics needed for controlling and planning strategies. Most epidemiological modelling studies of COVID-19 are based on WHO data. The studies on COVID-19 modelling done in Thailand 16 and 26 . The authors considered a mathematical model for the transmission dynamics of COVID-19. The data from Thailand, which considers the special features pertaining to Thailand and other neighboring countries 4 , 5 , 6 and 16 , 17 , 18 , and 24 . From the information obtained, we estimate the values of unknown parameters by statistical and mathematical methods. It should be noted that the effective parameters for the spread of the virus differ from country to country and that the effective control over the rate of virus transmission from country to country will be different. In necessary to stop the spread of the virus. It was found in other studies, that the spread of COVID-19 be managed by minimizing the contact rate of infected and increasing the quarantine of exposed individuals 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 . This study examined a mathematical model of the COVID-19 transmission dynamics by dividing it into two groups of coronavirus transmission. The research was organized as follows: explanation of the mathematical models, formulation of the differential equations, mathematical analysis of models, followed by numerical solutions of the differential equations, summarization and discussion.

Materials and methods

In this study, a deterministic mathematical model was created. It covers the well-known SEIR epidemic model 20 , 21 , 22 , 23 , 24 , 25 , 26 . By adding people who are in quarantine and do not have symptoms of the disease. Symptomatic and asymptomatic infected people will be collected. The SEIQR epidemic model was thus obtained, which evolved with the following subpopulations: Susceptible (S), Exposed (E- (people not yet infectious)), Infectious (I), Quarantined (Q- (setting aside individuals who are exposed), and Recovered (R). This is because people in the Q (Quarantined) group, which represents people who are required to stay in the hospital and at home for a period of time due to the disease, are concerned about their illness. The COVID-19 pandemic in Thailand, we used a ten-dimensional SEIQR (Susceptible, Exposed, Infected, Quarantined and Recovered) containing two populations \(S_{1, } E_{1, } I_{1, } Q_{1, } R_{1 }\) are Thais population respectively and \(S_{2, } E_{2, } I_{2, } Q_{2, } R_{2 }\) are Foreign (tourist) or migrant workers) population respectively of COVID-19 transmission model 21 , 22 , 23 , 24 , 25 , 26 .

The Recruitment term of the susceptible population in Thais and the rest if the Foreign (tourist) are given as \(\mu \) and \(C\) respectively. Only exposed and infectious are considered, it is assumed that those infected show symptoms of Thais and Foreign (tourist). The natural death rate of Thais population and the natural death rate of Foreign (tourist) population is assumed to be the same across the world are given as \(\delta_{1}\) and \(\delta_{2}\) . The force of infections in Thais population \(\varphi_{1 } S_{1 } \left( {E_{1} + I_{1} } \right)\) (Transmission rate of virus between population from Thais population to Foreign (tourist) population (in Thais) and \(\varphi_{12 } S_{1 } \left( {E_{2} + I_{2} } \right)\) (When Foreign (tourist) are present, a susceptible Thais can also be infected by an infected or exposed Foreign (tourist) (in Thais)) are the new infections caused by other infected individuals in Thais. The force of infection in rest of Foreign (tourist) population \(\varphi_{2 } S_{2 } \left( {E_{2} + I_{2} } \right)\) (Transmission rate of virus between population from Foreign (tourist) population to Thais population (in Foreign (tourist)) and \(\varphi_{21 } S_{2 } \left( {E_{1} + I_{1} } \right)\) (When Thais are present, the susceptible Thais can also be infected by an infected or exposed Thais (in Foreign (tourist)) are the new infections caused by other infected individuals in Foreign (tourist). Taking into consider the above discretion, the schematic flow diagram for COVID-19 model is appeared in Fig.  2 .

The flowchart illustration the dynamics of the model.

The host population was divided into five compartments: \(S_{1}\) number of Thais susceptible to COVID-19 infection at time \(t\) , \(E_{1}\) number of Thais exposed to COVID-19 infection at time \(t\) , \(I_{1}\) number of infectious Thais at time \(t\) , \(Q_{1}\) number of Thais quarantined for COVID-19 at time \(t\) , \(R_{1}\) number of recovered Thais at time \(t\) , \(S_{2}\) number of Foreign (tourist) susceptible at time \(t\) , \(E_{2}\) number of Foreign (tourist) exposed at time \(t\) , \(I_{2}\) number of Foreign (tourist) infected at time \(t\) , \(Q_{2}\) number of Foreign (tourist) quarantined at time \(t\) , \(R_{2}\) number of Foreign (tourist) recovered at time \(t\) .

A system of ordinary differential equations can be used to model the influence of two populations on each other as a set nonlinear differential equation 20 and 24 , 25 as follows:

with initial densities: \(S_{1} \ge 0, E_{1} \ge 0, I_{1} \ge 0, Q_{1} \ge 0, R_{1} \ge 0\) in Thais population and \(S_{2} \ge 0, E_{2} \ge 0, I_{2} \ge 0, Q_{2} \ge 0, R_{2} \ge 0\) in the Foreign (tourist) population.

All the parameters and corresponding biological meaning are defined in Table 1 .

The total Thais population \(N_{h}\) is \(S_{1} + E_{1} + I_{1} + Q_{1} + R_{1}\) . The equations for the human compartment are the following Eq. ( 1 ) and the total Foreign (tourist) population is \(N_{T} = S_{2 } + E_{2} + I_{2} + Q_{2} + R_{2 }\) . We assume that there are constant total number of human Thais population and of Foreign (tourist) population. Therefore the rate of change for total number of human Thais population and of Foreign (tourist) population are equivalent to zero. Thus, the Recruitment term of human and death rate are equivalent. We defined the new state variables as follows: \(\frac{{S_{1} }}{{N_{h} }} = {\mathop {S_{1} }\limits^{\prime } } , \frac{{E_{1} }}{{N_{h} }} = {\mathop {E_{1} }\limits^{\prime } } , \frac{{I_{1} }}{{N_{h} }} = {\mathop {I_{1} }\limits^{\prime } } , \frac{{Q_{1} }}{{N_{h} }} = {\mathop {Q_{1} }\limits^{\prime } } , \frac{{R_{1} }}{{N_{h} }} = R_{1}^{\prime } , \) \(\frac{{S_{2} }}{{N_{T} }} = S_{2}^{\prime } , \frac{{E_{2} }}{{N_{T} }} = E_{2}^{\prime } , \frac{{I_{2} }}{{N_{T} }} = I_{2}^{\prime } , \frac{{Q_{2} }}{{N_{T} }} = Q_{2}^{\prime } , \frac{{R_{2} }}{{N_{T} }} = R_{2}^{\prime }\) Renormalizing model ( 1 ) we obtain the following:

Positivity of solution

Model ( 2 ) must been found to be biologically and epidemiologically meaningful and well positioned. To do this, we needed to show that the solutions of all state variables were non-negative all the time. The following theorem 24 , 25 , 26 were required.

Theorem 1 : The given solution \(\left\{ {S_{1} ,E_{1} ,I_{1} ,Q_{1} ,R_{1,} S_{2 } ,E_{2} ,I_{2} ,Q_{2} ,R_{2 } } \right\}\) of the epidemiological systems (2) with non-negative initial data when \(S_{1} \ge 0, E_{1} \ge 0, I_{1} \ge 0, Q_{1} \ge 0, R_{1} \ge 0\) and \(S_{2} \ge 0, E_{2} \ge 0, I_{2} \ge 0, Q_{2} \ge 0,\)

\( R_{2} \ge 0\) stills non-negative for all time non-negative \(t > 0\) .

Proof of Theorem 1

Given the initial data \(S_{1} \left( 0 \right),E_{1} \left( 0 \right),I_{1} \left( 0 \right),Q_{1} \left( 0 \right),R_{1} \left( 0 \right)\) and \( S_{2 } \left( 0 \right),E_{2} \left( 0 \right),I_{2} \left( 0 \right),Q_{2} \left( 0 \right),R_{2 } \left( 0 \right)\) are non—negative. It is clear from the first sub-equation of the model (2) that

\(\frac{{d{\mathop {S_{1} }\limits^{\prime } } }}{dt} = \left[ {\varphi_{1} {\mathop {S_{1} }\limits^{\prime } } \left( {{\mathop {E_{1} }\limits^{\prime } } + {\mathop {I_{1} }\limits^{\prime } } } \right) + \delta_{1} {\mathop {S_{1} }\limits^{\prime } } + \varphi_{12} {\mathop {S_{1} }\limits^{\prime } } \left( {E_{2}^{\prime } + I_{2}^{\prime } } \right)} \right] \ge 0\) so that

Integrating ( 3 ) gives

Further, one sees from the second sub-equation of the model ( 2 ) that

\(\frac{{d{\mathop {E_{1} }\limits^{\prime } } }}{dt}\left[ { \delta_{1} {\mathop {E_{1} }\limits^{\prime } } + \frac{1}{{IIP_{1} }}{\mathop {E_{1} }\limits^{\prime } } } \right] \ge 0\) implies \(\frac{d}{dt}\left[ { {\mathop {E_{1} }\limits^{\prime } } {\text{exp }}(\delta_{1} + \frac{1}{{IIP_{1} }} \mathop \smallint \limits_{0}^{t} {\mathop {E_{1} }\limits^{\prime } } (\zeta_{1} ))d\zeta_{1} } \right] \ge 0\) which on integration yields

Further, one sees from the third sub-equation of the model ( 2 ) that

which upon integration yields

Further, one sees from the fourth sub-equation of the model ( 2 ) that

\(\frac{{d\mathop Q\limits^{\prime }_{1} }}{dt} = \left[ {\gamma_{1} + \delta_{1} } \right)\mathop Q\limits^{\prime }_{1} ] \ge 0\) implies \(\frac{{d\mathop Q\limits^{\prime }_{1} }}{dt}\left[ {\mathop Q\limits^{\prime }_{1} \exp (\gamma_{1} + \delta_{1} \mathop \smallint \limits_{0}^{t} \mathop Q\limits^{\prime }_{1} (\zeta_{1} } \right))d\zeta_{1} ] \ge 0\) which upon integration yields

Further, one sees from the fifth sub-equation of the model ( 2 ) that \(\frac{{d\mathop R\limits^{\prime }_{1} }}{dt} = \gamma_{1} \mathop Q\limits^{\prime }_{1} - (\delta_{1} + \alpha_{1} )\mathop R\limits^{\prime }_{1}\) , \(\frac{{d\mathop R\limits^{\prime }_{1} }}{dt}\left[ {(\delta_{1} + \alpha_{1} )\mathop R\limits^{\prime }_{1} } \right] \ge 0\) implies \(\frac{{d\mathop R\limits^{\prime }_{1} }}{dt}\left[ {\mathop R\limits^{\prime }_{1} \exp \left( {\delta_{1} + \alpha_{1} \mathop \smallint \limits_{0}^{t} \mathop R\limits^{\prime }_{1} \left( {\zeta_{1} } \right)} \right)d\zeta_{1} } \right] \ge 0\) which upon integration yields.

In a similar model, it can be shown that \( \mathop S\limits^{\prime }_{2} \left( t \right) \ge 0,\mathop E\limits^{\prime }_{2} \left( t \right) \ge 0,\mathop I\limits^{\prime }_{2} \ge 0,\mathop Q\limits^{\prime }_{2} \ge 0\) and \(\mathop R\limits^{\prime }_{2} \left( t \right) \ge 0\) for all time \(t > 0\) . This completes the proof. It is important to note that the model (2) has be analyzed in the region \(\beta\) given by

Divided into two groups \( \mathop S\limits_{1} + \mathop E\limits_{1} + \mathop I\limits_{1} + \mathop Q\limits_{1} + \mathop R\limits_{1} = 1\) and \( \mathop S\limits_{2} + \mathop E\limits_{2} + \mathop I\limits_{2} + \mathop Q\limits_{2} + \mathop R\limits_{2} = 1\) which can easily be shown to be positively univariate according to model ( 2 ). In the following, model ( 2 ) is epidemiologically and mathematically well-positioned in \( \beta\) .

The solution of system ( 1 ) is possible for all if entering an invariant region. \({\Omega } = {\Omega }_{1} \times {\Omega }_{2}\) , where \({\Omega }_{1} = \{ S_{1} ,E_{1} ,I_{1} ,Q_{1} ,R_{1} \in R_{ + }^{5} :0 < N_{h} \left( t \right) \le \frac{\mu }{{\delta_{1} }}\}\) as \(t \to \infty\) , when \(\theta = {\text{min}}\left\{ {\delta_{1} , \delta_{1} + \rho_{1} } \right\}\) and \({\Omega }_{2} = \left\{ { S_{2 } ,E_{2} ,I_{2} ,Q_{2} ,R_{2 } \in R_{ + }^{5} :0 < N_{T} \left( t \right) \le \frac{C}{{\delta_{2} }}} \right\}\) as \( t \to \infty\) , when \(\theta_{2} = \min \left\{ {\delta_{2} + \vartheta , \delta_{2} + \rho_{2} + \vartheta } \right\}.\)

Proof of Theorem 2

The invariant region is received from the bounded situation of the system. Here, \(N_{h} \left( t \right) = S_{1} \left( t \right) + E_{1} \left( t \right) + I_{1} \left( t \right) + Q_{1} \left( t \right) + R_{1} \left( t \right)\) and \(N_{T} \left( t \right) = S_{2} \left( t \right) + E_{2} \left( t \right) + I_{2} \left( t \right) + Q_{2} \left( t \right) + R_{2} \left( t \right)\) . It following that,

\(\begin{aligned} \frac{{dN_{h} }}{dt} & = \frac{{dS_{1} }}{dt} + \frac{{dE_{1} }}{dt} + \frac{{dI_{1} }}{dt} + \frac{{dQ_{1} }}{dt} + \frac{{dR_{1} }}{dt} \\ & = \mu - \left( {\rho_{1} + g_{1} } \right)I_{1} - g_{1} E_{1} - \delta_{1} N_{h} \\ & \le \mu - \delta_{1} N_{h} \\ \end{aligned}\)

This inequality can be expressed in a general solutions as

where \(N_{h} \left( 0 \right)\) is the initial values, i.e., \(N_{h} \left( t \right) = N_{h} \left( 0 \right)\) at \(t = 0\) .

In a similar model, it can be shown that \(N_{T} \left( t \right) = S_{2} \left( t \right) + E_{2} \left( t \right) + I_{2} \left( t \right) + Q_{2} \left( t \right) + R_{2} \left( t \right)\) for the bounded situation of the system all time \( t > 0\) . Moreover, every solution for systems ( 1 ) with initial conditions in \({{ \Omega }}\) remains in \({{ \Omega }}\) for all \(t > 0\) . Therefore, the dynamics of our model will be poised in \({\Omega }\) .

Analysis of the model

Basic reproduction number.

The next generation matrix method is used to calculate the basic reproduction number, \(R_{0}\) 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 the number of secondary infections caused by a single infected individual in a completely susceptible population (including of the local Thais and the Foreign (tourist)). The behavior of the disease in the total system defined by Eq. ( 2 ) 15 , 16 , 17 , 18 , 19 will be determined by \(R_{0}\) which has the form

where \(R_{0T} = \frac{{\left( {q_{2T} + \gamma_{1} + \delta_{1} } \right) \mu \left( {1 + IIP_{1} \left( {\delta_{1} + \rho_{1} } \right)} \right)\varphi_{1} }}{{\delta_{1} \left( {1 + IIP_{1} \delta_{1} } \right)\left( {g_{1} q_{2T} + \left( {q_{2T} + \gamma_{1} + \delta_{1} } \right)\left( {\delta_{1} + \rho_{1} } \right)} \right)}}\) is the basic reproduction number for Thais and \(R_{0F} = \frac{{\alpha_{3} C\left( {1 + \alpha_{2} IIP_{2} } \right)\varphi_{2} }}{{\alpha_{1} \left( {\alpha_{2} \alpha_{3} + g_{2} q_{3T} } \right)IIP_{2} \left( {\delta_{2} + \vartheta } \right)}}\) is the basic reproduction number for the Foreign (tourist) population only with \(\alpha_{1} = \delta_{2} + \vartheta + \frac{1}{{IIP_{2} }} , \alpha_{2} = \delta_{2} + \vartheta + \rho_{2} , \alpha_{3} = \delta_{2} + \vartheta + \gamma_{2} + q_{3T}\) and \(\alpha_{4} = \delta_{2} + \vartheta + \alpha_{2}\) , we have

To find the basic reproduction number of our proposed differential Eq. ( 2 ), using help of the next generation matrix formulas 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 . We initially define \( K = (\mathop E\limits_{1} ,\mathop I\limits_{1} ,\mathop Q\limits_{1} )^{T}\) and \( K_{1} = (\mathop E\limits_{2} ,\mathop I\limits_{2} ,\mathop Q\limits_{2} )^{T}\) . The model ( 2 ) is rewritten in the following form \(\frac{dy}{{dt}} = F\left( y \right) - V\left( y \right)\) , where \( F\left( y \right)\) is the non-negative matrix of the newly infected (Thais and Foreign (tourist) populations) and \( V\left( y \right)\) is the non-singular matrix for the transfers between the parts in the infective equations (Thais and Foreign (tourist) populations) (when \(y\) represents Thais populations and Foreign (tourist) populations) as follows:

for the Thais population.

for the Foreign (tourist) population.

The basic reproductive number \(\left( {R_{0} } \right)\) is the threshold for the stability of the disease-free equilibrium B 0 . It can be calculated by \(R_{0} = \rho \left( {FV^{ - 1} } \right)\) where, \( FV^{ - 1}\) is called the next generation matrix and \(\rho \left( {FV^{ - 1} } \right)\) is the spectral radius of the matrix \(FV^{ - 1}\) . Then we get reproduction number \(\left( {R_{0} } \right)\) where,

Finally, the Routh–Hurwitz criteria is used for determining the stabilities of the model. If \(R_{0} > 1\) , then the endemic equilibrium is local asymptotically stable, but if \(R_{0} < 1\) , then the disease free equilibrium point is local asymptotically stable.

Equilibrium point

The standard method is used to analyze the model. The equilibrium points are found by setting the right-hand side of Eq. ( 2 ) to zero. By doing this, the equilibrium points are determined as follows 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 .

The COVID-19 free equilibrium of the Eq. ( 2 ) exists and then given by

The COVID-19 endemic equilibrium of the Eq. ( 2 ) exists with infection and then given by

Local asymptotically stability of disease—free equilibrium point

(The Generalized Routh–Hurwitz Criterion). Given the charactistic equation

Define \(k\) matrices as follows:

where the \(\left( {l,m} \right)\) term in the matrix \(H_{j}\) is \(a_{2l - m}\) for \(0 < 2l - m < k\) , \(1\) for \(2l = m\) .

\(0\) for \(0 < 2l\) or \(2l < k + m.\)

Then all eigenvalues have negative real parts; that is, the steady-state \(\overline{N}\) is stable if and only if the determinants of all Hurwitz are positive:

When is \(\overline{N} = \overline{N}_{i} + a_{1} e^{{\lambda_{1} t}} + a_{2} e^{{\lambda_{2} t}} + \cdots + a_{k} e^{{\lambda_{k} t}}\) .

The local stability of disease-free equilibrium point is determined from the Jacobian matrix of the model of Eq. ( 2 ) evaluated at the equilibrium points. If \(R_{0} > 1\) , the point is stable and unstable otherwise. 12 , 13 , 14 , 15 , 16 , 17 , 18 , and 24 , 25 , 26 , 27 , 28 , 29 , and 31 .

Proof of Theorem 4

To determine the local stability of \(J_{0}\) , we evaluate the Jacobian matrix at the disease-free state to be

where \( \theta_{1} = \varphi_{1} \mathop S\limits_{1} - \left( { \delta_{1} + \frac{1}{{IIP_{1} }}} \right)\) , \( \theta_{2} = \varphi_{2} \mathop S\limits_{2} - \left( {\delta_{2} + \vartheta + \frac{1}{{IIP_{2} }}} \right)\) , \(\theta_{3} = - (\delta_{2} + \vartheta + q_{3T} + \rho_{2}\) ), \(\theta_{4} = (\delta_{2} + \vartheta + \gamma_{2} + \rho_{2}\) ) and \(\theta_{5} = - \left( {\delta_{2} + \vartheta + \alpha_{2} } \right)\) .

The eigenvalues of the \(J_{0} \) are obtained by solving \(Det \left( {J_{0} - \lambda I } \right) = 0\) . We obtain the characteristic equation, where \(\lambda\) is an eigenvalue of the matrix \( J_{0}\) . The, root of the model ( 2 ) i.e., eigenvalue of the matrix \(J_{0}\) are

The three eigenvalues from Eq. ( 15 ) were \(\lambda_{1} = - \delta_{2} - \vartheta - \alpha_{2}\) , \(\lambda_{2} = - \delta_{1} - \alpha_{1}\) and \(\lambda_{3} = - \delta_{1}\) and all of them must have negative real parts. For the other seven eigenvalues, we examine the stability of disease-free equilibrium state by using the Routh Hurwitz principle (R-H criterion) to show that all eigenvalues given by Eq. ( 14 ) has a negative real part, i.e., coefficients of the seventh order the polynomial appearing in Eq. ( 14 ) satisfies all R-H conditions when \(A_{1} ,A_{2} ,A_{3} ,A_{4} ,A_{5} ,A_{6} ,A_{7 } > 0 \) (The coefficients appearing in Eq.  15 from the Routh Hurwitz condition are plotted on the graph by the x axis being the coefficient \(A_{2}\) . and the Y-axis is the coefficient of \(A_{1} ,A_{2} ,A_{3} ,A_{4} ,A_{5} ,A_{6}\) and \(A_{7 }\) obtained by finding determinants from size nxn, parameter values from Table 1 by the use the Mathematica program.) This is displayed for \(R_{0} < 1\) , disease-free equilibrium point will be stable as showed in Fig.  3 .

The parameter areas for disease free equilibrium state which satisfies the Routh-Hurwitz criteria with the value of parameters: respectively, for with \((\lambda^{7} + A_{1} \lambda^{6} + A_{2} \lambda^{5} + A_{3} \lambda^{4} + A_{4} \lambda^{3} + A_{5} \lambda^{2} + A_{6} \lambda^{{}} + A_{7} = 0\) .

and \({\beta }_{7}= {A}_{8}({A}_{7}\left({A}_{6}\left(-{A}_{5}\left({A}_{3}^{2}{A}_{4}-{A}_{2}{A}_{3}{A}_{5}+{A}_{5}^{2}+{A}_{3}^{3}{A}_{6}\right)+\left({A}_{4}\left({A}_{3}^{2}{A}_{4}-{A}_{2}{A}_{3}{A}_{5}+{A}_{5}^{2}\right)+{A}_{3}\left(-2{A}_{2}{A}_{3}+ 3{A}_{5}\right){A}_{6}\right){A}_{7}+\left({A}_{2}^{2}{A}_{3}-2{A}_{3}{A}_{4}-{A}_{2}{A}_{5}\right){A}_{7}^{2}+{A}_{7}^{3}\right)+\left({A}_{5}^{4}-{A}_{3}{A}_{5}^{2}\left({A}_{2}{A}_{5}+4{A}_{7}\right)-\left({A}_{3}^{3}\left({A}_{5}{A}_{6}+2{A}_{4}{A}_{7}\right)+{ A}_{3}^{2}{A}_{4}{A}_{5}^{2}+3{A}_{2}{A}_{5}{A}_{7}+2{A}_{7}^{2}\right)\right){A}_{8}+{A}_{3}^{4}{A}_{8}^{2}+{A}_{1}\left({A}_{7}\left(-{A}_{6}\left({A}_{5}\left(-{A}_{2}{A}_{3}{A}_{4}+{A}_{2}^{2}-2{A}_{4}{A}_{5}\right)\right){A}_{7}- \left({A}_{3}^{2}-{3A}_{2}{A}_{4}+3{A}_{6}\right){A}_{7}^{2}\right)+\left(-2{A}_{4}{A}_{5}^{3}+3{A}_{3}{A}_{5}^{2}{A}_{6}+4{A}_{3}{A}_{4}{A}_{5}{A}_{7}+{A}_{3}^{2}{A}_{6}{A}_{7}+4{A}_{5}{A}_{7}^{2}+{ A}_{2}^{2}\left({A}_{5}^{3}-3{A}_{3}{A}_{5}{A}_{7}\right)+{A}_{2}\left({A}_{3}{A}_{5}\left(-{A}_{4}{A}_{5}+{A}_{3}{A}_{6}\right)+\left(2{A}_{3}^{2}{A}_{4}+{A}_{5}^{2}\right){A}_{7}-5{A}_{3}{A}_{7}^{2}\right)\right){A}_{8}-{A}_{3}^{2}\left({A}_{2}{A}_{3}+ 4{A}_{5}\right){A}_{8}^{2}\right)+\left({A}_{1}^{2}\left({A}_{7}\left(-{A}_{4}{A}_{5}{A}_{6}+{A}_{4}^{3}{A}_{7}+{A}_{6}^{2}\left(2{A}_{2}{A}_{5}+3{A}_{7}\right)+{A}_{4}{A}_{6}\left({A}_{3}{A}_{6}-3{A}_{2}{A}_{7}\right)\right)- \left({A}_{5}\left(-{A}_{4}^{2}{A}_{5}+{A}_{3}{A}_{4}{A}_{6}+2{A}_{2}{A}_{5}{A}_{6}\right)+\left(2{A}_{3}{A}_{4}^{2}-{A}_{2}{A}_{4}{A}_{6}\right)+{A}_{2}{A}_{3}{A}_{6}+5{A}_{5}{A}_{6}\right){A}_{7}+3\left(-{A}_{2}+{A}_{4}\right){A}_{7}^{2}\right){A}_{8}+{ A}_{3}^{2}{A}_{4}+3{A}_{2}{A}_{3}{A}_{5}+2{A}_{5}^{2}+4{A}_{3}{A}_{7}\right){A}_{8}^{2}\right)\) .

Local asymptotically stability of disease endemic equilibrium point

The disease endemic equilibrium point is set from the Jacobian matrix of the system of Eq. ( 2 ) evaluated at every equilibrium point. If \({R}_{0} <1\) the state is stable and unstable otherwise 12 , 13 , 14 , 15 , 16 , 17 , 18 , and 24 , 25 , 26 , and 30 , 31 .

Proof of Theorem 5

The Jacobian matrix of the model (2) at pandemic equilibrium point is

Where \( \eta_{1} = - \varphi_{1} \left( {\mathop E\limits_{1} + \mathop I\limits_{1} } \right) - \varphi_{12} \left( {\mathop E\limits_{2} + \mathop I\limits_{2} } \right) - \delta_{1}\) , \( \eta_{2} = \varphi_{1} \left( {\mathop E\limits_{1} + \mathop I\limits_{1} } \right) + \varphi_{12} \left( {\mathop E\limits_{2} + \mathop I\limits_{2} } \right)\) \( \eta_{3} = \varphi_{1} \mathop S\limits_{1} - \left( {\delta_{1} + \frac{1}{{IIP_{1} }}} \right)\) , \( \eta_{4} = - \varphi_{2} \left( {\mathop E\limits_{2} + \mathop I\limits_{2} } \right) - \varphi_{21} \left( {\mathop E\limits_{1} + \mathop I\limits_{1} } \right) - (\delta_{2} + \vartheta )\) , \( \eta_{5} = - \varphi_{2} \left( {\mathop E\limits_{2} + \mathop I\limits_{2} } \right) + \varphi_{21} \mathop S\limits_{2} \left( {\mathop E\limits_{1} + \mathop I\limits_{1} } \right)\) , \( \eta_{6} = - \varphi_{2} \mathop S\limits_{2} - (\delta_{2} + \vartheta + \frac{1}{{IIP_{2} }}\) , \( \eta_{7} = - \left( {q_{3T} + \delta_{2} + \rho_{2} + \vartheta } \right),\) \(\eta_{8} = - (\gamma_{2} + \delta_{2} + \rho_{2} + \vartheta ) \) and \(\eta_{9} = \left( {\delta_{2} + \vartheta + \alpha_{2} } \right)\) .

The endemic equilibrium point ( B 1) exists and is positive if \(R_{0} > 1\) . The eigenvalues of \(J_{1} \) are obtained by solving \(Det \left( {J_{1} - \lambda I} \right) = 0\) . The characteristic equation is as follows; we obtain the characteristic equation \(\lambda^{10} + W_{1} \lambda^{9} + W_{2} \lambda^{8} + W_{3} \lambda^{7} + W_{4} \lambda^{6} + W_{5} \lambda^{5} + W_{6} \lambda^{4} + W_{7} \lambda^{3} + W_{8} \lambda^{2} + W_{9} \lambda^{{}} + W_{10} = 0\) where \(\lambda\) are eigenvalues of the matrix \(J_{1}\) . To consider the local stability of the endemic equilibrium state, we check the stability of endemic equilibrium state by using the Routh-Hurwitz criteria required for all the eigenvalues defined by Eq. ( 16 ) to have negative real parts. We find that the Routh-Hurwitz conditions for the above the all the eigenvalues of the above 10th order polynomial to have negative real parts when \(W_{1} ,W_{2} ,W_{3} ,W_{4} ,W_{5} ,W_{6} ,W_{7} ,W_{8} ,W_{9} ,W_{10} > 0\) (The coefficients appearing in Eq. ( 15 ) from the Routh Hurwitz condition are plotted on a graph by the x axis being the coefficient \(W_{2}\) and the Y-axis is the coefficient of \( W_{1} ,W_{2} ,W_{3} ,W_{4} ,W_{5} ,W_{6} ,W_{7} ,W_{8} ,W_{9} ,W_{10}\) , obtained by finding determinants from size nxn, parameter values from Table 1 by the use the Mathematica program.) This is displayed for \({R}_{0}>1\) , endemic equilibrium point will be stable as showed in Fig.  4 .

The parameter areas for endemic equilibrium point which satisfies the Routh-Hurwitz criteria with the value of parameters: respectively, for with.

\(\lambda^{10} + W_{1} \lambda^{9} + W_{2} \lambda^{8} + W_{3} \lambda^{7} + W_{4} \lambda^{6} + W_{5} \lambda^{5} + W_{6} \lambda^{4} + W_{7} \lambda^{3} + W_{8} \lambda^{2} + W_{9} \lambda^{{}} + W_{10} = 0\) . When

Numerical results

Numerical simulations of the impact of the strategies to control the spread of coronavirus disease 19 (COVID-19) in the Thais population when there are Foreign (tourist) also present. Numerical values of various parameters and data points needed for the numerical calculations in Table 2 . The Data collected were from the official website of the Ministry of Public Health and World Health Organization (WHO) 1 , 2 , 3 , 4 , 5 , 6 and 24 , 25 , 26 , 27 , 28 , 29 , 30 . Using the numerical values in Table 2 , we obtained the time evolutions of a susceptible Thais individual, an exposed Thais, an infectious Thais, a quarantined Thais, a recovered Thais, a susceptible Foreigner (tourist), an exposed Foreigner (tourist), an infectious Foreigner (tourist), a quarantined Foreigner (tourist), and a recovered Foreigner (tourist). The values of the parameters were first chosen to lead to \(R_{0}\) to be less than one so the equilibrium state will be disease free State (0.80893). The time evolutions of the ten states were plotted in Figs.  5 . Next, we change the values of the parameters so that the value of \(R_{0}\) will be greater than one, meaning that the equilibrium state will be the endemic state (9.4175). In Fig.  6 , we see the evolution of the ten categories of individuals (susceptible Thais, exposed Thais, infectious Thais, quarantined Thais, recovered Thais, susceptible Foreign (tourist), exposed Foreign (tourist), infectious Foreign (tourist), quarantined Foreign (tourist), recovered Foreign (tourist)) converge to their epidemic equilibrium values (0.002951, 0.0000217, 0.0001608, 0.0000236, 0.000125, 0.0000001974, 0.00000134, 0.000001218).

Numerical simulations of each population for the disease free state. We will see that the solutions converge to the disease free state when it satisfy.

Numerical simulations of each population for the disease state.

The behavior’s of the endemic, we has plotted the 2-D trajectories of the following thirteen pairs (Thais susceptible-Thais exposed), (Thais susceptible-Thais infectious), (Thais susceptible-Thais quarantined), (Thais exposed-Thais infectious), (Thais exposed-Thais quarantined), (Thais infectious-Thais quarantined), (susceptible Foreign (tourist) -exposed Foreigner), (susceptible Foreign (tourist) -infectious Foreign (tourist)), (exposed Foreign (tourist) -quarantined Foreign (tourist)), (infectious Foreign (tourist) -quarantined Foreign (tourist)), (infectious Thais-exposed Foreign (tourist)), (infectious Thais-infectious Foreign (tourist)) and (quarantined Thais-quarantined Foreign (tourist). These 2D trajectories are shown in Fig.  7 . We can see that all the trajectories converge to a central point (the equilibrium pot).

The trajectories of the numerical projected onto the 2D ( a ) \((S_{1} ,E_{1} )\) , ( b ) \( (S_{1} ,I_{1} )\) , ( c ) \( (S_{1} ,Q_{1} )\) , ( d ) \( (E_{1} ,I_{1} )\) , ( e ) \( (I_{1} ,Q_{1} )\) , ( f ) \( (S_{2} ,E_{2} )\) , ( g ) \( (S_{2} ,Q_{2} )\) , ( h ) \( (I_{2} ,Q_{2} )\) , ( i ) \( (E_{1} ,E_{2} )\) , ( j ) \( (I_{1} ,I_{2} )\) and ( k ) \( (Q_{1} ,Q_{2} )\) . planes when there was no vertical transmission and equilibrium state the endemic state.

Global Stability of disease free equilibrium for model

The solutions to Eq. ( 2 ) were asymptotically stable locally in section “ Analysis of the model ”. We have now proved that the two equilibrium points are asymptotically stable globally through the following theorem.

If \(R_{0}^{{}} \le 1\) , then the disease—free equilibrium \(E^{*}\) is globally asymptotically stable, by

Proof of Theorem 6

The Lyapunov function may be constructed for the model (1) through the use of the function

Differentiating with respect to time yields.

Using the condition ( 17 ), Eq. ( 17a ) may be rewrite as,

Substitute with Eq. ( 17 ) we obtain

Note that on Ω, we have \(S_{1}^{*} = \frac{{\mu N_{h} }}{{\delta_{1} }}\) and \(S_{2}^{*} = \frac{{CN_{T} }}{{\delta_{2} + \vartheta }}\) with this in mind, Eq. ( 17 ) becomes

Hence, \(\dot{P}\left( t \right) \le 0\) . By using LaSalle’ s (1976) 31 , 32 , 33 , 34 , 35 , 36 , 37 extension to Lyapunov method, the limit of each solution is contained in the largest invariant set for which \(S_{1} = S_{1}^{*} , E_{1} = 0,I_{1} = 0, Q_{1} = 0, R_{1} = 0, S_{2} = S_{2}^{*} , E_{2} = 0,I_{2} = 0, Q_{2} = 0\) and \(R_{2} = 0\) which is the singleton \(\left\{ {E_{0} } \right\}\) . This means that the disease—free equilibrium \( E^{*} = \left\{ {S_{1}^{*} ,\mathop E\limits_{1}^{*} ,\mathop I\limits_{1}^{*} ,\mathop Q\limits_{1}^{*} ,\mathop R\limits_{1}^{*} ,\mathop S\limits_{2}^{*} ,\mathop E\limits_{2}^{*} ,\mathop I\limits_{2}^{*} ,\mathop Q\limits_{2}^{*} ,\mathop R\limits_{2}^{*} } \right\}\) is globally asymptotically stable on \(\Omega \) . This achieves the proof of the theorem.

If \(R_{0} > 1\) , then the positive endemic equilibrium state of system ( 1 ) exists and is globally asymptotically stable on \(\Omega\) , by assuming that \(\varphi_{1} = \frac{{\delta_{1} + \rho_{1} }}{{S_{1}^{*} }} \) , \(\rho_{1} = \gamma_{1}\) , \(\varphi_{2} = \frac{{\delta_{2} + \rho_{2} + \vartheta }}{{S_{2}^{*} }}\) ,

Proof of Theorem 7

We construct the Lypunov function from the model as follows

Substituting the relations in Eqs. ( 18 ), we have

Substituting the relations in Eq. ( 18a ), we have

Hence, the condition ( 16 ) show that \(\dot{\omega }\left( t \right) \le 0\) of all terms. Then the equilibrium steady state \( B_{1} = \left( {S_{1}^{*} ,\mathop E\limits_{1}^{*} ,\mathop I\limits_{1}^{*} ,\mathop Q\limits_{1}^{*} ,\mathop R\limits_{1}^{*} ,\mathop S\limits_{2}^{*} ,\mathop E\limits_{2}^{*} ,\mathop I\limits_{2}^{*} ,\mathop Q\limits_{2}^{*} ,\mathop R\limits_{2}^{*} } \right)\) is the globally asymptotically stable in the \(\Omega\) .

Sensitivity analysis

The model of the parameters will affect the spread and spread of COVID-19, the results of insertion into model ( 2 ) will be subjected to a sensitivity analysis. We begin by first introducing the following definitions of 30 , 31 , 32 , 33 , 34 , 35 , 36 .

Definition 1:

The normalized forward sensitivity index of the variable ( \(R_{0}\) ), depending on the parameter difference, is given as: \(E_{\zeta }^{\emptyset } = \frac{\partial \emptyset }{{\partial \zeta }} \times \frac{\zeta }{\emptyset }\) . A new expression for \(R_{0}\) is introduced as:

Then the sensitivity indices of the basic reproduction number ( \({R}_{0}\) ), with respect to the system model depends on the nineteenth parameter are computed as below.

We can estimate the sensitivity indices (S.I) of the basic reproduction number \({(R}_{0})\) , taking into account the parameter of the model ( 2 ). The signs of sensitivity indices (S.I) are shown in the Table 3 and bar chart Fig.  8 .

A bar chart showing the measurement of the sensitivity indices with various parameters of model (2) and the reference values as indicated in Table 3 .

The effects of changing parameter values on the functional value of the reproduction number \({R}_{0}\) are obtainable in this section. The necessary parameters must be found, which may be important criteria in disease management. The desirable changes of the occurred when their changes produce a positive effect, i.e., when their sensitivity indices have positive sign, i.e. \(\alpha_{1} ,\alpha_{2} ,IIP_{2} ,g_{1} ,g_{2} ,q_{3T} ,\mu_{{}} ,C_{{}} ,\varphi_{1} ,\varphi_{12} ,\varphi_{2}\) and \(\varphi_{21}\) have a positive effect on \(R_{0}\) . The determine that the increase in the number of two exposed population \(\left( {E_{1} ,E_{2} } \right)\) and two infectious host population \(\left( {I_{1} ,I_{2} } \right)\) with the value \(IIP_{1} , g_{1} , g_{2} ,IIP_{2}\) may lead to an outbreak. On the other hand, the negative sign of the sensitivity indices (S.I) in the \(R_{0}\) i.e. \(\delta_{1} , \delta_{2} , IIP_{1} ,q_{2T} ,\rho_{1} ,\gamma_{1} ,\vartheta_{{}} ,\gamma_{2}\) and \(\rho_{2}\) has a negative effect on the spread of disease according to the system ( 2 ). Thus, the sensitivity indices (S.I) of the Covid-19 ( 2 ) shows that there will been appreciable change at the beginning of the transmission the disease. This would help the public health official to plan on how best to develop a reasonable interference strategy to prevent and manage the spread of the disease.

Conclusions and discussion

Tourism has become an important source of foreign currency for many countries. This is especially true for Thailand. It is the second major source of currency. This means that tourists are coming to Thailand every year. When combined with the need for temporary, seasonal farmer workers to support the main source of income in Thailand, that is the agricultural industry, foreigners (tourists), and these foreign workers diseases can be brought into Thailand. Thailand must always be aware of the arrival of new infectious diseases. Most recently, the novel coronavirus COVID-19 appeared in China. From a few hundred infections in Wuhan, China, it is quickly evolved into a pandemic, which spread to five continents public health authorities in Thailand have initiated public health measures to control the spread and stop the spread of this virus in the United States. More than one million people have died from the disease. In this article a standard SEIQR model has been introduced for the transmission dynamics of COVID-19 infection in Thailand and for Foreign (tourists) entering the Thai population. Affecting the change of COVID-19 among Thais people, they took the SEIQR model for each population and linked them together, allowing members of each population to cross-infection with each other. The impact of factors causing changes in the spread of COVID-19 is examined. After that, we performed a basic reproductive number analysis and saw how homeostasis changes. Taking cross-infection (mixed) into account, we find that our model achieves an infection-free equilibrium. When the basic reproductive number is less than one. This model achieves local equilibrium at multiple points when the number is greater than one. Our analysis shows that the rate of recovery rate of both Thais and tourists would be affected by decreases in the recruitment rates and death rates. This result shows that the recovery rate for both Thais and foreigners has increased. This is because changes in recruitment and death rates will result in a decrease in the basic reproductive number. However, changes \(IIP_{1}\) (capita rate of progression of Thais population from the exposed state to the infectious state), \(IIP_{2}\) (capita rate of progression of foreign human from the exposed state to the infectious state), \(q_{2T}\) ( the number of infected Thais that leave the quarantine period with the virus intact) and \(q_{3T}\) (the number of infected Foreign that leave the quarantine period with the virus intact), \(g_{1}\) (the rate at which the exposed Thais are put into quarantine from the exposed and infected Thais) and \(g_{2}\) (the rate at which the exposed Foreign (tourists) are put into quarantine from the exposed and infected Foreign (tourists)), \(\varphi_{1}\) ( transmission rate of virus between population in Thais population), \(\varphi_{2}\) (transmission rate of virus between population in Foreign (tourists) population), \(\mu\) (recruitment term of the susceptible population in Thais) and \(C\) (recruitment term of the susceptible population in Foreign (tourists)) would cause the basic reproductive number to increase meaning increases in the severity of the pandemic, more people being infected by the COVID-19 coronavirus. Therefore, we controlled the number of new confirmed cases or new infections significantly by introducing a positive change in the parameter memory value in the sensitivity analysis.

In summary, from the study of the spread of the COVID-19 virus, which is an infectious disease. This disease is a health problem, leading to a rapid decline in the impact on the economy. Although governments and the World Health Organization have implemented international control measures and prevented interference. By creating a mathematical model that uses data from disease outbreaks between Thais population and Foreign (tourist) entering Thailand. To determine some parameters affecting the outbreak under proper control of the disease. By relying on the strategies of the government and the World Health Organization, including controlling the spread of infection, incubation, treatment and prevention of fever. It was found that controlling the disease transmission will be a guideline for reducing the spread and reducing the number of cases between Thai people and foreigners.

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Acknowledgements

The authors thank the handling editor and anonymous referees for their valuable comments and suggestions which led to an improvement of our original paper. R.S would like to thank Research and Development Institute and Faculty of Science and Technology, Phuket Rajabhat University. P.P would like to thank School of Science, King Mongkut’s Institute of Technology Ladkrabang.

Research and Development Institute and Faculty of Science and Technology, Phuket Rajabhat University.

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Sungchasit, R., Tang, IM. & Pongsumpun, P. Sensitivity analysis and global stability of epidemic between Thais and tourists for Covid -19. Sci Rep 14 , 21569 (2024). https://doi.org/10.1038/s41598-024-71009-x

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Sensitivity analysis of excited-state population in plasma based on relative entropy.

1. Introduction

2. theoretical backgrounds and methods, 3.1. glow discharge, 3.2. arc discharge, 3.3. recombining plasma, 4. discussion, 5. conclusions, author contributions, institutional review board statement, data availability statement, conflicts of interest.

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Shimada, Y.; Akatsuka, H. Sensitivity Analysis of Excited-State Population in Plasma Based on Relative Entropy. Entropy 2024 , 26 , 782. https://doi.org/10.3390/e26090782

Shimada Y, Akatsuka H. Sensitivity Analysis of Excited-State Population in Plasma Based on Relative Entropy. Entropy . 2024; 26(9):782. https://doi.org/10.3390/e26090782

Shimada, Yosuke, and Hiroshi Akatsuka. 2024. "Sensitivity Analysis of Excited-State Population in Plasma Based on Relative Entropy" Entropy 26, no. 9: 782. https://doi.org/10.3390/e26090782

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StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 Jan-.

StatPearls [Internet].

Diagnostic testing accuracy: sensitivity, specificity, predictive values and likelihood ratios.

Jacob Shreffler ; Martin R. Huecker .

Affiliations

Last Update: March 6, 2023 .

• Definition/Introduction

To make clinical decisions and guide patient care, providers must comprehend the likelihood of a patient having a disease, combining an understanding of pretest probability and diagnostic assessments. [1]  Diagnostic tools are routinely utilized in healthcare settings to determine treatment methods; however, many of these tools are subject to error.

• Issues of Concern

Benefits of Diagnostic Testing

The utilization of diagnostic tests in patient care settings must be guided by evidence. Unfortunately, many order tests without considering the evidence to support them. [1]  Sensitivity and specificity are essential indicators of test accuracy and allow healthcare providers to determine the appropriateness of the diagnostic tool. [2]  Providers should utilize diagnostic tests with the proper level of confidence in the results derived from known sensitivity, specificity, positive predictive values (PPV), negative predictive values (NPV), positive likelihood ratios, and negative likelihood ratios. [2]

The presentation of diagnostic exam results is often in 2x2 tables. The values within this table can help to determine sensitivity, specificity, predictive values, and likelihood ratios. A diagnostic test’s validity, or its ability to measure what it is intended to, is determined by sensitivity and specificity. [3]  See Table. Diagnostic Testing Accuracy Table.

Sensitivity

Sensitivity is the proportion of true positives tests out of all patients with a condition. [4]  In other words, it is the ability of a test or instrument to yield a positive result for a subject that has that disease. [2]  The ability to correctly classify a test is essential, and the equation for sensitivity is the following:

• Sensitivity=(True Positives (A))/(True Positives (A)+False Negatives (C))

Sensitivity does not allow providers to understand individuals who tested positive but did not have the disease. [5]  False positives are a consideration through measurements of specificity and PPV.

Specificity

Specificity is the percentage of true negatives out of all subjects who do not have a disease or condition [4] . In other words, it is the ability of the test or instrument to obtain normal range or negative results for a person who does not have a disease. [2]  The formula to determine specificity is the following:

Specificity=(True Negatives (D))/(True Negatives (D)+False Positives (B))

Sensitivity and specificity are inversely related: as sensitivity increases, specificity tends to decrease , and vice versa. [3] [6]  Highly sensitive tests will lead to positive findings for patients with a disease, whereas highly specific tests will show patients without a finding having no disease. [6]  Sensitivity and specificity should always merit consideration together to provide a holistic picture of a diagnostic test. [7]  Next, it is important to understand PPVs and NPVs.

PPV and NPV

PPVs determine, out of all of the positive findings, how many are true positives; NPVs determine, out of all of the negative findings, how many are true negatives. As the value increases toward 100, it approaches a ‘gold standard.’ [3]  The formulas for PPV and NPV are below.

Positive Predictive Value=(True Positives (A))/(True Positives (A)+False Positives (B))  

Negative Predictive Value=(True Negatives (D))/(True Negatives (D)+False Negatives(C))

Disease prevalence in a population affects PPV and NPV. When a disease is highly prevalent, the test is better at ‘ruling in' the disease and worse at ‘ruling it out.’ [1]  Therefore, disease prevalence should also merit consideration when providers examine their diagnostic test metrics or interpret these values from other providers or researchers. Providers should consider the sample when reviewing research that presents these values and understand that the values within their population may differ. [5]  Considering all of the diagnostic test outputs, issues with results (e.g., very low specificity) may make clinicians reconsider clinical acceptability, and alternative diagnostic methods or tests should be considered. [8]  

Likelihood Ratios

Likelihood ratios (LRs) represent another statistical tool to understand diagnostic tests. LRs allow providers to determine how much the utilization of a particular test will alter the probability. [4]  A positive likelihood ratio, or LR+, is the “probability that a positive test would be expected in a patient divided by the probability that a positive test would be expected in a patient without a disease.”. [4]  In other words, an LR+ is the true positivity rate divided by the false positivity rate [3] .  A negative likelihood ratio or LR-, is “the probability of a patient testing negative who has a disease divided by the probability of a patient testing negative who does not have a disease.”. [4]  Unlike predictive values, and similar to sensitivity and specificity, likelihood ratios are not impacted by disease prevalence. [9]  The formulas for the likelihood ratios are below.

Positive Likelihood Ratio=Sensitivity/(1-Specificity) 

Negative Likelihood Ratio=(1- Sensitivity)/Specificity

Now that these topics have been covered completely, the application exercise will calculate sensitivity, specificity, predictive values, and likelihood ratios.

Application Exercise

Example: A healthcare provider utilizes a blood test to determine whether or not patients will have a disease. 

The results are the following:

• A total of 1,000 individuals had their blood tested.
• Four hundred twenty-seven individuals had positive findings, and 573 individuals had negative findings.
• Out of the 427 individuals who had positive findings, 369 of them had the disease.
• Out of the 573 individuals who had negative findings, 558 did not have the disease.

Let’s calculate the sensitivity, specificity, PPV, NPV, LR+, and LR-. We first can start with a 2X2 Table. The information above allows us to enter the values in the table below. Notice that values in blue cells were not provided, but we can get them based on the numbers above and the utilization of total cells. See Image. Diagnostic Testing Accuracy Table 2.

The provider found that a total of 384 individuals actually had the disease, but how accurate was the blood test?

• Sensitivity=(369 (A))/(369(A)+15 (C))
• Sensitivity=369/384
• Sensitivity=0.961
• Specificity=(True Negatives (D))/(True Negatives (D)+False Positives (B))
• Specificity=(558 (D))/(558(D)+58 (B))
• Specificity=558/616                                                      
• Specificity=0.906

Positive Predictive Value

• PPV =(True Positives (A))/(True Positives (A)+False Positives (B))
• PPV =(369  (A))/(369 (A)+58(B))
• PPV =369/427

Negative Predictive Value

• NPV=(True Negatives (D))/(True Negatives (D)+False Negatives(C))  
• NPV=(558(D))/(558 (D)+15(C))
• NPV=(558 )/573

Positive Likelihood Ratio

• Positive Likelihood Ratio=Sensitivity/(1-Specificity)
• Positive Likelihood Ratio=0.961/(1-0.906)
• Positive Likelihood Ratio=0.961/0.094
• Positive Likelihood Ratio=10.22
• Negative Likelihood Ratio
• Negative Likelihood Ratio=(1- Sensitivity)/Specificity 
• Negative Likelihood Ratio=(1- 0.961)/0.906
• Negative Likelihood Ratio=0.039/0.906
• Negative Likelihood Ratio=0.043

The results show a sensitivity of 96.1%, specificity of 90.6%, PPV of 86.4%, NPV of 97.4%, LR+ of 10.22, and LR- of 0.043.

• Clinical Significance

Understanding that other diagnostic test data techniques do exist (e.g., receiver operating characteristic curves), the topics in this article represent essential starting points for healthcare providers. Diagnostic testing is a crucial component of evidence-based patient care. When determining whether or not to use a diagnostic test, providers should consider the benefits and risks of the test, as well as the diagnostic accuracy. [1]  By having a foundational understanding of the interpretation of sensitivity, specificity, predictive values, and likelihood ratios, healthcare providers will understand outputs from current and new diagnostic assessments, aiding in decision-making and ultimately improving healthcare for patients.

• Nursing, Allied Health, and Interprofessional Team Interventions

All interprofessional healthcare team members need to understand these values as applied to diagnostic testing, so they can better analyze a patient's condition based on testing results. Any lack of understanding in this area can lead to improper diagnostic interpretation leading to sub-optimal outcomes. Healthcare team members need to collaborate openly to facilitate proper diagnosis leading to properly targeted therapeutic interventions. [Level 5]

• Review Questions
• Access free multiple choice questions on this topic.
• Comment on this article.

Diagnostic Testing Accuracy Table. Contributed by M Huecker, MD, and J Shreffler, PhD

Diagnostic Testing Accuracy Table 2 Contributed by M Huecker, MD, and J Shreffler, PhD

Disclosure: Jacob Shreffler declares no relevant financial relationships with ineligible companies.

Disclosure: Martin Huecker declares no relevant financial relationships with ineligible companies.

This book is distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ), which permits others to distribute the work, provided that the article is not altered or used commercially. You are not required to obtain permission to distribute this article, provided that you credit the author and journal.

• Cite this Page Shreffler J, Huecker MR. Diagnostic Testing Accuracy: Sensitivity, Specificity, Predictive Values and Likelihood Ratios. [Updated 2023 Mar 6]. In: StatPearls [Internet]. Treasure Island (FL): StatPearls Publishing; 2024 Jan-.

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