Modeling the dynamics of COVID-19 with real data from Thailand

In recent years, COVID-19 has evolved into many variants, posing new challenges for disease control and prevention. The Omicron variant, in particular, has been found to be highly contagious. In this study, we constructed and analyzed a mathematical model of COVID-19 transmission that incorporates vaccination and three different compartments of the infected population: asymptomatic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(I_{a})$$\end{document}(Ia), symptomatic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(I_{s})$$\end{document}(Is), and Omicron \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(I_{m})$$\end{document}(Im). The model is formulated in the Caputo sense, which allows for fractional derivatives that capture the memory effects of the disease dynamics. We proved the existence and uniqueness of the solution of the model, obtained the effective reproduction number, showed that the model exhibits both endemic and disease-free equilibrium points, and showed that backward bifurcation can occur. Furthermore, we documented the effects of asymptomatic infected individuals on the disease transmission. We validated the model using real data from Thailand and found that vaccination alone is insufficient to completely eradicate the disease. We also found that Thailand must monitor asymptomatic individuals through stringent testing to halt and subsequently eradicate the disease. Our study provides novel insights into the behavior and impact of the Omicron variant and suggests possible strategies to mitigate its spread.

1. We propose an SVEI s I a I m R model by including a parameter that represents immunity development in vacci- nated individuals.This will help determine how well the vaccine works and help in the control of COVID-19 spread.2. The model is further extended to fractional order in the Caputo sense.3. Theoretical results are established 4. A global sensitivity analysis was performed.5. Numerical illustrations were also performed.
The fractional-order framework for conceptualizing COVID-19 offers several epidemiological advantages over classical integer-order models.The implementation offers several epidemiological advantages.First, it acknowledges the intricate dynamics of the virus, including super-spreading events, variable transmission rates, and the occurrence of multiple waves of infection 36,37 .Second, fractional-order models capture the persistence of memory, accounting for the past trajectory of an epidemic, thereby providing a more accurate representation of its dynamics [38][39][40][41][42][43][44] .These models also enable the evaluation of intervention effectiveness by considering the timedependent impact of measures, such as NPIs and vaccination campaigns.Moreover, fractional-order models enable multiscale analysis, capturing interactions at different levels, which enhances their reliability for informed decision-making in public health interventions and resource allocation 45 .
The remainder of this paper is arranged as follows: "Model formulation" section develops our model and provides mathematical preliminaries, as well as extending the model to fractional order."Qualitative properties of the model" section presents a qualitative analysis of the proposed model, including its stability and bifurcation properties."Numerical simulations of the model" section conducts numerical simulations of the Caputo fractional-order model and performs global sensitivity analysis to identify the key parameters affecting the dynamics."Discussion" section discusses the results and their implications for disease control and prevention."Conclusions" section concludes the paper with some remarks and future directions.

Model formulation
Here, we present a compartmental model that studies the spread of coronavirus in Thailand.The model is constructed as follows: we divide the infected class into three sub-classes I a , I s , I m that are asymptomatic infected, symptomatic infected (infected individuals showing clinical symptoms of COVID-19 most commonly: fever or chills, cough, shortness of breath, sore throat, loss of taste or smell), and infected individuals showing clinical symptoms unique to omicrons (sore throat, particularly a "scratchy" throat, persistent cough 46 ).Thus, the total population N is defined as At rate , susceptible individuals are recruited into the population, and recovered individuals lose immunity and move to this population at rate ρ .When certain people receive vaccination, the population decreases at rate ω(0 < ω ≤ 1) .This population further decreases when some individuals are exposed to the force of infection, as follows: where ν 1 and ν 2 are the probability of infectiousness of I s and I m respectively, and β is the effective contact rate.
τ indicates the rate of incubation of the exposed individuals that are infected with the virus without showing symptoms at the rate ̟ , showing symptoms of other variants at the rate σ or showing unique symptoms of Omi- cron at a rate of (1 − σ − ̟ ) .Parameters α 1 , α 2 and α 3 provide information on the recovery of asymptomatic, symptomatic, and omicron-infected individuals, respectively.δ 1 and δ 2 are the rates at which symptomatic people and those in the omicron class die due to the disease, respectively.
Figure 1 depicts the schematic diagram of the model, whereas Tables 1 and 2 provide the meanings of the variables and parameters, respectively.
(1) N = S + V + E + I s + I a + I m + R.
(2)    Additionally, the parameter values can be obtained from the existing literature and guided estimations.However, relying solely on this approach can occasionally result in erratic behavior.To collect authentic and reliable cases from the population infected with COVID-19, it is essential to determine the appropriate biological characteristics that characterize these cases.These types of real-life examples can be offered for a period ranging from days to weeks to months to even years.Due to uncertainties in data analysis, there is a possibility that the conclusion could be inaccurate.Although there are numerous methods in the literature that can be used to estimate parameter values, the least-squares method is the most frequently employed.The method uses the idea of minimizing residuals between available infections for real data ȳj = 0, 1, . . ., n and the discrete points obtained with the suggested set of simulation equations f (t j , y j ) as given below: The aforementioned objective has been accomplished by utilizing the built-in routines of NonlinearModelFit and ParametricNDSolve that are included in the programming language known as Wolfram Mathematica 12.1.These fitted parameters are displayed in Table 3 Using real data from Thailand, which ranges from 1st July 2022 to 30th September 2022 (see 19,47 ), the parameters of (3) were estimated.Thailand's initial population was estimated to be N(0) ≃ 70, 000, 000 48,49 .The initial populations of reported vaccinated and infectious individuals are given by V (0) = 107, 912, and I s (0) + I m (0) = 2354 respectively.The initial susceptible population was S(0) = 69, 688, 560 .We assume the initial population of the exposed, asymptomatic infected, and recovered individuals to be E(0) = 100, 000, I a (0) = 1177, and R(0) = 100, 000 .With the help of these conditions and the parameters listed in Table 3, the fitted curve is obtained in Fig. 2 where the statistical R 2 value is computed as ≈ 0.97 showing that the regression line perfectly fits the data.Moreover, the residuals in Fig. 3 are randomly distributed around a mean of zero, indicating a good fit.SVEI s I a I m R model in Caputo fractional operator form.In this section, we transform model (3) using the Caputo fractional derivatives in a similar manner 57 .First, we recall some preliminaries.
Definition 1 (See 58 ) Let g(t) be a function that satisfies some smoothness condition and α > 0 s.t α, t ∈ R , the derivative in Caputo form is define as: and for n = 1 and α ∈ (0, 1] .Also, α > 0 the corresponding fractional integral is defined as Definition 2 [Mittag-Leffler 59 ] This is defined as: and its general form Therefore, (3) in Caputo sense is defined as: where dξ n dξ , where n − 1 < α, n ∈ N.

Qualitative properties of the model
Analysis of ( 11) is carried out in this section.
Boundedness and positivity of the model system (11).The fractional-order system (11) must be positive because the solutions represent the densities of the populations that interact with each other, and from a biological perspective, the lowest possible value for each population in the model system is zero, which is relevant to establishing an upper bound.This is assured by the following outcomes: Theorem 3 Consider (11) with S(0) > 0, V (0) > 0, E(0) ≥ 0, I s (0) ≥ 0, I a (0) ≥ 0, I m (0) ≥ 0 and R(0) ≥ 0 as an initial condition, then all solutions are uniformly bounded and positive.
Proof First, we start by adding the population states which is possible by the linearity property of the Caputo fractional derivative: By using Laplace transform and its inverse on (12), after simplifying we obtain, where E α,1 (−µ α t α ) and E α,k+1 (−µ α t α ) are Mittag-Leffler functions 61 .Hence, solutions of the Caputo (11) con- fined in the region D , where Secondly, we show that the solutions of (11) are positive in the feasible region D .In order to show this, we begin by examining the first equation of the model ( 11) where b = β α (I a +ν α 1 I s +ν α 2 I m ) N + (ω α + µ α ) .Using the Laplace transform method and the positivity of the Mittag- Leffler function 62 we have In a similar manner, V (t), E(t), I s , I a , I m , R ≥ 0, ∀t ≥ 0 .
Existence and uniqueness of solutions of (11).Consider a Banach space on J = [0, T] of all continuous real-valued functions denoted as B(J, R) with the following norm: such as Applying ( 7) on both sides of ( 11) we obtain: The definition (7) then directs us to the following: with the respective kernels (15)    www.nature.com/scientificreports/An upper bound on S(t), V (t), E(t), I s (t), I a (t), I m (t), and R(t) is needed for the Lipschitz condition to be satis- fied by the kernels (G i , i = 1, 2, . . ., 7) in (18).Consider two distinct function S and S , then If we consider we have In a similar manner, we also have that for the remaining state variables where The Lipschitz constants for each kernel G i , i = 1, 2, . . ., 7 are asserted by ζ 1 , ζ 2 , . . ., ζ 7 respectively.As a result, the Lipschitz condition is satisfied.Using (17) the following recursive formulae can now be used in order to establish the uniqueness: In recursive formulas, the difference between the consecutive terms can be written as follows: (18)  www.nature.com/scientificreports/

Note that
Applying the norm, for each of the differences in (23) we formulate the recursive inequalities as follows: Since the kernel G 1 satisfies the Lipschitz condition with constant ζ 1 , then we can see that

Thus, we obtain
As a result, we can obtain the following: Theorem 4 Suppose for t ∈ [0, b] the following inequalities hold: Then the model (11) has a unique solution  25)-( 26) yields the following relations when the functions S(t), V (t), E(t), I s (t), I a (t), I m (t), and R(t) are assumed to be bounded and each kernel satisfies a Lipschitz condition: Thus, it can be observed that the sequence ( 27) satisfy ||ϒ i n (t)|| =⇒ 0 , for i = 1, 2, . . ., 7 as n =⇒ ∞ .Further, by applying the triangular inequality to equation ( 27) and for any k, we are able to find: where q ′ i s are by hypothesis . Thus, by (27), a Cauchy sequence in B is formed by S n , V n , E n , I s n , I a n , I m n , and R n .Hence as n =⇒ ∞ , the unique solution of ( 11) is obtained.

Equilibrium point and stability analysis.
In this section, we find the disease-free equilibrium point (DFE) and the endemic equilibria (EE).The next-generation matrix is used to calculate the effective reproduction number R eff .Lyapunov functions were constructed and used to establish the global stability of the equilibria.

Disease free equilibrium (DFE). First setting (11) to 0 we get the following distinct DFE solutions
Effective reproduction number ( R eff ).We use the method in 63 similarly to 64,65 to compute R eff .Let F represent non-negative matrix of the new infection, V is the transmission matrix, then: , 0, 0, 0, 0, 0 .
Vol:.( 1234567890) where Hence, we have the following lemma: The DFE is locally asymptotically stable if R eff < 1 , and unstable otherwise.
For global stability, consider the following theorem: The DFE is globally asymptotically stable if R eff ≤ 1.

Proof Consider the Lyapunov function:
where The Lyapunov derivative is calculated as Hence Existence of endemic equilibrium point (EE) and Bifurcation Analysis.The endemic equilibrium point (EE) represents the situation in which the disease continue to exist across the population.37) can now be written as After simplifying (39), we obtained the following: (η α s ) * = 0 as one of the solutions (which corresponds to the DFE and the quadratic equation): where Therefore, by simplifying (40) and substituting into EE, we then obtain a positive EE: Hence, we get the following result: Theorem 7 From (3): 1. R eff > 1 or a 2 < 0 implies a unique endemic equilibrium, 2. Also a 1 < 0 and R eff = 1 or = a 2 1 − 4a 0 a 2 = 0 implies a unique endemic equilibrium, 3. a 1 < 0, R eff < 1 and � > 0 implies two endemic equilibrium, and 4. no endemic equilibrium otherwise.
Since all the model parameters are positive it is obvious that a 0 > 0 and the proof complies with the charac- teristics of quadratic equation roots.a 2 is either positive or negative depending on whether R eff < 1 or R eff > 1 .Clearly, from the case (i) of Theorem 7 whenever R eff > 1 , (3) has a unique EE.From case (iii) of Theorem 7, we get a backward bifurcation, this is a scenario where stable DFE and stable EE coexist whenever R eff < 1 (see, [66][67][68] and references therein for discussions on bifurcation analysis).We verify the backward bifurcation (BB) in a ( 36) Vol:.( 1234567890 www.nature.com/scientificreports/similar manner in 69,70 by first letting discriminant a 2 1 − 4a 0 a 2 = 0 and simplifying for the critical value of R eff , denoted by R c eff and given by The BB would then occur for the values of R c eff such that R c eff < R eff < 1 .It can be seen in Fig. 4, this is illustrated by simulating the model with the following set of parameter values.It is important to note that the parameters used in this illustration are chosen for just demonstration purposes.The parameters used are given in Table 3, w i t h β = 0.3888, ω = 0.01202643, ǫ = 0.3, ν 1 = 0.3, ν 2 = 0.1, ̟ = 0.2, σ = 0.899, γ 1 = 0.3, γ 2 = 0.2, δ 1 = 0.028, δ 2 = 0.025, α 3 = 0.1203, ρ = 0.5, and α ∈ (0, 1] .So that, a 0 = 0.4849076636 × 10 −1 , a 1 = −0.1013622972×10 −2 , a 2 = 5.647464170×10 −7 , R c eff = 0.8764132312, and R eff = 0.9868237565(that is, R c eff < R eff < 1) .
Global stability analysis of the endemic equilibrium.From 71-73 , we have the following results: The unique endemic equilibrium is globally asymptomatically stable in D when R eff > 1 , provided that and are true.
See "Appendix" section for the proof.

Numerical simulations of the model
Using COVID-19 data obtained from Thailand, we carried out various simulations to show the transmission dynamics of the disease, considering many scenarios.

Numerical results.
Figure 5 shows the time-series simulation results for (3).
Next, using the Caputo operator (α) , numerical simulations based on the fractional model are presented.As a result, ( 11) is numerically solved as described in 74 using the biological parameter values presented in Table 3. Figure 6 shows the simulation results for varying α values of the state variables over time.
Figure 6a shows that the number of people who are susceptible (i.e., the number of people who are not immune to COVID-19) decreases over time.This is because as more people become infected or vaccinated, they become immune and reduce the pool of susceptible people.Figure 6b shows the exposed population for various values of α .The graph shows that this population reached a maximum peak and then began to decline.as NPIs and vaccines, were effective in reducing the number of infections in all three groups.This suggests that these measures can control the spread of the virus and mitigate the severity of the disease, regardless of whether individuals show symptoms or are infected with the Omicron variant.However, it is important to note that the asymptomatic group can still transmit the virus to others; therefore, they should be considered as a potential source of transmission.Figure 7 compares the three infected populations (symptomatic, asymptomatic, and Omicron-infected) and provides a comprehensive visual representation of the simulations.The graph enables for direct comparison of the trends and magnitudes of the three populations.These observations signal that the Omicron variant is more contagious and spreads the disease more rapidly than the other COVID-19 variants.It is also important to acknowledge that the asymptomatic infected population can contribute to the transmission of the virus despite the absence of symptoms.Consequently, controlling the spread of both symptomatic and asymptomatic infections is crucial for preventing further propagation of the virus.
Figure 8 shows the global stability of the disease-free equilibrium (DFE), where no one is infected and the disease cannot spread.The figure shows that the infected population always stays at zero over time, whereas the susceptible population increases to attain a steady state.Figure 9 illustrates the dynamic shifts in the susceptible and exposed populations as time progressed, specifically when the infection became endemic within the population.This captures fluctuating levels of susceptibility and exposure, highlighting the evolving nature of the impact of the disease on the population over time.

Vaccine intervention and global sensitivity analysis. The use of the COVID-19 vaccine helps reduce
the spread of the virus 51 .In this section, we evaluate the impact of the vaccine and its effectiveness on the reproduction number.We define the reproduction number in the absence of a vaccine as a basic reproduction number Using Eqs. ( 31) and ( 45), we get: Note that k 1 − k 2 = −ω α ǫ α , hence R eff − R 0 is strictly negative.The implication of this is that using the vaccine effectively will have a strong impact on the reduction of the spread of all COVID-19 variants including Omicron.Using the fitted and estimated parameters from Table 3, we can also estimate the basic reproduction number and effective reproduction number as R 0 = 1.828794173 and R eff = 0.9159273511 respectively.
Global sensitivity analysis using partial rank correlation coefficients.In epidemic modeling, errors occur when attempting to estimate parameter values for the model.Often there is a tendency for these parameters to be based on incomplete or limited data, which can lead to estimates that are inaccurately reflective of the actual population.In addition, the precise value of several parameters that are being evaluated is frequently uncertain.Variations between groups or areas, as well as personal circumstances that could not be considered, may lead to inaccuracies.Even with sufficient data and accuracy checks built in, parameter uncertainty is still probably caused by time-changing conditions within a given population or abrupt alterations because of unforeseeable occurrences like natural disasters or civil unrest.For these reasons, it's crucial to carry out sampling and sensitiv- www.nature.com/scientificreports/ity analysis to identify the variables that significantly affect model output.The Sampling and Sensitivity Analysis Tool (SaSAT) is a software tool developed for such purposes (see 75 ).In our model, 18 different COVID-19 epidemiological parameters, whose values varied from other research and model fitting to data were used to regulate the effective reproduction number.Each of these parameters had baseline values and ranges assigned to them following 66 .In order to create a 1000 by 18 matrix with each row defining a different parameter set, we utilized Latin hypercube sampling (LHS) to draw 1000 samples for each of the parameters.The effective reproduction numbers were calculated using the parameter sets, and the statistical contribution of each parameter to   www.nature.com/scientificreports/ the reproduction numbers was then described using the partial rank correlation coefficient (PRCC).Figure 10 shows the tornado plot of the results.The top five most sensitive parameters affecting the R eff are β, ̟ , ǫ, α 1 , and ν 2 in that order, as shown in Figure 10.To reduce the value of R eff , we need to reduce β, ̟ , and ν 2 or increase the values of ǫ and α 1 .It is important to note that a faster decline in the value of R eff will result from simultaneously increasing the values of parameters with negative PRCC values and decreasing the values of parameters with positive PRCC values.
Vaccines with a high level of efficacy have the potential to reduce the number of secondary infections in the community to a considerable extent.The graphical representation (response surface plot) of R eff in the parameter space (ǫ, α 1 ) see Fig. 11, which tends to suggest that the impact of the vaccine effectiveness is similar to the effect of recovery of the asymptomatic infected individuals.
Figure 12 depicts the impact of the effective contact rate β vs. vaccination rate of the susceptible individuals ω on the effective reproduction number.Figure 13 depicts the impact of the rate in progression from exposed individuals ̟ vs. infection reduction of the vaccinated individuals due to the vaccine effectiveness ǫ.

Discussion
Mathematical models provide efficient techniques for investigating how COVID-19 works and its potential behavior over time.Researchers and governments are using these models to predict how the virus will spread and help them make decisions about mitigation strategies, resource allocation, vaccine development, public health messaging, and other aspects of dealing with the virus.
In this study, we modeled the transmission and spread of COVID-19 by considering vaccination and vaccine effectiveness.Our model indicated that vaccination (a successful vaccine against COVID-19) should not  www.nature.com/scientificreports/be seen as the only solution that will eradicate this disease but rather as a valuable tool for containing its spread and reducing severe illnesses within populations.To attain true victory against this pandemic, attention must be given to the recovery of asymptomatic COVID-19-infected individuals.These individuals pose a unique challenge in controlling the spread of the disease.In other words, as a first step, and perhaps the most important, is to encourage individuals to get tested if they have any reason to believe that they have been exposed to someone who is recently diagnosed with COVID-19 or someone exhibiting symptoms.it is crucial that these individuals follow up with medical personnel and stay in contact regarding any further instructions that they may require.If there are any new symptoms that appear in the future, they should notify the medical personnel as soon as possible, so that they can get proper care and advice if necessary.Additionally, they should also practice proper hygiene measures, including washing their hands regularly and maintaining proper social distancing while out in public at all times.
From the results of our PRCC calculations, we found that the top five parameters that have the most influence on the disease transmission dynamics are effective contact rate, the rate of progression from exposed to asymptomatic infected individuals, infection reduction due to vaccine effectiveness, the recovery rate of asymptomatic infected individuals, and infectiousness of omicron individuals.These do not fully support the finding in 26 , which showed that the top 5 parameters of their model were the contact rate, infections of the omicron-infected individuals, incubation period, recovery rate of the asymptomatic infected individuals, and the rate of flow to omicron infected individuals.
The sensitivity analysis results revealed that significantly improving vaccine effectiveness and high recovery rate of the asymptomatic individuals can make the disease be eradicated.The current study agrees with the work of many authors that reported a significant decrease in COVID-19 transmission (see, for example, 76 ).
A response surface plot of the effective reproduction number, as a function of the recovery rate of the asymptomatic infected individuals and a fraction of the reduction of infections due to vaccine effectiveness, is depicted in Fig. 12.It follows from this figure if both parameters are increased, that could possibly eradicate the spread  www.nature.com/scientificreports/ of COVID-19.Another parameter in our model that can be targeted for interventions is the vaccination rate of susceptible individuals.According to official government data, more than half of Thailand's population has been vaccinated for the virus.An increase in this parameter can lead to a reduction in the effective reproduction number.The appearance of (ω) as one the top six most sensitive parameters is quite significant because it supports the work of 27 .

Conclusions
In this paper, we presented a comprehensive mathematical analysis of the transmission dynamics of COVID-19 infections in Thailand, taking into account the different clinical manifestations of the disease and the emergence of the omicron variant.We developed an SVEI s I a I m R model that captures the interactions among susceptible, vaccinated, exposed, symptomatic, asymptomatic, and omicron-infected individuals.The following are some of the main findings of this study: (1) We calculated the effective reproduction number (R eff ) and established the conditions for both local and global stability of the equilibrium points for the model ( 3). ( 2) We also showed that the model exhibits backward bifurcation at R eff = 1 which brings about a sudden change from a stable equilibrium to an unstable one.(3) To determine the most crucial parameters that control the dynamics of COVID-19 transmissions, we performed a global sensitivity analysis utilizing Latin Hypercube Sampling and Partial Rank Correlation Coefficient and we discovered the most important parameters in controlling this pandemic are effective contact rate, the rate of progression from exposed to asymptomatic infected individuals, infection reduction due to vaccine effectiveness, the recovery rate of asymptomatic infected individuals, and infectiousness of omicron individuals.(4) To demonstrate some of the aforementioned theoretical findings, numerical simulations are carried out using the fitted parameters to Thailand data or cited from existing literature.
Our study provides valuable insights into the epidemiology and control of COVID-19 in Thailand.Based on our findings, we suggest that public health authorities and policymakers should prioritize increasing vaccination coverage, enhancing testing and tracing capacities, enforcing social distancing and mask wearing measures, and monitoring the emergence and spread of new variants.These interventions can help reduce the transmission potential of COVID-19 and prevent future outbreaks.

Parameter Meaning and units 2 2
Recruitment rate (day −1 ) β Effective contact rate (per person • day −1 ) ω Vaccination rate (day −1 ) ǫ Infection reduction of vaccinated individuals (day −1 ) ν 1 Probability of infectiousness of symptomatic individuals I s ν Probability of infectiousness of omicron infected individuals I m µ Natural death rate (day −1 ) τ Disease incubation period (day −1 ) ̟ Rate of progression from exposed individuals to asymptomatic infected individuals (person • day −1 ) σ Rate of progression from exposed individuals to symptomatic infected individuals (person • day −1 ) γ 1 Rate at which asymptomatic infected person start showing symptoms of other variants (person • day −1 ) γ Rate at which asymptomatic infected start showing symptoms attributed to omicron (person • day −1 ) δ 1 COVID-19 induced mortality rate on I s (day −1 ) δ 2 COVID-19 Omicron induced mortality rate on I m (day −1 ) α 1 , α 2 , α 3 Rate of recovery of I a , I s and I m respectively (day −1 ) ρ The rate at which recovered individuals lost immunity (day −1 ) Parameter estimation and model fitting.Obtaining optimal parameter values and performing model validations are crucial when working with mathematical models that utilize real data.This is primarily because the accurate identification of parameter values from obtained data is often challenging.It is essential to obtain well-fitted parameter values for a specific model.Certain parameters associated with the epidemic can be computed by considering both the initial behavior of the epidemic and demographic factors linked to the disease.

Figure 2 .
Figure 2. The curve fitting of model (3) simulations with the real cases of the disease.

Figure 5 .
Figure 5. Dynamics of the whole population.

Figure 6 .Figure 7 .
Figure 6.Graphs for the nature of each state variable for the Caputo version of the fractional model at different values of α.

Figure 8 .
Figure 8. DFE of Susceptible (S) and Infected (I s , I a , I m ) population.

Figure 10 .
Figure 10.Tornado plot showing the sensitivities of the model parameters affecting the effective reproduction number R eff .

Figure 11 .
Figure 11.response surface plot of R eff with respect to α 1 versus ǫ.

Figure 12 .
Figure 12. response surface plot of R eff with respect to β versus ω.

Figure 13 .
Figure 13.response surface plot of R eff with respect to ̟ versus ǫ.
s − I * s − I * s ln I s I * s + J 5 I a − I * a − I * a ln I a I * a + J 6 I m − I * m − I * m ln I m I * m .

Table 1 .
Meaning of variables.

Table 2 .
Meanings of parameters and their units.