In this study, we propose a new mathematical model and analyze it to understand the transmission dynamics of the COVID-19 pandemic in Bangkok, Thailand. It is divided into seven compartmental classes, namely, susceptible S , exposed E , symptomatically infected I s , asymptomatically infected I a , quarantined Q , recovered R , and death D , respectively. The next-generation matrix approach was used to compute the basic reproduction number denoted as R cvd 19 of the proposed model. The results show that the disease-free equilibrium is globally asymptotically stable if R cvd 19 < 1 . On the other hand, the global asymptotic stability of the endemic equilibrium occurs if R cvd 19 > 1 . The mathematical analysis of the model is supported using numerical simulations. Moreover, the model’s analysis and numerical results prove that the consistent use of face masks would go on a long way in reducing the COVID-19 pandemic.

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A dynamical model for COVID-19 spread relating to non-pharmaceutical interventions and vaccination is mathematically generated by adding a gradual vaccination compartment for the susceptible population and considering only a symptomatic infectious stage. In our model, there are seven compartments dividing a given population into susceptible (S), vaccinated (V ), exposed (E), infected (I), quarantined (Q), recovered (R) and death (D) groups, respectively. Then, theoretically analysis is given by investigating the COVID-19 free and endemic equilibrium points, and computing the vaccination reproduction number of this model denoted as R vac using the next generation matrix. If R vac > 1, then the COVID-19 transmission increases exponentially and depends on vaccine efficacy. On the other hand, if R vac < 1, then there occurs the COVID-19 disease eradication. The risk from infection can be importantly reduced whenever the intake of COVID-19 vaccines exceeds one dose. The numerical results reveal that the nonpharmaceutical ways and the administered COVID-19 vaccines can be effective against the current variants of COVID-19, and the additional efforts such as a third vaccine booster shot should be considered and implemented to greatly mitigate the risks of emerging variants of the COVID-19 virus. Moreover, combining different types of COVID-19 vaccines can be appeared as a possible way to give better protection against COVID-19 as well.

Epidemic models are essential in understanding the transmission dynamics of diseases. These models are often formulated using differential equations. A variety of methods, which includes approximate, exact and purely numerical, are often used to find the solutions of the differential equations. However, most of these methods are computationally intensive or require symbolic computations. This article presents the Differential Transformation Method (DTM) and Multi-Step Differential Transformation Method (MSDTM) to find the approximate series solutions of an SVIR rotavirus epidemic model. The SVIR model is formulated using the nonlinear first-order ordinary differential equations, where S, V, I and R are the susceptible, vaccinated, infected and recovered compartments. We begin by discussing the theoretical background and the mathematical operations of the DTM and MSDTM. Next, the DTM and MSDTM are applied to compute the solutions of the SVIR rotavirus epidemic model. Lastly, to investigate the efficiency and reliability of both methods, solutions obtained from the DTM and MSDTM are compared with the solutions from the Runge-Kutta Order 4 (RK4) method. The solutions from the DTM and MSDTM are in good agreement with the solutions from the RK4 method. However, the comparison results show that the MSDTM is more efficient and converges to the RK4 method than the DTM. The advantage of the DTM and MSDTM over other methods is that it does not require a perturbation parameter to work and does not generate secular terms. Therefore the application of both methods can be extended to other compartmental models.

<abstract><p>The use of vaccines has always been controversial. Individuals in society may have different opinions about the benefits of vaccines. As a result, some people decide to get vaccinated, while others decide otherwise. The conflicting opinions about vaccinations have a significant impact on the spread of a disease and the dynamics of an epidemic. This study proposes a mathematical model of COVID-19 to understand the interactions of two populations: the low risk population and the high risk population, with two preventive measures. Unvaccinated individuals with chronic diseases are classified as high risk population while the rest are a low risk population. Preventive measures used by low risk group include vaccination (pharmaceutical way), while for the high risk population they include wearing masks, social distancing and regular hand washing (non-pharmaceutical ways). The susceptible and infected sub-populations in both the low risk and the high risk groups were studied in detail through calculations of the effective reproduction number, model analysis, and numerical simulations. Our results show that the introduction of vaccination in the low risk population will significantly reduce infections in both subgroups.</p></abstract>