In this research work, we present a mathematical model for novel coronavirus-19 infectious disease which consists of three different compartments: susceptible, infected, and recovered under convex incident rate involving immigration rate. We first derive the formulation of the model. Also, we give some qualitative aspects for the model including existence of equilibriums and its stability results by using various tools of nonlinear analysis. Then, by means of the nonstandard finite difference scheme (NSFD), we simulate the results for the data of Wuhan city against two different sets of values of immigration parameter. By means of simulation, we show how protection, exposure, death, and cure rates affect the susceptible, infected, and recovered population with the passage of time involving immigration. On the basis of simulation, we observe the dynamical behavior due to immigration of susceptible and infected classes or one of these two.

In this paper, we study a nonlinear mathematical model which addresses the transmission dynamics of COVID-19. The considered model consists of susceptible ([ ]), exposed ([ ]), infected ([ ]), and recovered ([ ]) individuals. For simplicity, the model is abbreviated as [ ]. Immigration rates of two kinds are involved in susceptible and infected individuals. First of all, the model is formulated. Then via classical analysis, we investigate its local and global stability by using the Jacobian matrix and Lyapunov function method. Further, the fundamental reproduction number [ ] is computed for the said model. Then, we simulate the model through the Runge–Kutta method of order two abbreviated as RK2. Finally, we switch over to the fractional order model and investigate its numerical simulations corresponding to different fractional orders by using the fractional order version of the aforementioned numerical method. Finally, graphical presentations are given for the approximate solution of various compartments of the proposed model. Also, a comparison with real data has been shown.

In this study, we considered a model for novel COVID-19 consisting on five classes, namely $\mathcal{S}$, susceptible; $\mathcal{E}$, exposed; $\mathcal{I}$, infected; $\mathcal{V}$, vaccinated; and $\mathcal{R}$, recovered. We derived the expression for the basic reproductive rate ${\mathcal{R}}_0$ and studied disease-free and endemic equilibrium as well as local and global stability. In addition, we extended the nonstandard finite difference scheme to simulate our model using some real data. Moreover, keeping in mind the importance of fractional order derivatives, we also attempted to extend our numerical results for the fractional order model. In this regard, we considered the proposed model under the concept of a fractional order derivative using the Caputo concept. We extended the nonstandard finite difference scheme for fractional order and simulated our results. Moreover, we also compared the numerical scheme with the traditional RK4 both in CPU time as well as graphically. Our results have close resemblance to those of the RK4 method. Also, in the case of the infected class, we compared our simulated results with the real data.

We develop a mathematical model for the SARAS-CoV-2 double variant transmission characteristics with variant 1 vaccination to address this novel aspect of the disease. The model is theoretically examined, and adequate requirements are derived for the stability of its equilibrium points. The model includes the single variant 1 and variant 2 endemic equilibria in addition to the endemic and disease-free equilibria. Various approaches are used for the global and local stability of the model. For both strains, we determine the basic reproductive numbers ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$. To investigate the occurrence of the layers (waves), we expand the model to include some analysis based on the second-order derivative. The model is then expanded to its stochastic form, and numerical outcomes are computed. For numerical purposes, we use the nonstandard finite difference method. Some error analysis is also recorded.

In this paper, we develop a new mathematical model for an in-depth understanding of COVID-19 (Omicron variant). The mathematical study of an omicron variant of the corona virus is discussed. In this new Omicron model, we used idea of dividing infected compartment further into more classes i.e asymptomatic, symptomatic and Omicron infected compartment. Model is asymptotically locally stable whenever ${\mathcal{R}}_0<1$ and when ${\mathcal{R}}_0\le 1$ at disease free equilibrium the system is globally asymptotically stable. Local stability is investigated with Jacobian matrix and with Lyapunov function global stability is analyzed. Moreover basic reduction number is calculated through next generation matrix and numerical analysis will be used to verify the model with real data. We consider also the this model under fractional order derivative. We use Grunwald–Letnikov concept to establish a numerical scheme. We use nonstandard finite difference (NSFD) scheme to simulate the results. Graphical presentations are given corresponding to classical and fractional order derivative. According to our graphical results for the model with numerical parameters, the population’s risk of infection can be reduced by adhering to the WHO’s suggestions, which include keeping social distances, wearing facemasks, washing one’s hands, avoiding crowds, etc.

The dynamics of co-infection model of SARS-Cov-2 and influenza is presented in this paper. A detailed analysis is conducted on the possible effects of the influenza vaccination alone as well as the combined effect of both vaccinations on the co-infection dynamics. The two diseases' basic reproduction numbers utilizing the next-generation matrix method.Endemic equilibrium point (EEP), and disease free equilibrium points (DFEP) are calculated for deterministic model. The global stability of the model equilibrium is demonstrated using the Lyapunov function function, and the local stability is discussed for deterministic as well as stochastic. There are supplied simulations to highlights the theoretical outcomes of the model. For each infection, we compute the basic reproductive numbers $\mathbb{R}_{01} <1$, and $\mathbb{R}_{02}<1$. To determine the realistic values of the model parameters, data regarding COVID-19 instances was obtained from China National Health Commission.Sensitivity study employing the PRCC technique to look into the key variables that affect $\mathbb{R}_{01}$, or $\mathbb{R}_{02}<1$ decrease or increase. We show the numerical simulation of the model using non-standard finite difference scheme (NSFD) and stochastic numerical method. On the basis of these mentioned numerical methods, some graphical results for the model with sensitive parameters are provided.