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The Novel Coronavirus which emerged in India on January/30/2020 has become a catastrophe to the country on the basis of health and economy. Due to rapid variations in the transmission of COVID-19, an accurate prediction to determine the long term effects is infeasible. This paper has introduced a nonlinear mathematical model to interpret the transmission dynamics of COVID-19 infection along with providing vaccination in the precedence. To minimize the level of infection and treatment burden, the optimal control strategies are carried out by using the Pontryagin's Maximum Principle. The data validation has been done by correlating the estimated number of infectives with the real data of India for the month of March/2021. Corresponding to the model, the basic reproduction number R 0 is introduced to understand the transmission dynamics of COVID-19. To justify the significance of parameters we determined the sensitivity analysis of R 0 using the parameters value. In the numerical simulations, we concluded that reducing R 0 below unity is not sufficient enough to eradicate the COVID-19 disease and thus, it is required to increase the vaccination rate and its efficacy by motivating individuals to take precautionary measures. [ABSTRACT FROM AUTHOR]

In this paper, we have proposed a nonlinear mathematical model of different classes of individuals for coronavirus (COVID-19). The model incorporates the effect of transmission and treatment on the occurrence of new infections. For the model, the basic reproduction number (ℛ 0) has been computed. Corresponding to the threshold quantity (ℛ 0) , the stability of endemic and disease-free equilibrium (DFE) points are determined. For ℛ 0 > 1 , if the endemic equilibrium point exists, then it is locally asymptotically stable, whereas the DFE point is globally asymptotically stable for ℛ 0 < 1 which implies the eradication of the disease. The effects of various parameters on the spread of COVID-19 are discussed in the segment of sensitivity analysis. The model is numerically simulated to understand the effect of reproduction number on the transmission dynamics of the disease COVID-19. From the numerical simulations, it is concluded that if the reproduction number for the coronavirus disease is reduced below unity by decreasing the transmission rate and detecting more number of infectives, then the epidemic can be eradicated from the population. [ABSTRACT FROM AUTHOR]

In this paper, a nonlinear population model for HIV-TB co-infection has been proposed. The model is incorporated with the effect of early and late initiation of HIV treatment in co-infectives already on TB treatment, on the occurrence of Immune Reconstitution Inflammatory syndrome (IRIS). A 15-dimensional (15D) mathematical model has been developed in this study. We begin with considering constant treatment rates and thereafter, proceed to time-dependent treatment rates for co-infectives as control parameters. The basic reproduction number, a threshold quantity, corresponding to each HIV and TB sub-model has been computed in case of constant controls. With constant values of control parameters, mathematical analysis shows the existence and local stability of the disease-free equilibrium point and the endemic equilibrium point for the model. Together with time-dependent parameters, an optimal control problem is introduced and solved using Pontryagin's maximum principle with an objective to minimize the number of infectives and disease induced deaths along with the cost of treatment. Numerical simulations are performed to examine the effect of reproduction numbers on control profiles and to identify, the ideal combination of treatment strategies which provides minimum burden on a society. Numerical results imply that if both HIV and TB are endemic in the population, then in order to bring in minimum burden from the co-infection, optimal control efforts must be enforced rather than constant treatment rate. [ABSTRACT FROM AUTHOR]

A non-linear SEIR mathematical model for coronavirus disease in India has been proposed, by incorporating the saturated incidence rate on the occurrence of new infections. In the model, the threshold quantity known as the reproduction number is evaluated which determines the stability of disease-free equilibrium and the endemic equilibrium points. The disease-free equilibrium point becomes globally asymptotically stable when the corresponding reproduction number is less than unity, whereas, if it is greater than unity then the endemic equilibrium point comes into existence, which is locally asymptotically stable under certain restrictions on the parameters value in the model. The impact of various parameters on the threshold quantity is signified by the sensitivity analysis. Numerical results imply that by implementing and strictly following the prevention measures a rapid reduction in the reproduction number for COVID-19 can be observed, through which the coronavirus disease can be controlled. [ABSTRACT FROM AUTHOR]

In this paper, a HIV/AIDS and TB co-infection model has been explored, which incorporates detection and treatment for both diseases. We begin with presenting a co-infection model and then start analyzing both the HIV and the TB sub-models separately. The basic reproduction numbers corresponding to both HIV and TB are computed. Both the HIV/AIDS and the TB sub-models have been shown to exhibit two equilibrium points, namely, the disease-free equilibrium point and the unique endemic equilibrium point. For both sub-models, the disease-free equilibrium point is locally as well as globally asymptotically stable when their corresponding reproduction number is less than unity. The endemic equilibrium point for both sub-models exists, when their corresponding reproduction numbers is greater than one. We also analyze the full HIV-TB co-infection model. With the aim of minimizing infectives and the cost of applying effort towards the detection and the treatment, optimal control analysis is performed for the full model using the Pontryagin's maximum principle. Numerical simulations with different combinations of efforts are then performed to explore the effect of detection in the presence of treatment for both diseases. Numerical simulations emphasize the fact that to reduce co-infection from the population, programs to accelerate the detection of both diseases are also required along with the treatment. • This paper presents a non-linear HIV-TB co-infection model. • Local stability analysis for the model has been done. • An optimal control problem to minimize the total cost is presented. • The existence and characterization of the optimal controls has been established. [ABSTRACT FROM AUTHOR]