In this work, we investigate the stability of an SIR epidemic model with a generalized nonlinear incidence rate and distributed delay. The model also includes vaccination term and general treatment function, which are the two principal control measurements to reduce the disease burden. Using the Lyapunov functions, we show that the disease-free equilibrium state is globally asymptotically stable if R 0 ≤ 1, where R 0 is the basic reproduction number. On the other hand, the disease-endemic equilibrium is globally asymptotically stable when R 0 > 1. For a specific type of treatment and incidence functions, our analysis shows the success of the vaccination strategy, as well as the treatment depends on the initial size of the susceptible population. Moreover, we discuss, numerically, the behavior of the basic reproduction number with respect to vaccination and treatment parameters.
MSC: 34D03; 92D30
We investigate the global behaviour of a SIRI epidemic model with distributed delay and relapse. From the theory of functional differential equations with delay, we prove that the solution of the system is unique, bounded, and positive, for all time. The basic reproduction number R 0 for the model is computed. By means of the direct Lyapunov method and LaSalle invariance principle, we prove that the disease free equilibrium is globally asymptotically stable when R 0 < 1. Moreover, we show that there is a unique endemic equilibrium, which is globally asymptotically stable, when R 0 > 1.
In this paper, we analyze the dynamics of a new proposed stochastic non-autonomous SVIR model, with an emphasis on multiple stages of vaccination, due to the vaccine ineffectiveness. The parameters of the model are allowed to depend on time, to incorporate the seasonal variation. Furthermore, the vaccinated population is divided into three subpopulations, each one representing a different stage. For the proposed model, we prove the mathematical and biological well-posedness. That is, the existence of a unique global almost surely positive solution. Moreover, we establish conditions under which the disease vanishes or persists. Furthermore, based on stochastic stability theory and by constructing a suitable new Lyapunov function, we provide a condition under which the model admits a non-trivial periodic solution. The established theoretical results along with the performed numerical simulations exhibit the effect of the different stages of vaccination along with the stochastic Gaussian noise on the dynamics of the studied population.
This paper is devoted to the study of an optimal control problem for a generalized multi-group reaction-diffusion SIR epidemic model, with heterogeneous nonlinear incidence rates. The proposed model incorporates a wide range of spatiotemporal epidemic models. Primarily, the ones used to describe the propagation of zoonotic and sexually transmitted diseases, as well as epidemics that propagate disparately within populations. For the aforementioned diseases, dividing the susceptible, infected and recovered populations into several subpopulations is necessary in order to capture all the possible ways of the disease transmission. This makes the problem of finding the possible optimal control strategies and the division of the available control resources complicated. To address this problem mathematically, for each subpopulation, we introduce two types of control variables, namely vaccination for the susceptible and treatment for the infected. The existence and uniqueness of a biologically feasible solution to the proposed model, for fixed controls, is derived by means of a truncation technique and a semigroup approach. Moreover, first-order necessary optimality conditions for the introduced optimal control problem are obtained using the adjoint state method. Finally, numerical simulations are performed for a two-group epidemic model with particular incidence rates and by considering three cases in the maximal control resources allowed for each subpopulation.
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