In order to model an epidemic, different approaches can be adopted. Mainly, the deterministic approach and the stochastic one. Recently, a huge amount of literature has been published using the two approaches. The aim of this paper is to illustrate the usual framework used for commonly adopted compartmental models in epidemiology and introduce variant analytic and numerical tools that interfere on each one of those models, as well as the general related types of existing, ongoing and future possible contributions.
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.
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.
The focal point of this paper is to further enhance the existing stochastic epidemic models by incorporating several new disease characteristics, such as the validation time of the vaccination procedure, the stages of vaccine required to gain a long-period immunity together with the time separating each stage, the deaths linked to the vaccine, and finally, the sudden environmental noise which is exhibited by sociocultural changes, such as antivaccination movements. To incorporate all the aforementioned characteristics, we extend the standard Susceptible-Vaccinated-Infected-Recovered (SVIR) epidemic model to a new mathematical model, which is governed by a system of coupled stochastic delay differential equations, in which the disease transmission rates are driven by Gaussian noise and Lévy-type jump stochastic process. First, under suitable conditions on the jump intensities, we address the mathematical well-posedness and biological feasibility of the model, by virtue of the Lyapunov method and the stopping-time technique. Then, by choosing an adequate positively invariant set for the considered model, we establish sufficient conditions guaranteeing the disease extinction and persistence. Lastly, to support the theoretical results, we provide the outcome of several numerical simulations which, together with our conducted analysis, indicate that the spread of the disease can be majorly altered by all the new considered characteristics.
This paper focuses on the study of an optimal control problem for a new spatio-temporal SIR epidemic model with nonlinear density dependent diffusion terms and a class of nonlinear incidence functions. We consider two types of control variables, vaccination for the susceptible and treatment for the infected. For fixed controls, by means of Schauder fixed point theorem, we prove that the proposed model admits a weak biologically feasible solution, the uniqueness of the latter is also investigated. Furthermore, using the state and adjoint problems, first order necessary optimal conditions are obtained. Finally, numerical simulations are carried out for particular diffusion terms incorporating the heard mentality of individuals, when it comes to the spatial movement, and for particular incidence functions, as well as by varying the parameters of the objective functional, to illustrate the possible optimal control strategies and their effect on the studied population.
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