Dynamics of a Multigroup SIR Epidemic Model with Nonlinear Incidence and Stochastic Perturbation

We introduce stochasticity into a multigroup SIR model with nonlinear incidence. We prove that when the intensity of white noise is small, the solution of stochastic system converges weakly to a singular measure (i.e., a distribution) if ℛ0≤1 and there exists an invariant distribution which is ergodic if ℛ0>1. This is the same situation as the corresponding deterministic case. When the intensity of white noise is large, white noise controls this system. This means that the disease will extinct exponentially regardless of the magnitude of ℛ0

Global Stability of Multigroup SIRS Epidemic Model with Varying Population Sizes and Stochastic Perturbation around Equilibrium

We discuss multigroup SIRS (susceptible, infectious, and recovered) epidemic models with random perturbations. We carry out a detailed analysis on the asymptotic behavior of the stochastic model; when reproduction number ℛ0>1, we deduce the globally asymptotic stability of the endemic equilibrium by measuring the difference between the solution and the endemic equilibrium of the deterministic model in time average. Numerical methods are employed to illustrate the dynamic behavior of the model and simulate the system of equations developed. The effect of the rate of immunity loss on susceptible and recovered individuals is also analyzed in the deterministic model

Extinction and recurrence of multi-group SEIR epidemic

In this paper, we consider a class of multi-group SEIR epidemic models with stochastic perturbations. By the method of stochastic Lyapunov functions, we study their asymptotic behavior in terms of the intensity of the stochastic perturbations and the reproductive number R0R0. When the perturbations are sufficiently large, the exposed and infective components decay exponentially to zero whilst the susceptible components converge weakly to a class of explicit stationary distributions regardless of the magnitude of R0R0. An interesting result is that, if the perturbations are sufficiently small and R0≤1R0≤1, then the exposed, infective and susceptible components have similar behaviors, respectively, as in the case of large perturbations. When the perturbations are small and R0>1R0>1, we construct a new class of stochastic Lyapunov functions to show the ergodic property and the positive recurrence, and our results reveal some cycling phenomena of recurrent diseases. Computer simulations are carried out to illustrate our analytical results

Analysis of a stochastic distributed delay epidemic model with relapse and Gamma distribution kernel

In this work, we investigate a stochastic epidemic model with relapse and distributed delay. First, we prove that our model possesses and unique global positive solution. Next, by means of the Lyapunov method, we determine some sufficient criteria for the extinction of the disease and its persistence. In addition, we establish the existence of a unique stationary distribution to our model. Finally, we provide some numerical simulations for the stochastic model to assist and show the applicability and efficiency of our results.Ministerio de Ciencia, Innovación y Universidades (MICINN). EspañaEuropean Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER

On the dynamics of a class of multi-group models for vector-borne diseases

The resurgence of vector-borne diseases is an increasing public health concern, and there is a need for a better understanding of their dynamics. For a number of diseases, e.g. dengue and chikungunya, this resurgence occurs mostly in urban environments, which are naturally very heterogeneous, particularly due to population circulation. In this scenario, there is an increasing interest in both multi-patch and multi-group models for such diseases. In this work, we study the dynamics of a vector borne disease within a class of multi-group models that extends the classical Bailey-Dietz model. This class includes many of the proposed models in the literature, and it can accommodate various functional forms of the infection force. For such models, the vector-host/host-vector contact network topology gives rise to a bipartite graph which has different properties from the ones usually found in directly transmitted diseases. Under the assumption that the contact network is strongly connected, we can define the basic reproductive number $\mathcal{R}_0$ and show that this system has only two equilibria: the so called disease free equilibrium (DFE); and a unique interior equilibrium---usually termed the endemic equilibrium (EE)---that exists if, and only if, $\mathcal{R}_0>1$. We also show that, if $\mathcal{R}_0\leq1$, then the DFE equilibrium is globally asymptotically stable, while when $\mathcal{R}_0>1$, we have that the EE is globally asymptotically stable

Dynamics of a Stochastic SIS Epidemic Model with Saturated Incidence

We introduce stochasticity into the SIS model with saturated incidence. The existence and uniqueness of the positive solution are proved by employing the Lyapunov analysis method. Then, we carry out a detailed analysis on both its almost sure exponential stability and its pth moment exponential stability, which indicates that the pth moment exponential stability implies the almost sure exponential stability. Additionally, the results show that the conditions for the disease to become extinct are much weaker than those in the corresponding deterministic model. The conditions for the persistence in the mean and the existence of a stationary distribution are also established. Finally, we derive the expressions for the mean and variance of the stationary distribution. Compared with the corresponding deterministic model, the threshold value for the disease to die out is affected by the half saturation constant. That is, increasing the saturation effect can reduce the disease transmission. Computer simulations are presented to illustrate our theoretical results

Matching Theory and Evidence on Covid-19 using a Stochastic Network SIR Model

This paper develops an individual-based stochastic network SIR model for the empirical analysis of the Covid-19 pandemic. It derives moment conditions for the number of infected and active cases for single as well as multigroup epidemic models. These moment conditions are used to investigate identification and estimation of recovery and transmission rates. The paper then proposes simple moment-based rolling estimates and shows them to be fairly robust to the well-known under-reporting of infected cases. Empirical evidence on six European countries match the simulated outcomes, once the under-reporting of infected cases is addressed. It is estimated that the number of reported cases could be between 3 to 9 times lower than the actual numbers. Counterfactual analysis using calibrated models for Germany and UK show that early intervention in managing the infection is critical in bringing down the reproduction numbers below unity in a timely manner

A comparison between random and stochastic modeling for a SIR model

In this article, a random and a stochastic version of a SIR nonautonomous model previously introduced in P. E. Kloeden and V. S. Kozyakin, The dynamics of epidemiological systems with nonautonomous and random coefficients, MESA: Mathematics in Engineering, Science and Aerospace, vol. 2, no. 2 (2011).is considered. In particular, the existence of a random attractor is proved for the random model and the persistence of the disease is analyzed as well. In the stochastic case, we consider some environmental effect on the model, in fact, we assume that one of the coefficients of the system is affected by some stochastic perturbation, and analyze the asymptotic behavior of the solutions. The paper is concluded with a comparison between the two different modeling strategies.Fondo Europeo de Desarrollo RegionalMinisterio de Economía y CompetitividadJunta de Andalucí