The stochastic extinction and stability conditions for nonlinear malaria epidemics

Abstract

The stochastic extinction and stability in the mean of a family of SEIRS malaria models with a general nonlinear incidence rate is presented. The dynamics is driven by independent white noise processes from the disease transmission and natural death rates. The basic reproduction number R * 0 , the expected survival probability of the plasmodium E(e −(µv T 1 +µT 2 ) ), and other threshold values are calculated. A sample Lyapunov exponential analysis for the system is utilized to obtain extinction results. Moreover, the rate of extinction of malaria is estimated, and innovative local Martingale and Lyapunov functional techniques are applied to establish the strong persistence, and asymptotic stability in the mean of the malaria-free steady population. Moreover, for either R * 0 < 1, or E(e −(µv T 1 +µT 2 ) ) < 1 R * 0 , whenever R * 0 ≥ 1, respectively, extinction of malaria occurs. Furthermore, the robustness of these threshold conditions to the intensity of noise from the disease transmission rate is exhibited. Numerical simulation results are presented.Recently, Wanduku[51] presented and studied the following novel family of SEIRS epi-

On the almost sure convergence of a stochastic process in a class of nonlinear multi-population behavioral models for HIV/AIDS with delayed ART treatment

“…Applying standard techniques in [31,33,35], it is proven that there exists a unique positive SSP fYðtÞ, t ! t 0 g that satisfies (4.3)-(4.9).…”

“…Proof. The proof of the existence and uniqueness of a positive solution for the system (4.3)-(4.9) follows the similar technique in [31,33,35], and is left out to maximize space. In the following the existence of the positive self-invariant space B R nþ6 ð0, rÞ is shown.…”

“…The properties of H j are given in Assumption 2.1. Using ideas in [32,33,35] we adopt assumptions for the nonlinear function H j , j 2 Ið0, nÞ:…”

“…That is, the proportions per unit time e j ¼ 1 À e j and e j , j 2 Ið0, nÞ are random variables, and likewise the natural death rates l k , k 2 S j , I, T, A, R È É , j 2 Ið0, nÞ are random variables per unit time. Denote these random variables, respectively, by e j ,ẽ j , j 2 Ið0, nÞ and l k , k 2 S j , I, T, A, R È É , j 2 Ið0, nÞ: We represent the environmental variabilities as independent white noise processes applying similar techniques in the earlier studies [33,35].…”

On the Approximated Reachability of a Class of Time-Varying Nonlinear Dynamic Systems Based on Their Linearized Behavior about the Equilibria: Applications to Epidemic Models

“…Such terms deviate the solution trajectory from the linear behavior about the equilibrium points. See, for instance, [5][6][7][8][9][10][11][12][13][14] and also [16][17][18][19][20][21] and some of the references therein. Example 1.…”