# The stochastic extinction and stability conditions for a class of malaria epidemic models

## TL;DR

This paper investigates the stochastic extinction and stability of a class of malaria models with nonlinear incidence, deriving threshold conditions for disease extinction and persistence using Lyapunov and martingale techniques.

## Contribution

It introduces new stochastic threshold criteria for malaria extinction and stability, incorporating noise effects in transmission and death rates, with rigorous mathematical analysis.

## Key findings

- Malaria extinction occurs when R*_{0}<1 or E(e^{-(μ_v T_1 + μ T_2)})<1/R*_{0}
- Threshold conditions are robust to noise intensity in disease transmission
- Numerical simulations support theoretical results

## 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 $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})$, 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. %The extinction of malaria depends on $R^{*}_{0}$, and $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})$. Moreover, for either $R^{*}_{0}<1$, or $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})<\frac{1}{R^{*}_{0}}$, whenever $R^{*}_{0}\geq 1$, 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.