# A study of the persistence and extinction of malaria in a family of epidemic models

## TL;DR

This paper analyzes conditions for the persistence or extinction of malaria in a family of complex SEIRS models with delays, providing criteria based on the basic reproduction number and survival probabilities, supported by numerical simulations.

## Contribution

It introduces new analytical techniques to determine disease persistence or extinction in delayed malaria models with nonlinear incidence rates.

## Key findings

- Extinction occurs if $R^{*}_{0}<1$ or if survival probability is low despite $R^{*}_{0}1$.
- Persistence is guaranteed when $R^{*}_{0}1$ and survival probability exceeds a threshold.
- Numerical simulations illustrate the impact of incidence and control response on malaria persistence.

## Abstract

This paper investigates the deterministic extinction and permanence of a family of SEIRS malaria models with multiple random delays, and with a general nonlinear incidence rate. The conditions for the extinction and permanence of the disease are presented. For the persistence of disease, improved analytical techniques are employed to obtain eventual lower bounds for the states of the system, while extinction results are obtained via analyzing the Lyapunov exponent. The basic reproduction number $R^{*}_{0}$ is calculated, and for $R^{*}_{0}<1$, extinction of disease occurs, while for $R^{*}_{0}\geq 1$, the disease persists. But if $R^{*}_{0}\geq 1$, and the expected survival probability of the plasmodium $E(e^{-(\mu_{v}T_{1}+\mu T_{2})})<\frac{1}{R^{*}_{0}}$, then extinction occurs. Results are interpreted, and numerical simulation examples are presented to (a) compare and evaluate the impacts of the intensity of the incidence rate, and the intensity of response to malaria control, on the persistence of malaria in the human population, and (2) to justify the asymptotic stability of the disease-free equilibrium.