Mathematical analysis for a new nonlinear measles epidemiological system using real incidence data from Pakistan

Abstract

Modeling of infectious diseases is essential to comprehend dynamic behavior for the transmission of an epidemic. This research study consists of a newly proposed mathematical system for transmission dynamics of the measles epidemic. The measles system is based upon mass action principle wherein human population is divided into five mutually disjoint compartments: susceptible S(t)-vaccinated V(t)-exposed E(t)-infectious I(t)-recovered R(t). Using real measles cases reported from January 2019 to October 2019 in Pakistan, the system has been validated. Two unique equilibria called measles-free and endemic (measles-present) are shown to be locally asymptotically stable for basic reproductive number $R_0<1$ and $R_0>1$ , respectively. While using Lyapunov functions, the equilibria are found to be globally asymptotically stable under the former conditions on $R_0$ . However, backward bifurcation shows coexistence of stable endemic equilibrium with a stable measles-free equilibrium for $R_0<1$ . A strategy for measles control based on herd immunity is presented. The forward sensitivity indices for $R_0$ are also computed with respect to the estimated and fitted biological parameters. Finally, numerical simulations exhibit dynamical behavior of the measles system under influence of its parameters which further suggest improvement in both the vaccine efficacy and its coverage rate for substantial reduction in the measles epidemic.