Eur. Phys. J. Plus (2020) 135: 775
https://doi.org/10.1140/epjp/s13360-020-00683-3

Regular Article

Mathematical analysis to control the spread of Ebola virus epidemic through voluntary vaccination

¹ Department of Mathematics, Government College University, Lahore, Pakistan
² Faculty of Engineering, University of Central Punjab, Lahore, Pakistan

^a call4waheedahmad@gmail.com

Received: 6 July 2020
Accepted: 8 August 2020
Published online: 6 October 2020

Abstract

Ebola virus disease (EVD) is one of the deadliest viral infections that has caused a serious global health problem in the known human history. It was first transmitted to humans through domestic and wild animals, and then, the spread was through direct and indirect contacts among individuals. To control this spread is one of the most challenging aspects of epidemic studies carried out so far. The aim of this paper is to propose a transmission model called Susceptible-Vaccinated-Exposed-Infected-Recovered (SVEIR) by incorporating an additional class of vaccinated individuals. We investigate the rate of spread of EVD in population when the voluntary vaccination is introduced at the level of its susceptibility. Our proposed model helps in better understanding of the dynamical behavior of EVD and explains its stability pattern. We present two main equilibrium points and their stability analysis. It is proven that disease-free equilibrium (DFE) $F^0$ is both locally and globally stable when the value of threshold parameter ${\mathcal{R}}_0$ we obtain through our model is strictly less than one. Moreover, for ${\mathcal{R}}_0>1$, $F^0$ is not stable, and the endemic equilibrium (EE) $F^1$ is locally and globally stable. Hence, EVD spreads uniformly among individuals. We also study the effect of threshold parameter ${\mathcal{R}}_0$ at different vaccination coverage levels to validate our conclusions in this paper. The theory of Lyapunov functions is employed to study global stabilities at both levels. We use Runge–Kutta method of order 4 (RK4) and Non-Standard Finite Difference (NSFD) scheme for the proposed model to confirm our obtained theoretical results through numerical simulations. Furthermore, the discretized SVEIR model obtained by applying NSFD scheme is dynamically consistent with the continuous model for any step size h thus used. A quantitative analysis of an epidemic model for different vaccination coverage levels is also presented. It is concluded that eradication of Ebola virus is possible if a human population adopts voluntary vaccination coupled with concentrated efforts of public education at various coverage levels. The effect of vaccination coverage levels on threshold parameter ${\mathcal{R}}_0$ is executed numerically.