Mathematical analysis to control the spread of Ebola virus epidemic through voluntary vaccination
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Mathematical analysis to control the spread of Ebola virus epidemic through voluntary vaccination Waheed Ahmad1,a , Muhammad Rafiq2 , Mujahid Abbas1 1 Department of Mathematics, Government College University, Lahore, Pakistan 2 Faculty of Engineering, University of Central Punjab, Lahore, Pakistan
Received: 6 July 2020 / Accepted: 8 August 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 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 R0 we obtain through our model is strictly less than one. Moreover, for R0 > 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 R0 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 R0 is executed numerically.
1 Introduction Ebola virus disease if left untreated has become a serious life threat to humans. On an average, the case fatality rate of EVD is around 50% [1]. However, in the recent past, the case fatality rates varied from 25% (Uganda 2007) to 90% (Congo 2003) as reported by World Health
a e-mail: [email protected] (corresponding author)
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