Complex Ebola Virus Disease fractional order model in the Caputo sense with Laplace Adomian Decomposition Method model solution

Abstract

Ebola Virus Disease is a life-threatening disease that is transmitted between humans and animals. The aim of this study is to develop a fractional order Ebola Virus Disease model for the dynamics of Ebola considering the control measures;quarantine,isolation,treatment, vaccine and protection (condom use) with a view to study the effects of these control measures on the spread of Ebola in a given population of humans and animals in the Caputo Sense. The fractional order Ebola Virus Disease model in the Caputo sense to control the spread of the disease with eighteen compartments is considered in this paper.This model considered vaccination, condom use, quarantine, use of treatment drugs and isolation as control measures combined together. We analyzed the model where the validity of the model was proven by establishing a region of invariance and the positivity of solutions. We also established the model disease-free equilibrium (DFE)and endemic equilibrium (EE). The reproduction number (a threshold value) (B) was obtained using the next generation matrix. The result of the study produced an expression for B, where if the reproduction number B, is less than 1 then the model DFE point is stable, so the Ebola virus disease dies out. If the reproduction number B is greater than 1, then the model DFE point is unstable, so the Ebola Virus disease persists in the populations.We also analysed the local stability of the DFE point and found out it will be stable when the reproduction number B < 1. The novel application of the Laplace–Adomian decomposition method to the model obtained a result of the complex fractional model in an infinite series form which converges further to its exact value. Comparing the solutions of the fractional Ebola virus disease model with that of the classical case using simulation plots we found out that the case of fractional order has more degree of freedom in such a way that can be varied as the fraction (α) could be varied to get different results. We proved convergence of the Laplace Adomian Decomposition method using the Cauchy-Kovalevskaya theorem for differential
equations with analytic vector fields and then obtained a new result on the convergence rate of the Laplace Adomian Decomposition Method.