Abstract:
Emergence of infectious diseases has renewed research interests in disease transmission modelling. Susceptible-Exposed-Infected-Recovered (S-E-I-R) model has been used for such studies. A major assumption of infectious disease modelling using S-E-I-R is that the population is closed to migration. However, this was violated with the outbreak of Ebola virus of 2014 that spread across national boundaries. This study was therefore designed to formulate a modified S-E-I-R model which incorporates migration and to ascertain its effects on control of the spread of infectious diseases during outbreaks.
Migration rate was introduced into the susceptible population of S-E-I-R non linear differential equations to model disease transmission. The equilibrium points of the modified model and basic reproduction number were investigated using the next generation matrix. The local stability was analysed and global stability of disease free equilibrium was conducted by applying the Lyapunov function. Furthermore, the sensitivity analysis of the parameters was studied to determine their sensitivity to reproduction number by finding the derivative of each parameter with respect to reproduction number. Consequently, optimal control of the model was considered using Pontryagin maximum principle to determine the best control strategy to stem out the effect of the disease. Effect of environmental noise in the model was also studied by applying the stochastic differential equation. The current and the modifiedS-E-I-R(with migration) models were both demonstrated on numerical simulation and on 2014 Ebola Virus outbreak data in West Africa retrieved from the WHO website to analyse the effect of migration on the disease transmission.
The equilibrium points of the model were and S=((μ+ﻁ+δ)(ﻁ+e+μ))/aδ, while the basic reproduction number was, R_0 = aΛδ/(μ(μ+ﻁ+δ)(ﻁ+e+μ)) where S = susceptible,
E= Exposed, μ = natural death rate , Ʌ= migration rate, a = transmission rate,
ﻁ= recovery rate, e = disease induced death rate and δ = progression into infected. The parameter estimates gave a = 0.000025, b = 0.48941, c = 1.963907, δ = 0.0498,
µ = 0.002165, Ʌ =1034, e = 0.05019. The system was asymptotically stable with R0 < 1. Migration and disease transmission rates were most sensitive. The reproduction number
(2. 027),revealed persistence of the disease over model without migration (1.88). The 95% confidence interval of 1.9399 R0 2.0346 accommodated the value of R0.
The formulated model predicted persistence of the infectious diseases as a result of migration into susceptible population. In view of this development, stringent migration controls should be deployed during infectious diseases outbreaks to enable early containment.
