Compartmental Disease Transmission Models for Smallpox

Abstract

Mathematical models are importants tool to develop emergency response plans and mitigating strategies for an anticipated epidemic or pandemic attack. With the objective of minimizing total number of deaths, total number of infected people, and total health care costs, mitigation decisions may include early response, vaccination, hospitalization, quarantine or isolation. In this study, smallpox disease transmission is analyzed with a compartmental model consisting of the following disease stages: susceptible, exposed, infectious, quarantined and recovered. Considering natural birth and death rates in a population, two models are developed to study the control and intervention of smallpox under the assumption of imperfect quarantine. First model is an ordinary differential equation (ODE) model assuming exponential distribution function and the latter is a general integral equation model with gamma distribution. Numerical results are provided and model results are compared to demonstrate the differences in the reproduction numbers, which can be described as the threshold for stability of a disease-free equilibrium related to the peak and final size of an epidemic.