Journal of Applied Mathematics
Volume 2013 (2013), Article ID 381286, 8 pages
Research Article

Extinction of Disease Pathogenesis in Infected Population and Its Subsequent Recovery: A Stochastic Approach

1Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University, Kolkata, West Bengal 700032, India
2Department of Mathematics, Barasat College, Kolkata 700126, India
3Department of Chemistry, Narula Institute of Technology, Kolkata 700109, India
4Agricultural and Ecological Research Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India
5Department of Stochastics, Institute of Mathematics, Budapest University of Technology and Economics, Budapest 1521, Hungary

Received 5 February 2013; Accepted 21 May 2013

Academic Editor: Xinyu Song

Copyright © 2013 Priti Kumar Roy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A stochastic mathematical model of host-pathogen interaction has been developed to estimate the time to extinction of infected population. It has been assumed in the model that the infected host does not grow or reproduce but can recover from pathogenic infection and move to add to the susceptible host population using various drugs or vaccination. Extinction of infected population in host-pathogen interaction depends significantly upon the total population. Here, we consider an extension of our previous work with the stochastic approach to predict the time to extinction of disease pathogenesis. The optimal control approach helped in designing an innovative, safe therapeutic regimen where the susceptible host population enhanced with simultaneous decrease in the infected population. By means of an optimal control theory paradigm, it has also been shown in our preceding research paper that the cost-effective combination of treatment may depend on the population size. In this research paper, we have studied an approximation which is derived in favor of quasi-stationary distribution along with the expected time to extinction for the model of host-pathogen interactions. The complete study has been roofed through the stochastic approach in context that disease pathogenesis is to be extinct and infected population are going to be recovered. Numerical simulation is also done to confirm the analysis.