Host Pathogen Interactions: Insight of Delay Response Recovery and Optimal Control in Disease Pathogenesis

Research in the infection and recovery, based on mathematical understanding is aimed at developing how the immune system protects against infectious diseases, with a major focus on host-pathogen interactions [1]. In recent years greatest challenges in front of humanity is learning how the immune response protects against variety of breathing and emerging infectious disease. A lot of theoretical and experimental investigations of natural selections in host-pathogen system been explored and it has been established that vaccination is the most cost effective public health tool in human history [2]. In this research articles we attempt to offers an opportunity to examine the mathematical relationship that forming a model between host and pathogen as in biology and manifests the time rate of change of these species for better understanding of the recovery of the disease. On that outlook a recovery rate of the infected host to that of susceptible host is incorporated in our proposed mathematical model. Then we introduce a discrete time delay to the model to describe the time requirement for recovery from infected host to susceptible host by using various drugs or vaccinations. Again we use optimal control that is characterized in terms of the optimality of the system, to achieve improved quality of treatment. The model is analyzed theoretically and numerically. Numerical solution of the model is in conformity with those obtained theoretically. An effort is made to relate the model to prototype biological system by exploring a probable disease free parametric region. We also studied the effect of time delay on the stability of the infected equilibrium and optimal control process which can reduce the infected host populations.