Dynamics of an Infectious Disease in the Presence of Saturated Medical Treatment of Holling Type Iii and Self-Protection

Abstract

A nonlinear SEIR model is formulated and analyzed. This model accounts for three important interventions — the saturated treatment on infective individuals, the screening on the exposed individuals and the information induced self-protection on susceptible individuals. Existence and stability of equilibria are discussed. A sensitivity analysis for the model parameters is performed and we identified the parameters which are more sensitive to the model system. The sensitivity analysis is further followed up with the two parameters heat plot that determines the regions for the parametric values in which the system is either stable or unstable. Further, an optimal control problem is formulated by considering screening and treatment as control variables and corresponding cost functional is constructed. Using Pontryagin’s Maximum Principle, paths of optimal controls are obtained analytically. A comparative study is conducted numerically to explore and analyze analytical results. We note that in absence of treatment, screening policy may be a cost-effective choice to keep a tab on the disease. However, comprehensive effect of both screening and treatment has a huge impact, which is highly effective and least expensive. It is also noted that treatment is effective for mild epidemic whereas screening has a significant effect on the disease burden while epidemic is severe. For a range of basic reproduction number, effect of self-protection and saturation in treatment is also explored numerically.

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“…It has been noted by many authors that the saturation in medical treatment leads to complex behavior of the diseases such as existence of multiple endemic equilibria, backward bifurcation, Hopf bifurcation, etc. [12,29,41,55]. In [55] backward bifurcation and global dynamics of an epidemic model with saturated incidence rate and saturated treatment function is studied.…”