Study of an SIQR model with optimal control techniques: A mathematical approach

Abstract

In this article, a SIQR deterministic model with a non-linear incidence function for an infectious disease is formulated and analyzed. The well-posedness of the model is ensured with the help of the non-negativity and boundedness of the solution of the system. The basic reproduction number (R0) of the model is found. The sensitivity analysis of the basic reproduction number is discussed. The local and global stability of different equilibria of the model are discussed in terms of basic reproduction number. We use the center manifold theory to analyze the transcritical bifurcation exhibited by the system at R0=1 and found that the disease does not vanish even if R0<1 due to the occurrence of backward bifurcation. Furthermore, this model is extended to the optimal control model and is analyzed to determine the effect of screening and treatment as controls on the population dynamics through Pontryagin’s Maximum Principle. Moreover, we performed numerical analysis to illustrate our theoretical findings. We aim at determining the best intervention strategy to control the infectious diseases among screening, treatment and quarantine. We conclude that only screening is economically effective as compared to only treatment but the combined effort of screening and treatment is most effective and less expensive than single implementation of any control measure. Our study reveals that both the control interventions not only reduce the severity of disease burden but also minimize the economic burden to control the disease.

Keywords

• Basic reproduction number
• Sensitivity analysis
• Saturated incidence rate
• Screening
• Quarantine
• Optimal control