Time delayed SIR model under the effect of pollution: mathematical model and analysis

  • Naina Arya AIAS, Amity University, Uttar Pradesh, Noida
  • Sumit Kaur Bhatia AIAS, Amity University Uttar Pradesh
  • Amrita Kumar AIAS, Amity University, Uttar Pradesh, Noida
  • Puneet Sharma Indian Institute of Technology, Jodhpur.


This paper deals with the time delayed SIR model with saturated treatment function under the effect of pollution. As the pollution rate has hiked in this era and the presence of toxicants are found in various resources, therefore it is important to study their effects on disease dynamic of population. Hence, we consider the populace affected by the pollution. As the effect of treatment is not immediate, rather involves time lag to show its effect, therefore we consider time that is taken by infective to recover as the time delay. Existence of equilibrium points and the boundedness of the system has been obtained. Stability analysis using reproduction number R0 has been done. Global stability of the endemic equilibrium point is established using Lyapunov Lasalle’s theorem. Considering time delay as the critical parameter, existence of hopf bifurcation has been proved. Also, taking into account the normal form theory, direction and stability of Hopf Bifurcation has been obtained. Numerical simulations are done in support of results obtained analytically. It is observed that as pollution increases, infective take more time to recover and we also note that recovered individuals decrease with increase in the pollution. Therefore, it important to reduce pollution and provide timely treatment to the patients. Thus, we prove that, pollution and time delay play a critical role in shaping the dynamics of the system.