Global stability and control strategies of a SIQRS epidemic model with time delay

The existing research models of SIQRS are often defined by linear incidence, ignoring the impact of lagging public awareness during outbreaks of infectious diseases. Their research results often deviate from the actual situation. For decision-makers, the delay of quarantine measures according to its formulation is liable to bring losses. Given this, we establish a new class of SIQRS models based on the nonlinear incidence and focus on the effect of time delay on the propagation characteristics of the system. Firstly, we calculate the basic reproduction number of the transmission dynamics model to obtain the expression of the transmission threshold. Secondly, by constructing suitable Lyapunov functions, we prove the stability of the disease-free equilibrium point and endemic equilibrium point of the system. Thirdly, we add model immune parameters to study three different immunization strategies, namely, uniform immunization, targeted immunization, and acquaintance immunization. Studies show that compared with uniform immunization, targeted and acquaintance immunization have better control effects. When the immunization ratio increases, the number of infected people decreases, and immune factors can control the spread of the disease. Meanwhile, we calculate the optimal quarantine level value u * ≈ 0.56 to solve the problem of financial costs in quarantine measures to control infectious diseases. Finally, we verify conclusions of the theoretical research by numerical simulation and further discover that when an infection is in the high-density outbreak, the time delay will lead to a slower convergence rate of the disease. And with the increase of time delay, the control cost J u * will also increase gradually.