Hopf bifurcation and stability switches in an infectious disease model with incubation delay, information, and saturated treatment

Abstract

In this paper, a delay SIR model with a nonlinear incidence function and saturated treatment function is proposed and analyzed. The non-monotonous incidence function accounts for the behavior response of susceptible individuals towards the infective density in the population due to the impact of information about the disease. We obtain a threshold for disease persistence in the population. The system shows the possibility of multiple endemic equilibria. We analytically study the stability of endemic equilibrium, when it is unique. This stability of the unique endemic equilibrium depends on the delay parameter and is found to be locally asymptotically stable conditionally along with the existence of Hopf bifurcation. The direction and stability of Hopf-bifurcation are established analytically. Further, numerical results are provided in support of analytical findings. Numerically, we observe stability switch of unique endemic equilibrium and existence of endemic bubble due to change in delay parameter. We further exhibited the impact of delay when multiple equilibria are present in the system. In this case, we observe that bi-stable endemic equilibria in absence of delay can be transformed to oscillatory bi-stable limit cycles in presence of delay (when the delay is sufficiently large). Here the delay causes the change in stability of both the equilibria by generating two local Hopf bifurcations around both the equilibria which were stable in absence of delay. We also investigated the effect of information and disease transmission rate on disease persistence along with the incubation delay length. Thus, we conclude that the nonlinearity and delays in the model system account for rich and complex dynamics.

Appendix

1.1 Basic reproduction number

Following [38], we obtain the next generation matrix at the disease free equilibrium point $E_1=(K,0,0)$ for the model system (1) which is given by,

$$\begin{aligned} FV^{-1}=\begin{pmatrix} \beta K \end{pmatrix}\begin{pmatrix} \frac{1}{d+\delta +\alpha } \end{pmatrix}=\frac{\beta K}{d+\delta +\alpha }.\end{aligned}$$

Here note that I(t) is the single infective compartment. Now, the basic reproduction number is given by spectral radius of $FV^{-1}$ which is $\frac{\beta K}{d+\delta +\alpha }$. Therefore,

$$\begin{aligned} \mathcal {R}_0=\frac{\beta K}{d+\delta +\alpha }. \end{aligned}$$