Delay Infectivity and Delay Recovery SIR model

Abstract

We have derived the governing equations for an SIR model with delay terms in both the infectivity and recovery of the disease. The equations are derived by modelling the dynamics as a continuous time random walk, where individuals move between the classic SIR compartments. With an appropriate choice of distributions for the infectivity and recovery processes delay terms are introduced into the governing equations in a manner that ensures the physicality of the model. This provides novel insight into the underlying dynamics of an SIR model with time delays. The SIR model with delay infectivity and recovery allows for a more diverse range of dynamical behaviours. The model accounts for an incubation effect without the need to introduce new compartments.

Figures (6)

• Figure 1: Plot of the delay exponential function, Eq. (\ref{['def_dexp']}). The green, orange and blue curves correspond to $\tau_2=e^{-1}, e^{-2}$ and $e^{-3}$, respectively, with $\mu=1$.
• Figure 2: Plot of the infectivity rate, $\rho(t)$, with $\mu=1$ and $\tau_2=e^{-1}$. The red, orange and green curves correspond to $\tau_1= 0.2, e^{-1}$ and $0.5$, respectively.
• Figure 3: An SIR model with constant vital dynamics and a time-delay on the infectivity and recovery terms.
• Figure 4: Plot of the Infective compartment when $\tau_1$ ranges from $0.5$ and $2.5$, with $\tau_2=0.1$ and $\mu=10e^{-1}$. The birth rate is $\lambda=0.5$, the death rate is $\gamma=0.001$, the infectivity rate is $\omega=0.02$, with initial values $S(0)=498$, $I(0)=2$ and $R(0)=0$.
• Figure 5: Plot of the Infective compartment with $\tau_1=1$ and $\tau_2=0.1$, with values for $\mu$ between $0.6$ and $3.6$. The birth rate is $\lambda=0.5$, the death rate is $\gamma=0.001$, the infectivity rate is $\omega=0.02$, with initial values $S(0)=498$, $I(0)=2$ and $R(0)=0$.