Delay infectivity and delay recovery SIR model

The governing equations for an SIR model with delay terms in both the infectivity and recovery rate of the disease are derived from an underlying stochastic process. By modelling the dynamics as a continuous time random walk, where individuals move between the classic SIR compartments, the delays can be introduced in a way that retains physical aspects of the model. Delays are introduced through an appropriate choice of distribution for the infectivity and recovery processes. The selected stochastic process has a one-to-one correspondence with the resulting delay SIR model. This approach ensures the physicality of the model and provides insight into the underlying dynamics of existing SIR models with infectivity delays. We show that a delay in the infectivity term arises from taking the infectivity to be related to a hazard function of the disease recovery rate, and a delay in the recovery arises from taking the corresponding survival function as a delay exponential distribution. The delays in the derived model can represent a latency period or incubation effect.