Fractional-order SIR models have become increasingly popular in the literature in recent years, however unlike the standard SIR model, they often lack a derivation from an underlying stochastic process. Here we derive a fractional-order infectivity SIR model from a stochastic process that incorporates a time-since-infection dependence on the infectivity of individuals.The fractional derivative appears in the generalised master equations of a continuous time random walk through SIR compartments, with a power-law function in the infectivity. We show that this model can also be formulated as an infection-age structured Kermack-McKendrick integro-differential SIR model. Under the appropriate limit the fractional infectivity model reduces to the standard ordinary differential equation SIR model.

Over the past several decades, there has been a proliferation of epidemiological models with ordinary derivatives replaced by fractional derivatives in an ad hoc manner. These models may be mathematically interesting, but their relevance is uncertain. Here we develop an SIR model for an epidemic, including vital dynamics, from an underlying stochastic process. We show how fractional differential operators arise naturally in these models whenever the recovery time from the disease is power-law distributed. This can provide a model for a chronic disease process where individuals who are infected for a long time are unlikely to recover. The fractional order recovery model is shown to be consistent with the Kermack-McKendrick age-structured SIR model, and it reduces to the Hethcote-Tudor integral equation SIR model. The derivation from a stochastic process is extended to discrete time, providing a stable numerical method for solving the model equations. We have carried out simulations of the fractional order recovery model showing convergence to equilibrium states. The number of infecteds in the endemic equilibrium state increases as the fractional order of the derivative tends to zero.

Abstract:The introduction of fractional-order derivatives to epidemiological compartment models, such as SIR models, has attracted much attention. When this introduction is done in an ad hoc manner, it is difficult to reconcile parameters in the resulting fractional-order equations with the dynamics of individuals. This issue is circumvented by deriving fractional-order models from an underlying stochastic process. Here, we derive a fractional-order infectivity and recovery Susceptible Infectious Recovered (SIR) model from the stochastic process of a continuous-time random walk (CTRW) that incorporates a time-since-infection dependence on both the infectivity and the recovery of the population. By considering a power-law dependence in the infectivity and recovery, fractional-order derivatives appear in the generalised master equations that govern the evolution of the SIR populations. Under the appropriate limits, this fractional-order infectivity and recovery model reduces to both the standard SIR model and the fractional recovery SIR model.

Compartment models have been used to describe the time evolution of a system undergoing reactions between populations in different compartments. The governing equations are a set of coupled ordinary differential equations. In recent years fractional order derivatives have been introduced in compartment models in an ad hoc way, replacing ordinary derivatives with fractional derivatives. This has been motivated by the utility of fractional derivatives in incorporating history effects, but the ad hoc inclusion can be problematic for flux balance. To overcome these problems we have derived fractional order compartment models from an underlying physical stochastic process. In general, our fractional compartment models differ from ad hoc fractional models and our derivation ensures that the fractional derivatives have a physical basis in our models. Some illustrative examples, drawn from epidemiology, pharmacokinetics, and in-host virus dynamics, are provided.

Many physical phenomena occur on domains that grow in time. When the timescales of the phenomena and domain growth are comparable, models must include the dynamics of the domain.