A Fractional Order Recovery SIR Model from a Stochastic Process

Abstract

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.

A Fractional-Order Infectivity and Recovery SIR Model

“…In a similar fashion, we obtain the fractional recovery SIR model [10] by taking the limit β → 1 whilst leaving 0 < α ≤ 1. Making use of the limit in Equation (42) and the functional form of ρ(t) from Equation (47), we obtain…”

“…Since Kermack and McKendrick, the SIR model has become widely used for modelling a range of diseases and has been extended to allow for re-infection, latent infections and the interaction of species [4,5]. More recently, there has been increased interest in the extension of SIR models through the incorporation of fractional derivatives [6][7][8][9][10][11][12].…”

“…For the standard SIR model, these probabilities are related to exponential waiting-time densities. The model can be constructed as a directed continuous-time random walk (CTRW) through the SIR compartments [10,11,13].…”

“…The classic SIR model cannot accomodate this; however, the age-structured models of Kermack and McKendrick [2,3] can model such diseases. Fractional time derivatives, which include integrals over the function's history, can also be used to incorporate the system's history [6][7][8][9][10][11]. The generalisation of an integer-order derivative to a fractional derivative is not unique, with typical examples being the Caputo derivative and Riemann-Liouville derivative [15].…”

“…Fractional derivatives have recently been included in compartment models in a physically consistent way by modelling the dynamics of transitions between compartments as a stochastic process, a CTRW [10,11,13]. The parameters in the resulting fractional-order equations are well-defined and consistent with the dynamics of the individuals in the population.…”