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.
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This work was supported by the Australian Research Council (DP130100595, DP140101193).
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Appendices
Appendix 1: Derivation of the Kermack–McKendrick Model for Infecteds in a Age-Structured System
Let i(a, t) denote the number of individuals who are infected at time t and who have been infected for time a. The total number of infected individuals at time t is given by
where
is the product of the probability of surviving death, \(\theta (t,t')\), and of ‘surviving’ the transition out of the next stage, \(\phi (t-t')\). If we differentiate Eq. (127) with respect to t using Leibniz rule, we have
We also have
But
so that if we add Eqs. (129) and (130) then we obtain
We can expand the partial derivative of \(\varPhi (t,t')\) using the product rule in Eq. (128) with
and
to obtain
Using Eq. (128) and then Eq. (127) in the second term on the RHS of Eq. (135), we now have
We now define
and use Eq. (128) in the first term on the RHS of Eq. (136) to obtain
In the first integral on the RHS of Eq. (138), it is convenient to make a change in variables \(t'=t-a\) with \(\hbox {d}t'=-\hbox {d}a\); then, we have
Finally, we note that
so that Eq. (139) can be written as
This is consistent with Eq. (27) for the change in infectives in the age-structured Kermack McKendrick model.
Appendix 2: Limits to Continuous Time
The discrete time fractional recovery SIR model can be shown to limit to the frSIR model by identifying \(t=n\Delta t\) and taking the limit \(\Delta t \rightarrow 0\) with \(r/\Delta t^{\alpha }\) finite. The continuous time equations can be obtained from the discrete time equations using Z star transform methods. The Z star transform of Y(n) is given by
It follows that
where we have introduced \(\tilde{Y}(t)\) as a function defined over a continuous variable t. We can now take the inverse Laplace transform from s to t
where \(\delta (t)\) is the Dirac delta function. Here, and in the following, we use the notation \( \mathcal {L}^{-1}_s\left[ Y(s)\Bigg \vert t\right] \) to denote the inverse Laplace transform from s to t and we use the notation \( \mathcal {L}_t\left[ Y(t)\Bigg \vert s\right] \) to denote the Laplace transform from t to s.
It is useful to define the function
In a similar fashion, we have
Note that, with \(t'=n\Delta t\), in Eq. (146), we have
This formally identifies
provided that the limit exists.
We further note the product rule
which equates to \(\tilde{X}(t)\tilde{Y}(t)\) in each case, with \(t'=n\Delta t\), provided that both \(\tilde{X}(t)\) and \(\tilde{Y}(t)\) exist.
We now take the inverse Laplace transform of the Z star transform of Eq. (105) and multiply by \(\frac{\Delta t}{\Delta t}\) to write
We now take the continuous time limit of Eq. (155) using \(t'=n\Delta t\) and the product rule in Eq. (154) to obtain
where we have defined continuous time rate parameters
and
Equation (156) simplifies further to
The further reduction in this equation depends on the specific form of the memory kernel \(\kappa (n)\). In the case of a jump at each time step, the memory kernel is
In this case, we can perform the sum over k explicitly in Eq. (159) to arrive at
In order for the continuous time limit of the above equation to exist, we define
Note that r is a free parameter in the range [0, 1], and hence, \(\mu \) is only well defined in this limit if we take r to be a function of \(\Delta t\). With this definition of \(\mu \), we can now perform the limit \(\delta t\rightarrow 0\) to obtain the continuous time equation
This further simplifies to
Equation (164) recovers the corresponding equation in the classic SIR model.
We now consider the continuous time limit of Eq. (159) with the Sibuya memory kernel, given by Eq. (104). First, we simplify the double sum in Eq. (159) using Laplace transforms and Z star transforms as follows:
The last line in the above follows from the convolution theorem for Z star transforms.
To proceed further, we use the Z transform of the Sibuya memory kernel in Eq. (114) to write
The result in Eq. (165) can now be written as
where
Finally, we substitute the result of Eq. (168) into Eq. (159) and use the known result (Podlubny 1999)
to invert the Laplace transform and obtain
Equation (171) recovers the continuous time frSIR model equation.
Note that in order for the continuous time limit of the frSIR model equation to exist, we defined
which requires \(r\in [0,1]\) to be a function of \(\Delta t\). This is important for numerical simulations based on this DTRW method where we take \(r=\mu \Delta t^\alpha \) and then the requirement that \(r\in [0,1]\) places restrictions on \(\Delta t\) for given \(\alpha \) and \(\mu \).
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Angstmann, C.N., Henry, B.I. & McGann, A.V. A Fractional Order Recovery SIR Model from a Stochastic Process. Bull Math Biol 78, 468–499 (2016). https://doi.org/10.1007/s11538-016-0151-7
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DOI: https://doi.org/10.1007/s11538-016-0151-7