For principled model fitting in mathematical biology

Abstract

I argue for a principled approach to model fitting in mathematical biology that combines statistical and mechanistic insights.

Appendix A: Technical appendix

1.1 A.1: Individual-based model and likelihood

The underlying stochastic process is the general stochastic epidemic model (Bailey 1957), which consists of two integer-valued non-independent random variables in continuous time $S(t)$ and $I(t)$, such that $S(t) + I(t) \le N$, where the integer $N$ is the population size. This model has two real-valued parameters, $\lambda$ and $\gamma$, which have dimensions of inverse time and determine the rates of processes of the Markov chain:

$$\begin{aligned} (S,I) \rightarrow (S-1,I+1)\,\, \text { at rate }\,\, \lambda S I,\quad (S,I) \rightarrow (S,I-1) \,\,\text { at rate }\,\, \gamma I. \end{aligned}$$

(1)

And so if the current state is $(S,I)$ then the probability densities for the next event being an infection or recovery after time $t$ are respectively

$$\begin{aligned} \rho _1(t) = \lambda S I \mathrm{e}^{-(\lambda S + \gamma )I t},\quad \rho _2(t) = \gamma I \mathrm{e}^{-(\lambda S + \gamma )I t}. \end{aligned}$$

(2)

We will consider the case where observations are a set of times $T$ and events $\{e(t) \; | \; t \in T \; \& \; e(t) \in \{1,2\}\}$, so that a likelihood function can be defined as

$$\begin{aligned} \mathcal {L}(\beta ,\gamma ) = \prod _{t\in T} \rho _{e(t)}(t). \end{aligned}$$

(3)

1.2 A.2: Early diffusion limit

For a population with large size $N$, the early epidemic prevalence $I(t) \ll N$ converges $I(t) \rightarrow Y(t)$, where $Y(t)$ obeys the stochastic differential equation

$$\begin{aligned} \frac{\mathrm {d}Y}{\mathrm {d}t} = (\beta - \gamma ) Y + \left( \beta ^2 + \gamma ^2\right) ^{1/2} Y \xi , \end{aligned}$$

(4)

where $\beta := N\lambda$. If we make a series of observations $\{y_m\}$ of infectious prevalence at times $\{t_m\}$, then we can approximate the likelihood using a Gaussian process:

$$\begin{aligned}&\displaystyle \mathcal {L}(\beta ,\gamma ) = \prod _m \mathbb {P}[Y(t_{m+1}) =y_{m+1} | Y(t_m)=y_m], \nonumber \\&\displaystyle \mathbb {P}[Y(t+\delta ) =y' | Y(t)=y] \approx \mathcal {N}\bigg (y' \bigg | \mu = y \mathrm{e}^{(\beta - \gamma ) \delta } , \; \sigma = \left( \beta ^2 + \gamma ^2\right) ^{1/2}\mu \delta \bigg ), \nonumber \\&\displaystyle \mathcal {N}(x | \mu , \sigma ) := \frac{1}{(2\pi )^{1/2} \sigma } \mathrm{e}^{-(x-\mu )^2/(2\sigma ^2)}. \end{aligned}$$

(5)

1.3 A.3: Deterministic limit

In the limit of large $N$ (or more strictly $I(t) \gg 1$) the stochastic process (1) converges on the well-known SIR equations

$$\begin{aligned} \frac{\mathrm {d}s}{\mathrm {d}t} = -\beta s i, \quad \frac{\mathrm {d}i}{\mathrm {d}t} = \beta s i - \gamma i, \end{aligned}$$

(6)

where

$$\begin{aligned} s(t):= \frac{1}{N} \mathbb {E}[S(t)],\quad i(t):= \frac{1}{N} \mathbb {E}[I(t)]. \end{aligned}$$

(7)

If one approached data of the kind discussed in Sect. A.1 with the Eq. (6), then an alternative to likelihood-based fitting would be to employ a ‘least squares’ approach, and choose parameters to minimise the function

$$\begin{aligned} \mathcal {E}(\beta , \gamma , i_0) = \sum _{t\in T} \left( I(t) - i(t; \beta , \gamma , i_0)\right) ^2. \end{aligned}$$

(8)