Approximation of epidemic models by diffusion processes and their statistical inference

Abstract

Multidimensional continuous-time Markov jump processes $(Z(t))$ on $\mathbb {Z}^p$ form a usual set-up for modeling $SIR$-like epidemics. However, when facing incomplete epidemic data, inference based on $(Z(t))$ is not easy to be achieved. Here, we start building a new framework for the estimation of key parameters of epidemic models based on statistics of diffusion processes approximating $(Z(t))$. First, previous results on the approximation of density-dependent $SIR$-like models by diffusion processes with small diffusion coefficient $1{/}{\sqrt{N}}$, where $N$ is the population size, are generalized to non-autonomous systems. Second, our previous inference results on discretely observed diffusion processes with small diffusion coefficient are extended to time-dependent diffusions. Consistent and asymptotically Gaussian estimates are obtained for a fixed number $n$ of observations, which corresponds to the epidemic context, and for $N\rightarrow \infty$. A correction term, which yields better estimates non asymptotically, is also included. Finally, performances and robustness of our estimators with respect to various parameters such as $R_0$ (the basic reproduction number), $N$, $n$ are investigated on simulations. Two models, $SIR$ and $SIRS$, corresponding to single and recurrent outbreaks, respectively, are used to simulate data. The findings indicate that our estimators have good asymptotic properties and behave noticeably well for realistic numbers of observations and population sizes. This study lays the foundations of a generic inference method currently under extension to incompletely observed epidemic data. Indeed, contrary to the majority of current inference techniques for partially observed processes, which necessitates computer intensive simulations, our method being mostly an analytical approach requires only the classical optimization steps.

Appendix A

1.1 A.1 Time changed Poisson process representation of a Markov jump process

First, the process satisfying (4) is obtained recursively as follows. Let $Z_0(t)\equiv Z(0)$ and set $Z_1(t)= Z(0)+ \sum _{l \in \mathbb {Z}^p} l\; P_l\left( \int _0^t \alpha _l(Z_{0}(u))du\right) .$ For $k >1$, define $Z_k(t)= Z(0)+ \sum _{l \in \mathbb {Z}^p} l\; P_l(\int _0^t \alpha _l(Z_{k-1}(u))du).$ Then, if $\tau _k$ is the $k$th jump of $Z_k(t)$, $Z_k(t)= Z_{k-1}(t)$ for $t< \tau _k$. The process $Z(t)= {lim}_{k\rightarrow \infty }\; Z_k(t)$ exists and satisfies (4).

Second, a characterization of these random time changed processes is mainly based on the property: given a positive measurable function $\eta : E \rightarrow (0, +\infty )$ and a Markov process $Y(.)$ such that $\int _0^{\infty }\frac{du}{\eta (Y(u))}= \infty$ a.s., one can define the random time change $\tau (t)$ by $\int _0^{\tau (t)}\frac{du}{\eta (Y(u))} =t \Longleftrightarrow \dot{\tau }(t)= \eta (Y(\tau (t)).$ The process $R(t)$ defined as $R(t) := Y(\tau (t))$ satisfies the equation $R(t) =Y(\int _0^t \eta (R(u))du)$. Moreover, if $A$ is the generator of $Y(.)$, the generator of $R(t)$ is equal to $\eta A$. Now, if $(Y(t))$ is the Poisson process $P_l(t)$ with rate $1$ (generator $Af(k)= f(k+1)-f(k)$), and $\eta (.)= \alpha _l(.)$, the process $Z_l(t) = P_l( \tau _l(t))$ has generator $A_lf(k)= \alpha _l(k)(f(k+1)-f(k))$ and satisfies, $Z_l(t) =P_l(\int _0^t \alpha _l(Z_l(s))ds)$. This allows to prove that the solution of (4) has the generator $Af(k) =\sum _{l \in \mathbb {Z}^p} \alpha _l(k)(f(k+l)-f(k))=\alpha (k)\sum _{l \in \mathbb {Z}^p} (f(k+l)-f(k))\frac{ \alpha _l(k)}{\alpha (k)}.$ We identify this generator as the one of $(Z(t))$ defined by (2).

1.2 A.2 Diffusion approximation for non-homogeneous Markov jump processes

We extend the approximation results from Ethier and Kurtz (2005) to the time dependent case. Their approach consists in using a Poisson time changed representation of the Markov jump process, a Brownian motion time changed representation of the diffusion process, and to compare them with an appropriate theorem from Komlós et al (1976). The extension of the proof of Ethier and Kurtz (2005) detailed in Appendix A.1 relies on the existence of (4) for time dependent Markov processes. The main problem is that the natural characterization of the random time change stated in Appendix A.1 now writes $\int _0^{\tau (t)}\frac{du}{\eta (\tau ^{-1}(u),Y(u))} =t$, and the time change becomes implicit. We rather use the general convergence results from Jacod and Shiryaev (1987) to obtain the diffusion approximation.

We consider the pure Jump Markov process $Z(t)$ with state space $E=\{0,..,N\}^p$ and transitions rates $q_{x,x+l}(t)=\alpha _l(t,x)$. This process has for generator $\mathcal {A}_tf(x)=\int _{\mathbb {R}^p} K_t(x,dy)\left( f(x+y)-f(x)\right)$ with the transition kernel $K_t(x,dy)=\sum _{l\in E^-}\alpha _l(t,x)\delta _l(y)$ where $\delta _l$ is the Dirac measure at point $l$. Within the framework developed by Jacod and Shiryaev (1987), it is a semimartingale with a random jump measure integrating $\parallel y\parallel$. So its characteristics in the sense of Definition 2.6 in Chapter II are $(B,C,\nu _t)$, where

• 1. $B=(B_i(t))_{1 \le i \le p}$ is the predictable process, $B(t)=\int _{0}^{t}b(s,Z(s))ds$, with $b(s,x)=\int _{\mathbb {R}^p}y\; K_s(x,dy)=\sum \nolimits _{l} l \alpha _l(s,x).$

• 2. $C=(C(t))$ is the quadratic variation of the continuous martingale part of $Z(t)$, $C(t)=(C_{i,j}(t))_{1\le i,j \le p}$. For a pure jump process, $C(t)\equiv 0$ .

• 3. $\nu _t$ is the compensator of the jumps random measure of $(Z_t)$, $\nu _t(dt,dy)= dt\; K_t(Z(t),dy)= dt\;\sum _{l}\alpha _l(t,Z(t))\delta _l(y)$.

• 4. The quadratic variation of the $p$-dimensional martingale $M(t)=Z(t)-B(t)$ is, for $1\le i,j \le p$, $[M_{ij}](t)= \int _{0}^{t} m_{ij}(s) ds$ with $m_{ij}(s)=\int _{\mathbb {R}^p}\;y_i\;y_j\; K_s(Z(s),dy)= \sum \nolimits _{l} l_i\;l_j \alpha _l(s,Z(s))ds.$

Consider now the sequence of normalized pure jump processes $Z_N(t)=\frac{Z(t)}{N}$ indexed by $N$. The state space of $Z_N$ is $E_N=\{0,\frac{1}{N},..,1\}^p$, its transition kernels are $K^{N}_t(x,dy)= \sum _{l\in E^-}\alpha _l(t,Nx)\delta _{\frac{l}{N}}(dy)$. Hence, its characteristics are $(B^N,C^N\nu _t^N)$ with

• 1. $B^N(t)= \int _{0}^{t} b^N(s,Z_N(s))ds$, with $b^N(s,x)= \sum _{l}\alpha _l(s,Nx)\frac{l}{N}$

• 2. $C^N(t) \equiv 0$,

• 3. $\nu ^{N}_{t}(dt,dy)=dt\; K^N_t(Z_N(t),dy)= dt\;\sum _{l}\alpha _l(t,NZ_N(t))\delta _{\frac{l}{N}}(y)$,

• 4. $[M_{i,j}^N](t)= \int _{0}^{t} m^N_{ij}(s) ds$ , with $m^N_{ij}(s)=\int _{\mathbb {R}^p}\;y_i\;y_j\; K^N_s(Z(s),dy) =\sum _{l}\alpha _l(s,NZ_N(s))\frac{l_i}{N}\frac{l_j}{N}$.

Under (H1), (H2), recall that $b(t,x)=\sum _{l\in E^-}l\beta _l(t,x)$ and $x_{x_0}(t) =x_0 +\int _0^t b(s,x_{x_0}(s)) ds.$

We first prove the convergence of the process $(Z_N(t))$ to $x_{x_0}(t)$ (which has characteristics $(\int _{0}^{t}b(s,x_{x_0}(s))ds,0,0)$) by applying Theorem 3.27 of Chapter IX in Jacod and Shiryaev (1987). We have to check the following conditions:

• (i) $\forall t\in [0,T]$, $\underset{0\le s \le t}{sup}||B^{N}(t)-\int _{0}^{t}b(s,x_{x_0}(s))ds ||\underset{{N}\rightarrow {\infty }}{\longrightarrow }0$,

• (ii) $\forall t\in [0,T]$, $[M^N](t) \rightarrow 0$ in probability,

• (iii) for all $\eta >0$, $\underset{a\rightarrow +\infty }{lim}\underset{N}{limsup}\,{\mathbb {P}}\left\{ \int _{0}^{t}\int _{\mathbb {R}^p}||y ||^2 1_{||y ||>a} (y) K^N_s(Z_N(s),dy)>\eta \right\} =0$,

• (iv) $\forall t\in [0,T]$, $\int _{0}^{t}ds\int _{\mathbb {R}^p}y \;K^N_s(Z_N(s),dy)\underset{{N}\rightarrow {\infty }}{\longrightarrow } 0$ in probability.

• (v) $Z_N(0)\underset{{N}\rightarrow {\infty }}{\longrightarrow }x_0$ a.s.

Using (H1t), we obtain the uniform convergence of $b^N(t,x)\underset{{N}\rightarrow {\infty }}{\longrightarrow }b(t,x)$ and $[M^N_{ij}](t)\underset{{N}\rightarrow {\infty }}{\longrightarrow }0$ on $[0,T]\times [0,1]^p$, which ensures conditions (i) and (ii). Condition (v) is satisfied by assumption. Since $\int _{\mathbb {R}^p}||y ||^2 K^N_s(x,dy) <\infty$, (iii) is satisfied. Using now that $\int _{\mathbb {R}^p}||y || K^N_s(x,dy) <\infty$ yields (iv).

Therefore, $Z_{N}(t) \rightarrow x_{x_0}(t)$ in distribution. Noting that $b^N(t,x)$ and $[M^{N}](t)$ converge uniformly towards $b(t,x)$ and $0$ respectively, and using that the Skorokhod convergence coincides with the uniform convergence when the limit is continuous, we get

$$\begin{aligned} \underset{t\in [0,T]}{sup}||Z_{N}(s)-x_{x_0}(s) || \underset{{N}\rightarrow { \infty }}{\longrightarrow } 0 \text{ in } \text{ probability. } \end{aligned}$$

(21)

It remains to study the process $Y_N (t)=\sqrt{N}\left( Z_{N}(t)-x_{x_0} (t)\right)$.

For sake of clarity, we omit in the sequel the index $x_0$ in $x_{x_0}(t)$. The jumps of $Y_N$ have size $l/\sqrt{N}$, the transition kernel of the jumps random measure is ${\tilde{K}}_t^N(y,du)= \sum _{l} \alpha _l(t, Nx(t)+\sqrt{N}y) \delta _{\frac{l}{\sqrt{N}}}(u)$, $Y_N$ is a semimartingale with characteristics $({\tilde{B}}^N,{\tilde{C}}^N, {\tilde{\nu }}_t^N)$

• 1. $\tilde{B}^N(t)= \int _{0}^{t}{\tilde{b}}^N(s,Y_N(s)) ds$, with ${\tilde{b}}^N(s,y) = \int _{\mathbb {R}^p}^{} u\; {\tilde{K}}_s^N(y,du)-\sqrt{N}\; b(s,x(s)) = \sum _{l} \alpha _l(s, Nx(s)+\sqrt{N}y) \frac{l}{\sqrt{N}}- \sqrt{N}\; b(s,x(s)).$

• 2. ${\tilde{C}}^N(t) =0$.

• 3. ${\tilde{\nu }}^N_t(dt,du)=dt\; \tilde{K}^N_t(Y_N(t),du)=dt\; (\sum _{l} \alpha _l(s,Nx(t)+\sqrt{N}Y_N(t)) \delta _{\frac{l}{\sqrt{N}}}(u)).$

• 4. $[\tilde{M}^N_{ij}](t)= \int _{0}^{t} \tilde{m}^N_{ij}(s) ds$, with $\tilde{m}^N_{ij}(s)=\int _{\mathbb {R}^p}\;u_i\;u_j\; \tilde{K}^N_s(Y_N(s),du) =\sum _{l}\alpha _l(s,Nx(s)+ \sqrt{N}Y_N(s))\frac{l_i}{\sqrt{N}}\frac{l_j}{\sqrt{N}}$.

Let us first study $\tilde{B}^N(t)$. Using (H1) $\frac{1}{N}\alpha _l(t, N(x(t)+\frac{y}{\sqrt{N}}))= \beta _l(t, x(t)+\frac{y}{\sqrt{N}})+ r_N(t)$, with $r_N(t)\rightarrow 0$, by (H2), $\beta _l(t,.)$ is differentiable and expanding $\beta _l(t,.)$ around $x(t)$ yields

$\beta _l(t,x(t)+\frac{y}{\sqrt{N}})= \beta _l(t,x(t))+\sum _{1}^{p} \frac{y_i}{\sqrt{N}} \frac{\partial \beta _l}{\partial x_i}(t, x(t)) +\;\frac{1}{\sqrt{N}} r'_N(t),$ with $r'_N(t)\rightarrow 0$. Therefore $\frac{1}{\sqrt{N}}\alpha _l(t, N(x(t)+\frac{y}{\sqrt{N}}))= \sqrt{N}\beta _l(t, x(t))+\sum _{i=1}^{p} y_i\frac{\partial \beta _l}{\partial x_i}(t, x(t))+\sqrt{N}r_N(t)+ r'_N(t).$ Hence, we need the additional assumption:

(H1t${^{\prime }}$) $\sqrt{N}(\frac{1}{N}\alpha _l(t, Nx)-\beta _l(t,x))\rightarrow 0$ uniformly w.r.t. $(t,x) \in [0,T]\times [0,1]^p$ as $N\rightarrow \infty$. Then, $\tilde{b}^N(t,y) \rightarrow \sum _{i=1}^{p} y_i \sum _{l}l\frac{\partial \beta _l}{\partial x_i}(t,x(t))= \sum _{i=1}^{p}y_i\frac{\partial b}{\partial x_i}(t,x(t))$ and $\tilde{B}^N(t)= \int _{0}^{t}\sum _{i=1}^{p}\frac{\partial b}{\partial x_i}(s,x(s)) Y_N(s) ds.$

Therefore, $[\tilde{M}^N_{ij}](t) \rightarrow \int _{0}^{t}\Sigma _{ij}(s,x(s)) ds.$ Checking conditions (iii),(iv),(v) is straightforward. Finally, we obtain that $Y_N$ converges in distribution to the process $Y(t)$ with continuous sample paths, predictable process $\int _{0}^{t}\nabla b(s,x(s)) Y(s)ds$ and quadratic variation $\int _{0}^{t}\Sigma _{ij}(s,x(s)) ds$. This is the diffusion process satisfying the SDE,

$$\begin{aligned} dY(t)= (\frac{\partial b_i}{\partial x_j})_{i,j}(t,x(t)) Y(t) dt + \sigma (t,x(t))dB(t)\;;\;Y(0)=0, \end{aligned}$$

where $\sigma ()$ satisfies $\sigma (t,x)\,{}^{t}\!{\sigma (t,x)}= \Sigma (t,x)$ and $B(t)$ is a $p$-dimensinal Brownian motion. This is an Ornstein-Uhlenbeck type SDE, which can be solved explicitly, leading to the Gaussian process $G(t)$ previously introduced.

1.3 A.3 Extending the contrast approach (for non autonomous diffusion processes and for non constant sampling intervals)

Here, we provide the main line for the extension of the results in Guy et al (2014) for non autonomous diffusions and non constant sampling intervals. The complete proof is omitted. The main point of the proof of Proposition 3.2 in Guy et al (2014) relies on the relations (3.7) and (3.8) $\frac{1}{\epsilon \sqrt{\Delta }}A_k(\theta _0)\underset{{\epsilon }\rightarrow {0}}{\longrightarrow }V_k^{\theta _0}$ and $\frac{1}{\Delta }\frac{\partial {A_k(\theta )}}{\partial {\theta _i}}\underset{{\epsilon }\rightarrow {0}}{\longrightarrow }D_{k,i}(\theta _0)$. The proof of these relations is based on Taylor stochastic expansion and the fundamental relation of our contrast approach (18) . The Taylor stochastic expansion of the diffusion was considered in Freidlin and Wentzell (1978) only for autonomous models, but has been extended for time dependent processes by Azencott (1982) and consequently holds when $b$ and $\Sigma$ are time dependent. Relation (18) is supported in the autonomous case by the semi-group property of function $\Phi _\theta$ which leads to an associated analytic expression of $g_\theta (t)=\int _{0}^{t}\Phi _\theta (t,s)\sigma (\theta ,s)dB_s$. Since the semi-group property is stated for non-autonomous cases in Cartan (1971), the extension is immediate. For non constant sampling interval, the simple fact that relation (18) holds for any sequence $t_0<t_1<\dots <t_n$ ensures that the results of Proposition 3.2 in Guy et al (2014) hold.