Conditional propagation of chaos in a spatial stochastic epidemic model with common noise

Abstract

We study a stochastic spatial epidemic model where the N individuals carry two features: a position and an infection state, interact and move in ${\mathbb {R}}^d$. In this Markovian model, the evolution of infection states are described with the help of the Poisson Point Processes , whereas the displacement of individuals are driven by mean field interactions, a (state dependence) diffusion and also a common noise, so that the spatial dynamic is a random process. We prove that when the number N of individual goes to infinity, the conditional propagation of chaos holds : conditionally to the common noise, the individuals are asymptotically independent and the stochastic dynamic converges to a “random” nonlinear McKean-Vlasov process. As a consequence, the associated empirical measure converges to a measure, which is solution of a stochastic mean-field PDE driven by the common noise.

Appendix

1.1 Definition of poisson point process and poisson random measure

Definition 6.1

An ${\mathbb {N}}$-valued process $(P_t)_{t\ge 0}$ is said to be a standard Poisson Point Process if $P_0=0$, its increments over disjoint intervals are independent, and for $0\le s\le t$, $P_t-P_s$ follows the $Poi(t-s)$ distribution. This implies in particular that all the jumps of $P_t$ have size 1.

Definition 6.2

A measure Q on ${\mathbb {R}}_{+}^2$ is said to be a standard Poisson Random Measure if it is a sum of Dirac measures located at countably many points of ${\mathbb {R}}_{+}^2$, in such a way that the numbers of points in disjoint Borel sets are mutually independent, and the number of points in a Borel subset $A\in {\mathbb {R}}_{+}^2$ follows the Poi(Lebesgue(A)) distribution. The mean measure of a standard Poisson Random Measure is the Lebesgue measure. Non standard Poisson Random Measures can have more general mean measures.

1.2 Proof of the equivalences in definition 2.3

Proof

$(i) \Rightarrow (ii)$ is obvious.

$(ii)\Rightarrow (iii)$. Taking any $\phi \in C_b(\Pi )$,

$$\begin{aligned} {\mathbb {E}}\left[ \left\langle \mu ^N-\mu , \phi \right\rangle ^2 \, \bigg \vert \, {\mathcal {F}}^0\right] =&\dfrac{1}{N^2}\sum _{i,j=1}^N{\mathbb {E}}\left[ \phi (Z^{i,N})\phi (Z^{j,N})\vert {\mathcal {F}}^0\right] \\&-\dfrac{2}{N}\sum _{i=1}^N{\mathbb {E}}\left[ \phi (Z^{i,N})\left\langle \mu ,\phi \right\rangle \vert {\mathcal {F}}^0\right] + {\mathbb {E}}\bigl [\left\langle \mu ,\phi \right\rangle ^2\,\big \vert \, {\mathcal {F}}^0\bigr ]. \end{aligned}$$

Using the conditional exchangeability we rewrite the r.h.s. as

$$\begin{aligned}&\dfrac{1}{N}{\mathbb {E}}\left[ \phi (Z^{1,N})^2 \,\vert \, {\mathcal {F}}^0\right] +\dfrac{N-1}{N}{\mathbb {E}}\left[ \phi (Z^{1,N})\phi (Z^{2,N}) \,\vert \,{\mathcal {F}}^0\right] \\ {}&\qquad -2 \left\langle \mu ,\phi \right\rangle {\mathbb {E}}\left[ \phi (Z^{1,N}) \, \Big \vert \, {\mathcal {F}}^0\right] + \left\langle \mu ,\phi \right\rangle ^2, \end{aligned}$$

which tends to 0 by (ii). Therefore,

$$\begin{aligned} {\mathbb {E}}\left[ \left\langle \mu ^N,\phi \right\rangle \, \vert \, {\mathcal {F}}^0\right] \rightarrow \left\langle \mu ,\phi \right\rangle . \end{aligned}$$

Since $\mu$ is ${\mathcal {F}}^0$-measurable and the above statement is true for any bounded continuous $\phi$, we conclude that $\mu ^N$ converges weakly to $\mu$ ${\mathbb {P}}^0$-a.s.

$(iii) \Rightarrow (i)$. We consider only the case $k=2$ of that implication, but the general case $k \ge 2$ could be handled in a similar way.

Let $\phi _1,\phi _2 \in C_b(\Pi )$. We will prove the convergence in law of $(Z^{1,N},Z^{2,N})$ using only functions of the form $\phi (z_1,z_2) = \phi _1(z_1) \phi _2(z_2)$, whose linear combinations are dense in $C_p(\Pi ^2)$. By the triangular inequality, one has

$$\begin{aligned}&\bigg |{\mathbb {E}}\Big [\phi _1(Z^{1,N})\phi _2(Z^{2,N}) \, \Big \vert \,{\mathcal {F}}^0\Big ]-\left\langle \mu ,\phi _1\right\rangle \left\langle \mu ,\phi _2\right\rangle \bigg | \end{aligned}$$

(6.1)

$$\begin{aligned}&\le \bigg |{\mathbb {E}}\Big [\phi _1(Z^{1,N})\phi _2(Z^{2,N}) \, \Big \vert \,{\mathcal {F}}^0\Big ] -{\mathbb {E}}\Big [\left\langle \mu ^N,\phi _1\right\rangle \left\langle \mu ^N,\phi _2\right\rangle \, \Big \vert \, {\mathcal {F}}^0\Big ]\bigg |\nonumber \\&\quad +\bigg |{\mathbb {E}}\Big [\left\langle \mu ^N,\phi _1\right\rangle \left\langle \mu ^N,\phi _2\right\rangle \, \Big \vert \,{\mathcal {F}}^0\Big ]-{\mathbb {E}}\Big [\left\langle \mu ,\phi _1\right\rangle \left\langle \mu ,\phi _2\right\rangle \, \Big \vert \, {\mathcal {F}}^0\Big ]\bigg |. \end{aligned}$$

(6.2)

Using the conditional exchangeability, we have an upper bound for (6.1),

$$\begin{aligned}&\bigg |{\mathbb {E}}\Big [\phi _1(Z^{1,N})\phi _2(Z^{2,N})\vert {\mathcal {F}}^0\Big ]-{\mathbb {E}}\Big [\left\langle \mu ^N,\phi _1\right\rangle \left\langle \mu ^N,\phi _2\right\rangle \vert {\mathcal {F}}^0\Big ]\bigg |\\&\quad =\Bigg |\frac{1}{N(N-1)}\sum _{i,j=1,i\ne j}^{N}{\mathbb {E}}\left[ \phi _1(Z^{i,N})\phi _2(Z^{j,N}) \vert {\mathcal {F}}^0\right] \\ {}&\qquad -\frac{1}{N^2}\sum _{i,j=1}^{N}{\mathbb {E}}\left[ \phi _1(Z^{i,N})\phi _2(Z^{j,N}) \vert {\mathcal {F}}^0\right] \Bigg |\\&\quad \le \Bigg |\left( \frac{1}{N(N-1)}-\frac{1}{N^2}\right) \sum _{i,j=1,i\ne j}^{N}{\mathbb {E}}\Big [\phi _1(Z^{i,N})\phi _2(Z^{j,N})\vert {\mathcal {F}}^0\Big ]\Bigg |\\&\qquad +\Bigg |\frac{1}{N^2}\sum _{i=1}^{N}{\mathbb {E}}\left[ \phi _1(Z^{i,N})\phi _2(Z^{i,N})\vert {\mathcal {F}}^0\right] \Bigg |\\&\quad \le \frac{1}{N} \Vert \phi _1\Vert _{\infty }\Vert \phi _2\Vert _{\infty }+\frac{1}{N} \Vert \phi _1\Vert _{\infty }\Vert \phi _2\Vert _{\infty }, \end{aligned}$$

where the last quantity tends to 0 as $N\rightarrow \infty$.

Considering (6.2), we can prove that it converges to 0, using the point (iii) and the fact that the function $\tilde{\mu }\mapsto \langle \ \tilde{\mu }, \phi _1\rangle \langle \tilde{\mu }, \phi _2 \rangle$ is continuous on $\mathcal {P}(\Pi )$. $\square$

1.3 Proof of lemma 2.4

Proof

The sytem of Eq. (1.3) is symetric under any permutation of the individuals and their individual noises. Hence, if $Z^N_t:=\left( Z^{1}_t,\dots , Z^N_t\right)$ is a strong solution of the system with initial condition $\left( Z^1_0, \dots , Z^N_0\right)$ then $Z^{\sigma , N}_t:=\left( Z_{t}^{\sigma (1), N}, \dots , Z_{t}^{\sigma (N), N}\right)$ is also a strong solution of the equation with initial condition $\left( Z^{\sigma (1)}_0, \dots , Z^{\sigma (N)}_0\right)$ with permuted individual noise. Moreover, by the assumption that the initial laws of $Z^{N}_0$ and $Z^{\sigma , N}_0$ are equal, the conclusion follows using the weak uniqueness of solutions to the SDE system (1.3) with jumps. In fact, the standard approach of [16, Theorem 9.1] applies to our particular case, since all the coefficients appearing in the system (1.3) are regular enough.

1.4 Rate of convergence in the empirical law of large number: The conditionally independence case

Proposition 6.3

For any $d \in {\mathbb {N}}$ and $q \in (2, \infty )$, there exists a constant $C_{d,q}$ such that the following holds. For any $\mathbb {P}_0$-measurable random probability $\mu$ on $\Pi$ satisfying

$$\begin{aligned} {\mathbb {E}}\biggl [ \int _{{\mathbb {R}}^d} |z|^q \mu (dz) \biggr ] = {\mathbb {E}}\bigl [|Z_i|^q \bigr ] < \infty , \end{aligned}$$

if conditionally upon ${\mathcal {F}}_0$, the $\Pi$-valued random variables $(Z_i)_{i \ge 1}$ are i.i.d. with conditional law $\mu$, and denoting the random empirical measure by $\mu ^N_Z= \frac{1}{N}\sum _{i=1}^{N}\delta _{Z_i}$, we have

$$\begin{aligned} {\mathbb {E}}\left[ W_1\left( \mu ^N_Z ,\mu \right) \right] \le C_{d,q} {\mathbb {E}}\left[ |Z_1|^q\right] ^{1/q} {\left\{ \begin{array}{ll} N^{-1/2},&{} d =1, \\ N^{-1/2}\log N,&{} d=2, \\ N^{-1/d},&{} d \ge 3. \end{array}\right. } \end{aligned}$$

Proof

For a given realisation of the common noise, we rely first on a result by Fournier Guillin [12, Theorem 1] (which is the best result so far after a long sequence of previous partial or less accurate result) in the case of i.i.d. random variables with a given (deterministic) common law $\mu$. Applying it (with $p=1$) in our special case,⁴ it gives the following estimate conditionally on ${\mathcal {F}}^0$,

$$\begin{aligned}&{\mathbb {P}}^0-\text {a.s.}, \quad {\mathbb {E}}\left[ W_1\left( \mu ^N_Z ,\mu \right) \,\big \vert \, {\mathcal {F}}^0 \right] \\&\quad \le C_{d,q} {\mathbb {E}}\left[ |Z_1|^q \,\big \vert \, {\mathcal {F}}^0 \right] ^{1/q} {\left\{ \begin{array}{ll} N^{-1/2}+N^{-(q-1)/q},&{} d =1, \\ N^{-1/2}\log N+N^{-(q-1)/q},&{} d=2, \\ N^{-1/d}+N^{-(q-1)/q},&{} d \ge 3. \end{array}\right. } \end{aligned}$$

First, we could remove all the second power of N involving q in the three cases since it is always smaller that the first term when $q>2$.

Taking then the expectation w.r.t. to $\mathbb {P}_0$ in the above inequality, and using since $q>1$ the Jensen inequality on the q moment

$$\begin{aligned} {\mathbb {E}}\Bigl [{\mathbb {E}}\left[ |Z_1|^q \,\big \vert \, {\mathcal {F}}^0 \right] ^{1/q} \Bigr ]\le {\mathbb {E}}\left[ |Z_1|^q \right] ^{1/q}, \end{aligned}$$

we get the desired result. $\square$