Some epidemic systems are long range interacting particle systems

We present some recent results about dynamical interacting particle systems in the setting of epidemiology. The individuals are particles whose states (of health) depend on their relative positions. These individuals interact since they fall ill more ofte

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1. INTRODUCTION We present some recent results about dynamical interacting particle systems in the setting of epidemiology. The individuals are particles whose states (of health) depend on their relative positions. These individuals interact since they fall ill more often when their neighbours are infectious. We begin with a description of the interaction between two individuals at the microscopic level. Then we study the behaviour of the whole system at the macroscopic level, when the number of individuals tends to infinity. Let ur(w) be the random state of the individual i in the system of N individuals: {j; 1 $ j $; N}. For instance ur may specify its position in a geographical space, its state of health and its deterministic type. Let us denote X the fiber, that is the set of the possible values of each Uf (w). We suppose that all individuals of the same type are similar, therefore the configuration (Ur(w); 1 $; i $ N) E XN is completely described by the empirical probability measure 1 N N

L Ouf

(w) E

II( X)

i=l

Here Ox is the Dirac measure at point x. In all what follows, if M is a topological space equipped with its Borel 0'-algebra, II( M) stands for the set of all probability measures on M and it is,endowed with its natural topology q(II(M), Cb(M)). When studying the behaviour of the system as N tends to infinity, it is worth representing it in terms of its empirical measure. Indeed, it allows us to imbed the sequence of sets (X N, N ~ 1) in the unique set II(X) . Therefore, II(X) is the natural set of all configurations. One could consider X = {position at time t} x {state of health} x {type}

c

1Rk

and study the evolution of the measure-valued stochastic process

1" N

(w,t)

==?

k N LJoUf(w,t) E 11(1R)

i=l

Instead of this, we shall look at the random empirical measure N

w ==?

~L

Ouf(w,-) E II((1Rk)[o,T])

i=l

Hence, the fiber X is the set (1Rk)[o,T] of all the paths from (O,T] into 1Rk. Other measure-valued stochastic processes have already been used to modelize spatial branching processes arising in biology. One should have a look to the survey paper, written J.-P. Gabriel et al. (eds.), Stochastic Processes in Epidemic Theory © Springer-Verlag Berlin Heidelberg 1990

171 by D.A. Dawson (1984), on this subject. The first result we shall present, is a law of large numbers. Under some regularity assumptions, it states the existence of a unique probability measure fl on X= (IRk)[o,T] of all the paths from (O,T] into IRk such that:

(1.1)

almost surely in w , lim N1 N-+oo

N

~ 8uN(w ·) L.....t • '

= fl

in II(X)

i=l

Then, we shall give an estimate for the probability of the large deviations from this almost sure event. This result is of the type:

(1.2)

1 logP(w; Nl lim N

N-oo

N

~ 8uN(w ·) • '

~

E A)=- inf{I(v); v E A}

i=l

for "regular enough" subsets A of II(X) . The function I(·) on II(X) takes its values in lR+ U { +oo} and is such that I( v) = 0 if and only if v = fl (which is given by (1.1)). It is called the rate function for the large deviations, since the probability of t