The possible states of our model are 0 = susceptible, 1 = infected, and 2 = removed. Here, we are thinking of non-fatal diseases such as measles or influenza, so removed means that the individual has had the disease and is immune to further infection. Rescaling time to make our first constant one, we can formulate the dynamics as follows:

A susceptible individual becomes infected at a rate equal to the fraction of neighbors that are infected.

An infected individual becomes removed at rate *delta*.

A removed individual becomes susceptible at rate *alpha*.

Our model has a fixed population size so the last transition can be thought of as: a removed individual dies (or leaves town) and is replaced by a new susceptible. To understand this model we begin with the case

**alpha = 0.** Suppose there is one infected at the origin in
an otherwise susceptible population.
If the recovery rate is too large or if we consider a one
dimensional population, the disease will die out. However, on the
square lattice there is a critical value *delta*_c approximately
0.22 so that if *delta* < *delta*_c then the epidemic has
positive probability of persisting for all time. A shape theorem
has been shown by Cox and Durrett (1988) for the nearest neighbor case
and by Zhang (1993) for a general finite range interaction:

**Theorem.** If *delta* < *delta*_c
and the epidemic does not die out then it spreads
linearly in time, has an asymptotic shape, and all the
infection is near the boundary of the growing region.

We will not state that result precisely here but suggest instead the

**s3 Exercise. Set delta = 0.1, and start the epidemic from
a 3x3 square of infecteds and watch the result.**

**
alpha > 0. As usual we begin by considering the behavior
of the corresponding mean field ODE**

**
**

**
In this case if delta < 1, there is an attracting fixed point
at**

**
**

**
For example, if delta = 0.1 and alpha = 0.04 then the
fixed point is u[1] = 0.257, u[2] = 0.643 and the ODE looks like:**

**
**

**
With an attracting fixed point in the ODE, one should expect coexistence
in the particle system, i.e., despite the placement of the example
on the contents page we are in Case 1.
Durrett and Neuhauser (1991) have shown:**

**
Theorem. Suppose delta < delta_c and
alpha > 0. Then there is a translation invariant stationary
distribution in which infected individuals have positive density.**

**
s3 Exercise. To compare with the ODE, set
delta = 0.1 and alpha = 0.04, and start from the
random initial condition. After a few oscillations,
the densities settle down to an equilibrium in which
about 20% of the sites are infected, and 53% are removed.
The difference comes from the fact that in the spatial
model (and in real life) the state of an individual has a significant
correlation with the states of its neighbors.**

**
Finite Size Effects. In the epidemic literature it is a
well known phenomenon that if the population size is too small
(e.g. measles in Iceland) then there will be moments when there
are no infecteds in the population. This phenomenon appears in
the spatial model as well. If the regrowth rate alpha is too
small then the epidemic dies out, while if alpha is close to this
critical level then the densities oscillate. To see this effect
in simulations set delta = 0.1 and alpha = 0.01.
For more discussion of this phenomenon, see Durrett (1995).**

**
**

**
Cox, J.T. and Durrett, R. (1988) Limit theorems for the
spread of epidemics and forest fires. Stoch. Proc. Appl.
30, 171--191**

**
Durrett, R. and Neuhauser, C. (1991) Epidemics with recovery in d=2.
Ann. Applied. Probab. 1, 189--206**

**
Boccara, N., and Cheong, K. (1992) Automata network SIR models fo the
spread of infectious diseases in populations of moving individuals.
J. Phys. A. 25, 2447--2461**

**
Zhang, Y. (1993) A shape theorem for epidemics and forest fires with
finite range interactions. Ann. Prob. 21, 1755-1781**

**
Durrett, R. (1995) Spatial epidemic models. Pages 187--201 in
Epidemic Models: Their Structure and Relation to Data.
Edited by D. Mollision. Cambridge U. Press.**

**
Rhodes, C.J. and Anderson, R.M. (1996) Power laws governing epidemics in
isolated populations. Nature 381, 600--602**

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