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 = 0.257, u = 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
On to the next page or back to the survey contents page