The temptation to compare this behaviour with that produced by an explicit space stochastic sim- ulation is hard to resist. Despite the fact that the pair model is best viewed as a model in its own right, its spatial structure can reect that of appropriately congured lattice models in a similar way to the hawk dove game example of Chapter 2. Consider an even more transmissible and long lived infection specied by = 500 = 0:02 = 0:3, released on a population of size N = 10000 which is arranged in a regular hexagonal lattice of 100 100 individuals. Each site neighbours its six nearest neighbours (using toroidal boundary conditions at the edges) and hence the spatial structure is equivalent to m = 6 and = 2=5. Stochastic simulation of this system was performed using an initial condition consisting of a random scatter of individuals in the proportions 10% susceptible, 5% infected and 85% recovered. Figure (6.5) shows the results graphically as snapshots of the spatial structure, and gure (6.6) shows the corresponding time series for the total susceptible, infected
and recovered populations. The pattern is one of repeated rapid epidemics, each followed by a more gradual replenishment of susceptibles. Each epidemic typically covers the whole spatial domain, and despite some stochastic noise (most easily seen in the time series of the less abundant infecteds) the result obtained in the majority of simulations (as shown) is surprisingly regular oscillations with a period of the order of 40 time units.
Establishing the oscillations from the random initial condition was not a certain process on about half of the attempts the infection did not `take hold' and died o. However, it did appear that an infection established for the period of one cycle was usually followed by several more oscillations, indicating that the spatial structure had to be correctly formatted, in some sense, to avoid the infection burning out. Of course the system is always vulnerable to stochastic fade-out, especially in the infection troughs where the number of infected individuals falls very low, in which case re- maining recovered individuals decay exponentially back into susceptibles (not shown in gure (6.6)). Simulations on a smaller 50 50 lattice suered fade-outs much more frequently, and maintained oscillations were harder to support.
Figure (6.7) shows the time series output of the pair model SIR equations for the same parameter values as used in the stochastic simulation of gures (6.5) and (6.6) using equivalent initial conditions and displayed on identical axes. Also shown is the original non-spatial SIR equation output, again using the same initial conditions and parameter values, with the exception of the transmissibility rate, . Recall from section (1.4.2) that in the mean-eld model, = Mwas really a composite rate
of the density of S-I contacts (relative to the product SI) multiplied by the probability of eective disease transmission. In both the pair model and lattice simulation, however, the contacts are explicitly known and = P is purely the eective transmission rate per contact. To compensate for
this, we arrange for the net transmission rates to be identical when there are no spatial correlations: there are (almost) N2 contacts in a model where everyone interacts with everyone (implied by the
mass-action principle), but only mN in the pair and lattice models. With identical species densities and no spatial correlations, the fraction of these that are S-I contacts in the dierent models will be in the same ratio so we expect the relation between mean-eld M and pair model P transmission
rates to be
MN2= PmN
This gives M = 0:3 as the proper value to correspond with P = 500, and this value is used in
gure (6.7).
On comparing gures (6.6) and (6.7), the rst impression is very favourable for the pair model. It produces cycling similar to that observed in the lattice simulation, whereas the mean-eld equations, as they must do, only approach an equilibrium. A closer inspection reveals that the match between the two oscillating models is not perfect: the pair model has more rapid cycling than the stochastic
a b c
d e f
g h i
Figure 6.5: Snapshots of the spatial structure shown by the hexagonal grid SIR model at = 500 = 0:02 = 0:3N = 10000 and = 0:4. Black represents sus- ceptible individuals, red is infected individuals and the recovereds are blue. Viewing from top left to bottom right, the system can rst be seen recovering from a large epidemic (a,b,c). Eventually a large body of susceptibles builds up, but (e) some of the few remaining infected individuals begin another large epidemic which rapidly sweeps through the population again. Typically, this cycle repeats over and over again, giving successive epidemics, until the infection is eliminated by chance, usu- ally in one of its troughs.
0 50 100 150 200 250 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 time number
Figure 6.6: Time series corresponding to gure (6.5) for the hexagonal lattice simulation, showing several subsequent epidemic outbursts. Susceptibles, infect- eds and recovereds are respectively black, red and blue and were initially present in a random mixture in the proportions 10%5%85%. The parameters are = 500 = 0:02 = 0:3N = 10000 and = 0:4.
simulation, with a period of approximately 28 time units, and the amplitude of each cycle is smaller than the lattice model. On the other hand, the mean values of the number of susceptible, infected and recovered individuals are broadly correct (unlike the mean-eld equations which predict only a tiny fraction of the observed number of susceptibles, and correspondingly too many recovereds). Of course, the cycles are perfectly regular in the deterministic pair model and it shows none of the stochasticity that will eventually inevitably disrupt the dynamics of the lattice simulation.
There are undoubtedly many factors involved in producing oscillations in the lattice simulation. Spatial structure is clearly important based on the visual evidence of gure (6.5) alone, but just how much of this is the result of local pair correlations and how much is dependent upon larger spatial structures or indeed the size of the grid is unknown. No thorough attempt was made to investigate the eect of grid size beyond the 100 100 and 50 50 lattice simulations, but it is likely that on sig- nicantly larger grids (too large to simulate with comparable computing power) the neat oscillations seen here will disappear as spatial structure is averaged away over widely separated and out of phase areas. One might also expect to see travelling waves of infection spreading rapidly through space, leaving areas of recovereds and then new susceptibles in their wake. (Such behaviour is occasionally seen on truly large scales in the real world, for example with human inuenza epidemics sweeping through continents). The choice of population size (N = 10000) and high transmissibility ( = 500) studied here probably made it easier for the pair model to reproduce similar dynamics to the lattice
0 50 100 150 200 250 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 time number
Figure 6.7: Time series (solid) for the pair model SIR equations for = 500 = 0:02 = 0:3N = 10000 = 0:4 and m = 6. The numerical integration starts from an initially random distribution (i.e. no pair correlations) of 10% S, 5% I and 85% R
which is close to the stable limit cycle to which it is quickly attracted. Susceptibles are black, infected red and recovered blue. The dashed lines are output from the non-spatial SIR equations (Chapter 1) with the same parameters except for = 0:3 - see the text for an explanation. In this case, the susceptibles are present only in extremely small numbers and are hardly visible on the graph).
simulation because they correspond to epidemics that are able to infect virtually all areas of the spatial domain almost simultaneously. This reduces the amount of important large scale spatial structure. It is more dicult to imagine how pair models could accommodate parameter regimes for which very large scale spatial heterogeneities are present.
Nevertheless, the achievement of the SIR pair model is substantial in accommodating oscillations at all, and all the more interesting because of the undoubted importance and frequency with which such oscillations appear in the real world. It is interesting to note that the pair model numerically solved for identical parameters to those of gure (6.7) except for the simpler case of = 0 predicts only an equilibrium solution, so the contribution of closed triangular connections is important to the model predictions. Because we are interested in the pair equations as a stand alone model, dis- crepancies with lattice model simulations (which are of course inadequate in many ways themselves, for example by not incorporating any migration in an otherwise very rigid spatial structure) are less signicant. The key point is that many real systems are likely to exist in a spatial environment that is more similar to the lattice model that to the mean-eld equations, so the behaviour of the pair equations is a step in the right direction.