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4.1- Deterministic Analysis

To begin the analysis, imagine imbedding a graph in a d-dimensional space by associating each node j arbitrarily with a position . We can derive a deterministic approximation for , the fraction of infected individuals at position at time t, as follows. Let represent the rate at which the node at attempts to infect the node at and represent the rate at which a node at is cured (if it was infected). Then, following the same considerations as were used to derive the deterministic approximation for random graphs, we obtain:

 

Note that Eq. 19 is a slightly generalized form of Eq. 1, with node indices replaced by positions.

Now assume that a node can only interact with nodes lying within some small local neighborhood, and that the infection rates between two nodes depend only upon the distance between them. Furthermore, assume that the cure rate is independent of . Then, if and vary sufficiently slowly from one node to another, we can treat them as continuous functions, in which case it can be shown that Eq. 19 becomes approximately:

 

where

The first two terms in Eq. 20 are the familiar growth and decay terms that appear in Eq. 1. By themselves, they describe growth or decay of the level of infection at each point in space independently of the dynamics at any other point in space. The last term is something new. It is a second-order spatial derivative which accounts for diffusion of infection between different points in space. The diffusion coefficient D can be derived for various assumptions about the influence of nodes upon their neighbors. For example, if is uniform within a hypercubic volume with integral , . If is gaussian-distributed with standard deviation , .

Due to the assumed radial symmetry of , will remain radially symmetric if the initial condition is. Such a choice greatly simplifies both the calculation and the presentation of the results. Figure 10 depicts the typical course of an epidemic in two dimensions as predicted by Eq. 20, where i(x,0) is a narrow gaussian with volume . The population inhabits a circle of radius 1, so this initial distribution constitutes 1/10000 of the population.

  

Figure 10: Density of infected individuals i as a function of radius r from the initially source of infection at times t = 0, 4, 20, 50, and 80. As usual, the infection and cure rates are

and

. The diffusion coefficient

. Initially, 0.0001 of the population is infected, represented by a narrow gaussian with a standard deviation of 0.02 near r=0. At first, the height of the gaussian grows, until it saturates at the homogeneous limit of 0.8. Then, the infection enters a diffusive phase, growing outward at constant velocity in a circle with a fairly sharp boundary of fixed shape. Eventually, the spatial distribution of infection becomes uniform, with 80% of the individuals being infected.

In its first phase of growth, the pulse grows in height (t=4). When the pulse saturates at the equilibrium value of 0.8, it remains pinned at that value but keeps spreading outward radially. The leading front of the expanding circle develops a sharp edge, with a radius that increases at a constant velocity (note the positions at t = 20, 50, and 80). Thus the number of infected individuals, proportional to the area of the circle, increases quadratically with time. In d dimensions, the infection expands outward at constant velocity as a sharp-edged sphere, so the number of infected individuals grows as . This is of course much slower than the exponential growth of the random graph model. Finally, when the infection reaches the entire population, the total fraction of infected individuals reaches the same limit as in the random graph model, and its distribution is spatially uniform.

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