Table 2 A stochastic SI approximation to the risk of dengue for visitors

Let us start by calculating the probability that x individuals are in the state S and y individuals are in the state I at time t + Δt:

P _x,y (t + Δt) = P _x,y (t)(1 − λxΔt) + P _{x + 1,y − 1} λ(x + 1)Δt (2)

In equation (2) the first term refers to the probability that there were x and y individuals at time t in the states S and I respectively, and that no susceptible individuals x acquired the infection in the period. The second term refers to the probability that there were (x + 1) and (y-1) individuals at time t in the states S and I respectively, and that one susceptible individual acquired the infection in the period.

From equation (2) it follows that:

\( {\scriptscriptstyle \frac{dPx,y(t)}{dt}}=-\lambda x{P}_{x,y}(t)+\lambda \left(x+1\right){P}_{x+1,y-1}(t). \) (3)

The general expression for the Probability Generation Function (PGF), G(u,v,t), is given by:

G(u, v, t) = ∑ ^N _y = 0 u ^x v ^y P _x,y (t). (4)

For the particular model expressed in equation (4) it is possible to deduce that the PGF is:

G(u, v, t) = [(u − v)e ^− λt + v]^N.. (5)

Now the average number of infected individuals, y, at time t can be calculated by taking the first partial derivative of the PGF with respect to v at u,v = 1:

\( {\scriptscriptstyle \frac{\partial G\left(u,v,t\right)}{\partial v}}\left|{}_{u,v=1}=N\left(1-{e}^{-\lambda t}\right)\right.. \) (6)

Hence, the average per capita risk of infection, π is given by:

π = 1 − e ^− λt. (7)

The variance of the probability distribution for the number of infected individuals at time t is given by:

\( var\left[y\right]={\scriptscriptstyle \frac{\partial^2G\left(u,v,t\right)}{\partial {v}^2}}\left|{}_{u,v=1}\right.+{\scriptscriptstyle \frac{\partial G\left(u,v,t\right)}{\partial v}}{\left|{}_{u,v=1}-\left[{\scriptscriptstyle \frac{\partial G\left(u,v,t\right)}{\partial v}}\left|{}_{u,v=1}\right.\right]\right.}^2 \) (8)

which results in:

var[y] = Ne ^− λt[1 − e ^− λt]. (9)