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) |