Disease Spread in Coupled Populations: Minimizing Response Strategies Costs in Discrete Time Models

Figure 1

The impact of the variation in 𝜌^𝐼_𝑥𝑦 (here = 𝜏^𝐼_𝑥𝑦) on the total number of infected individuals along 30 days for different initial values of infected individuals in patch 𝑥: (a) 𝐼₀^𝑥𝑥=1, (b) 𝐼₀^𝑥𝑥=10, and (c) 𝐼₀^𝑥𝑥=100. Varying individuals’ flow intensity from one patch to another causes a notably increased appearance of infectives in the SIR model, due to the delayed entrance of infecteds. For this example we assume that patch 𝑦 is initially to be disease-free and that we are interested in controlling the flow of infected individuals towards it. Accordingly, we take 𝐼₀^𝑦𝑦=𝐼₀^𝑥𝑦=𝐼₀^𝑦𝑥=0. The other initial values are 𝑆^𝑥𝑥=1,500, 𝑆^𝑦𝑥=𝑆^𝑥𝑦=0, and 𝑆^𝑦𝑦=1,000. The parameter values are chosen from a particular influenza episode modeled with an SIR scheme [20], 𝛽=10⁻³/day, 𝑑=0.44/day. The values for the fractions of individuals moving between patches are chosen in a range estimated for patches with high interconnectivity [9]: 𝜏^𝑆_𝑥𝑦=𝜌^𝑆_𝑥𝑦=0.03, 𝜏^𝑆_𝑦𝑥=𝜌^𝐼_𝑦𝑥=0.02, and 𝜌^𝑆_𝑦𝑥=𝜌^𝐼_𝑦𝑥=0.02.

(a)

(b)

(c)