A new approach for designing disease intervention strategies in metapopulation models

Figures & data

Figure 1. Morbidity curves of patch 1 (red) and patch 2 (blue), without control (solid curves) and with control (dashed curves). We let ${\mathcal{R}}_1=1.2$ (${\beta }_1=0.240047$), ${\mathcal{R}}_2=1.05$, $m_{12}=0.015$, and $m_{21}=0.015$ for (a) and $m_{21}=0.1$ for (b). Other parameters are as described in the text. Figure (a): When $m_{21}=0.015$, then ${\mathcal{R}}_0=1.153>1$ (solid curves), the condition (2(1) $\frac{1}{2}\left(\frac{{\beta }_2}{{\gamma }_2+m_{12}}+\sqrt{{\left(\frac{{\beta }_2}{{\gamma }_2+m_{12}}\right)}^2+\frac{4m_{12}{m}_{21}}{({\gamma }_2+m_{12})({\gamma }_1+m_{21})}}\right)<1.$(1) ) is satisfied ($0.981714<1$), so we calculate ${\mathcal{T}}_S=1.41186$ and ${\beta }_1^c=0.170022$. Choosing ${\beta }_1=0.1<{\beta }_1^c$ (dashed curves), the reproduction number drops below 1 (see in the bracket) and the outbreak is prevented. Figure (b): When $m_{21}=0.1$, then ${\mathcal{R}}_0=1.07455>1$ (solid curves), the condition (4(3) $\rho (\tilde{\mathbf{K}}-{\tilde{\mathbf{K}}}_{\mathbf{W}})=\frac{{\beta }_2}{{\gamma }_2+m_{12}}<1$(3) ) is satisfied ($0.976758<1$), so we calculate ${\mathcal{T}}_W=1.80031$ and ${\beta }_1^c=0.109093$, $m_{21}^c=0.0454465$. Choosing ${\beta }_1=0.1<{\beta }_1^c$ and $m_{21}=0.04<m_{21}^c$ (dashed curves), the reproduction number drops below 1 (see in the bracket) and the outbreak is prevented.

Figure 2. Morbidity curves of patch 1 (red) and patch 2 (blue), without control (solid curves) and with control (dashed curves). We let ${\mathcal{R}}_1=0.95$ (${\beta }_1=0.190037$), ${\mathcal{R}}_2=1.05$, $m_{12}=0.015$, $m_{21}=0.015$. Other parameters are as described in the text. These parameters make ${\mathcal{R}}_0=1.01495>1$ (solid curves). Figure (a): The condition (4(3) $\rho (\tilde{\mathbf{K}}-{\tilde{\mathbf{K}}}_{\mathbf{W}})=\frac{{\beta }_2}{{\gamma }_2+m_{12}}<1$(3) ) is satisfied ($0.976758<1$), so we calculate ${\mathcal{T}}_W=1.01495$, and ${\beta }_1^c=0.172752$, $m_{21}^c=0.0136356$. Choosing ${\beta }_1=0.15<{\beta }_1^c$ and $m_{21}=0.012<m_{21}^c$ (dashed curves), the reproduction number drops below 1 (see in the bracket) and the outbreak is prevented. Figure (b): The condition (5(4) $\rho (\breve{\mathbf{K}}-{\breve{\mathbf{K}}}_{\mathbf{Z}})=\frac{{\beta }_1}{{\gamma }_1}<1,$(4) ) is satisfied (${\mathcal{R}}_1<1$), so we calculate ${\mathcal{T}}_Z=1.66667$, and $m_{12}^c=0.025$. Choosing $m_{12}=0.03>m_{21}^c$ (dashed curves), the reproduction number drops below 1 (see in the bracket) and the outbreak is prevented.