In this paper, an SIR patch model with vaccination is formulated to investigate the effect of vaccination coverage and the impact of human mobility on the spread of diseases among patches. The control reproduction number $\mathfrak{R}_v$ is derived. It shows that the disease-free equilibrium is unique and is globally asymptotically stable if $\mathfrak{R}_v < 1$ , and unstable if $\mathfrak{R}_v>1$ . The sufficient condition for the local stability of boundary equilibria and the existence of equilibria are obtained for the case $n=2$ . Numerical simulations indicate that vaccines can control and prevent the outbreak of infectious in all patches while migration may magnify the outbreak size in one patch and shrink the outbreak size in other patch.
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Figure 2.
Time evolution of system (3) with multi-initial conditions in the case of
Figure 3.
Contour plot of the control reproduction number
Figure 4.
Comparison of the second peak size and second peak time when there is no migration between patches. Figures (a), (b) and (c) respectively represent the trajectory of infectious vary with time for patch 1, patch 2 and the entire population with different vaccination coverage. Direct calculation implies that
Figure 5. Comparison of the second peak size and peak time with different migration rate in absence of vaccination. The trajectory of infectious varying with time for patch 1, patch 2 and the whole population are depicted in (a), (b) and (c), respectively. Figs. (a) and (c) show that human movement advanced the second peak time and increased the second peak size for patch 1 and the whole population when more individuals move to patch 1 (higher transmission rate). Fig. (b) illustrates that individuals migration can reduce the second outbreak size even no second outbreak during the first 2000 days
Figure 6.
Comparison of the residual value of first peak size (i.e., peak size with migration minus the case without migration) under different vaccination coverage. Figure (a) is for patch 1, (b) is for patch 2 and (c) is for the entire population. Here, case 1, case 2 and case 3 respectively represent
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