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An SIR epidemic model with vaccination in a patchy environment

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  • 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.

    Mathematics Subject Classification: Primary: 34D23, 37N25; Secondary: 92B05.


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  • Figure 1.  Diagram of transitions between epidemiological classes in the $i$-th patch

    Figure 2.  Time evolution of system (3) with multi-initial conditions in the case of $n=2$ and the initial values of $(S_1(0),S_2(0),I_1(0),I_2(0),R_1(0),R_2(0))$ are chose as (6000,1000,320, 20,1980,780), (2000,3000, 10,230,1980,2780), (5000,3000, 5, 35,980,980), (3600,2400,700,300,1500,1500), (3000,4000, 32, 8,980,1980). Here $l_1=0.1,\ l_2=0.23$ and the rest parameters are default values.

    Figure 3.  Contour plot of the control reproduction number $\mathfrak{R}_v$ in the $p_1-p_2$ plane and in $l_{12}-l_{21}$ plane. Here, (a) represents the relationship between vaccination rate (i.e., $p_1,p_2$) and $\mathfrak{R}_v$ if there is no migration between patches, (b) and (c) depict the relationship between migration rate (i.e., $l_{12},l_{21}$) and $\mathfrak{R}_v$ under the two specific vaccination rate marked in red point in figure (a). The red curves $\mathfrak{R}_v=0.83$ in (b) and $\mathfrak{R}_v=1.39$ in (c) respectively correspond to the red points $p_1=0.8,p_2=0.6$ and $p_1=0.6,p_2=0.8$ in (a). In these three figures, the blue curves represent the case of $\mathfrak{R}_v=1$. Other parameters are default values

    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 $\mathfrak{R}_{v1}$ equal to 3.48, 2.179, 1.22 and $\mathfrak{R}_{v2}$ equal to 2.09, 1.88, 0.627 are respectively corresponding to the vaccination coverage $p_1=p_2=0$, $p_1=0.2, p_2=0.1$ and $p_1=0.65, p_2=0.7$. The results show that lower vaccination coverage delay the second peak time and slightly reduce the second peak size for patch 1, patch 2 even the entire population. Whereas the higher coverage will not generate a second outbreak during the first 2000 days

    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 $l_{12}=l_{21}=0.2$, $l_{12}=0.3, l_{21}=0.1$ and $l_{12}=0.04,l_{21}=0.36$, and other parameters are default values

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