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December  2017, 14(5&6): 1141-1157. doi: 10.3934/mbe.2017059

An SIR epidemic model with vaccination in a patchy environment

Qianqian Cui , Zhipeng Qiu , and  Ling Ding

School of Science, Nanjing University of Science and Technology, Nanjing 210094, China

* Corresponding author: smoller_1@163.com

Received  July 2016 Accepted  November 2016 Published  May 2017

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.

Keywords: SIR model, vaccination, patchy environment, transmission dynamics, reproduction number.
Mathematics Subject Classification: Primary: 34D23, 37N25; Secondary: 92B05.
Citation: Qianqian Cui, Zhipeng Qiu, Ling Ding. An SIR epidemic model with vaccination in a patchy environment. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1141-1157. doi: 10.3934/mbe.2017059
References:
[1]

M. E. Alexander, C. S. Bowman, S. M. Moghadas, R. Summers, A. B. Gumel and B. M. Sahai, A vaccination model for transmission dynamics of influenza, SIAM J. Appl. Dyn. Syst., 3 (2004), 503-524.  doi: 10.1137/030600370.  Google Scholar

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P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[11]

M. C. Eisenberg, Z. Shuai, J. H. Tien and P. van den Driessche, A cholera model in a patchy environment with water and human movement, Math. Biosci., 246 (2013), 105-112.  doi: 10.1016/j.mbs.2013.08.003.  Google Scholar

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H. Guo, M. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q., 14 (2006), 259-284.   Google Scholar

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Vaccination, website: https://en.wikipedia.org/wiki/Vaccination. Google Scholar

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Immunisation Advisory Centre, A Brief History of Vaccination, website: http://www.immune.org.nz/brief-history-vaccination. Google Scholar

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K. E. Jones, N. G. Patel, M. A. Levy, A. Storeygard, D. Balk, J. L. Gittleman and P. Daszak, Global trends in emerging infectious diseases, Nature, 451 (2008), 990-993.  doi: 10.1038/nature06536.  Google Scholar

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J. P. LaSalle, The Stability of Dynamical systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976.  Google Scholar

[21]

S. Ruan, W. Wang and S. A. Levin, The effect of global travel on the spread of SARS, Math. Biosci. Eng., 3 (2006), 205-218.  doi: 10.3934/mbe.2006.3.205.  Google Scholar

[22]

H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[23]

C. Sun, W. Yang, J. Arino and K. Khan, Effect of media-induced social distancing on disease transmission in a two patch setting, Math. Biosci., 230 (2011), 87-95.  doi: 10.1016/j.mbs.2011.01.005.  Google Scholar

[24]

W. Wang and X. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112.  doi: 10.1016/j.mbs.2002.11.001.  Google Scholar

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W. Wang and X. Zhao, An epidemic model with population dispersal and infection period, SIAM J. Appl. Math., 66 (2006), 1454-1472.  doi: 10.1137/050622948.  Google Scholar

show all references

References:
[1]

M. E. Alexander, C. S. Bowman, S. M. Moghadas, R. Summers, A. B. Gumel and B. M. Sahai, A vaccination model for transmission dynamics of influenza, SIAM J. Appl. Dyn. Syst., 3 (2004), 503-524.  doi: 10.1137/030600370.  Google Scholar

[2]

J. Arino, C. C. Mccluskey and P. van den Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math., 64 (2003), 260-276.  doi: 10.1137/S0036139902413829.  Google Scholar

[3]

J. Arino, R. Jordan and P. van den Driessche, Quarantine in a multi-species epidemics model with spatial dynamics, Math. Biosci., 206 (2007), 46-60.  doi: 10.1016/j.mbs.2005.09.002.  Google Scholar

[4]

P. Auger, E. Kouokam, G. Sallet, M. Tchuente and B. Tsanou, The Ross-Macdonald model in a patchy environment, Math. Biosci., 216 (2008), 123-131.  doi: 10.1016/j.mbs.2008.08.010.  Google Scholar

[5]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Philadelphia, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

[6]

World Health Organization, Ebola response roadmap -Situation report update 3 December 2014, website: http://apps.who.int/iris/bitstream/10665/144806/1/roadmapsitrep_3Dec2014_eng.pdf Google Scholar

[7]

F. Brauer, P. van den Driessche and L. Wang, Oscillations in a patchy environment disease model, Math. Biosci., 215 (2008), 1-10.  doi: 10.1016/j.mbs.2008.05.001.  Google Scholar

[8]

C. Castillo-Chavez and H. Thieme, Asymptotically autonomous epidemic models, in O. Arino, D. Axelrod, M. Kimmel, M. Langlais (Eds.), Mathematical Population Dynamics: Analysis of Heterogeneity, Springer, Berlin, 1995, 33–35. Google Scholar

[9]

O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface, 7 (2010), 873-885.  doi: 10.1098/rsif.2009.0386.  Google Scholar

[10]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[11]

M. C. Eisenberg, Z. Shuai, J. H. Tien and P. van den Driessche, A cholera model in a patchy environment with water and human movement, Math. Biosci., 246 (2013), 105-112.  doi: 10.1016/j.mbs.2013.08.003.  Google Scholar

[12]

How Many Ebola Patients Have Been Treated Outside of Africa? website: http://ritholtz.com/2014/10/how-many-ebola-patients-have-been-treated-outside-africa/. Google Scholar

[13]

D. Gao and S. Ruan, A multipathc malaria model with Logistic growth populations, SIAM J. Appl. Math., 72 (2012), 819-841.  doi: 10.1137/110850761.  Google Scholar

[14]

D. Gao and S. Ruan, Malaria Models with Spatial Effects, Analyzing and Modeling Spatial and Temporal Dynamics of Infectious Diseases, Wiley, 2015. doi: 10.1002/9781118630013.ch6.  Google Scholar

[15]

D. Gao and S. Ruan, An SIS patch model with variable transmission coefficients, Math. Biosci., 232 (2011), 110-115.  doi: 10.1016/j.mbs.2011.05.001.  Google Scholar

[16]

H. Guo, M. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. Appl. Math. Q., 14 (2006), 259-284.   Google Scholar

[17]

Vaccination, website: https://en.wikipedia.org/wiki/Vaccination. Google Scholar

[18]

Immunisation Advisory Centre, A Brief History of Vaccination, website: http://www.immune.org.nz/brief-history-vaccination. Google Scholar

[19]

K. E. Jones, N. G. Patel, M. A. Levy, A. Storeygard, D. Balk, J. L. Gittleman and P. Daszak, Global trends in emerging infectious diseases, Nature, 451 (2008), 990-993.  doi: 10.1038/nature06536.  Google Scholar

[20]

J. P. LaSalle, The Stability of Dynamical systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976.  Google Scholar

[21]

S. Ruan, W. Wang and S. A. Levin, The effect of global travel on the spread of SARS, Math. Biosci. Eng., 3 (2006), 205-218.  doi: 10.3934/mbe.2006.3.205.  Google Scholar

[22]

H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[23]

C. Sun, W. Yang, J. Arino and K. Khan, Effect of media-induced social distancing on disease transmission in a two patch setting, Math. Biosci., 230 (2011), 87-95.  doi: 10.1016/j.mbs.2011.01.005.  Google Scholar

[24]

W. Wang and X. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112.  doi: 10.1016/j.mbs.2002.11.001.  Google Scholar

[25]

W. Wang and X. Zhao, An epidemic model with population dispersal and infection period, SIAM J. Appl. Math., 66 (2006), 1454-1472.  doi: 10.1137/050622948.  Google Scholar

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