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

An SIR epidemic model with vaccination in a patchy environment

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

* Corresponding author: smoller_1@163.com

Received  July 2016 Revised  November 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.

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, B. M. Sahai, A vaccination model for transmission dynamics of influenza, SIAM J. Appl. Dyn. Syst., 3 (2004), 503-524. doi: 10.1137/030600370.

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J. Arino, C. C. Mccluskey, 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.

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J. Arino, R. Jordan, 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.

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

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

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O. Diekmann, J. A. P. Heesterbeek, 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.

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P. van den Driessche, 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.

[11]

M. C. Eisenberg, Z. Shuai, J. H. Tien, 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.

[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/.

[13]

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

[14]

D. Gao and S. Ruan, Malaria Models with Spatial Effects, Analyzing and Modeling Spatial and Temporal Dynamics of Infectious Diseases, Wiley, 2015.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

S. Ruan, W. Wang, 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.

[22]

H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University, 1995.

[23]

C. Sun, W. Yang, J. Arino, 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.

[24]

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

[25]

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

show all references

References:
[1]

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

[2]

J. Arino, C. C. Mccluskey, 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.

[3]

J. Arino, R. Jordan, 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.

[4]

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

[5]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Philadelphia, 1994.

[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

[7]

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

[8]

C. Castillo-Chavez, H. Thieme, Asymptotically autonomous epidemic models, Mathematical Population Dynamics: Analysis of Heterogeneity, (1995), 33-35.

[9]

O. Diekmann, J. A. P. Heesterbeek, 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.

[10]

P. van den Driessche, 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.

[11]

M. C. Eisenberg, Z. Shuai, J. H. Tien, 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.

[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/.

[13]

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

[14]

D. Gao and S. Ruan, Malaria Models with Spatial Effects, Analyzing and Modeling Spatial and Temporal Dynamics of Infectious Diseases, Wiley, 2015.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

S. Ruan, W. Wang, 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.

[22]

H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University, 1995.

[23]

C. Sun, W. Yang, J. Arino, 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.

[24]

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

[25]

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

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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