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Effects of isolation and slaughter strategies in different species on emerging zoonoses
An SIR epidemic model with vaccination in a patchy environment
School of Science, Nanjing University of Science and Technology, Nanjing 210094, China |
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.
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. |
[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. |
[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.
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[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.
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[5] |
A. Berman and R. J. Plemmons,
Nonnegative Matrices in the Mathematical Sciences, Philadelphia, 1994.
doi: 10.1137/1.9781611971262. |
[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 and 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 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. |
[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. |
[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. |
[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. |
[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 and 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,
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doi: 10.1002/9781118630013.ch6. |
[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. |
[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.
|
[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 and P. Daszak,
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[20] |
J. P. LaSalle,
The Stability of Dynamical systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976. |
[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. |
[22] |
H. L. Smith and P. Waltman,
The Theory of the Chemostat, Cambridge University, 1995.
doi: 10.1017/CBO9780511530043. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[5] |
A. Berman and R. J. Plemmons,
Nonnegative Matrices in the Mathematical Sciences, Philadelphia, 1994.
doi: 10.1137/1.9781611971262. |
[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 and 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 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. |
[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. |
[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. |
[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. |
[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 and 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.
doi: 10.1002/9781118630013.ch6. |
[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. |
[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.
|
[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 and 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 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. |
[22] |
H. L. Smith and P. Waltman,
The Theory of the Chemostat, Cambridge University, 1995.
doi: 10.1017/CBO9780511530043. |
[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. |
[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. |
[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. |






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