American Institute of Mathematical Sciences

Advanced
December  2017, 14(5&6): 1119-1140. doi: 10.3934/mbe.2017058

Effects of isolation and slaughter strategies in different species on emerging zoonoses

Jing-An Cui , and  Fangyuan Chen

School of Science, Beijing University of Civil Engineering and Architecture, No. 1, Zhanlanguan Road, Xicheng District, Beijing 100044, China

* Corresponding author: Jing-An Cui

Received  July 2016 Accepted  October 2016 Published  May 2017

Fund Project: This work was supported by the National Natural Science Foundation of China (11371048).

Zoonosis is the kind of infectious disease transmitting among different species by zoonotic pathogens. Different species play different roles in zoonoses. In this paper, we established a basic model to describe the zoonotic pathogen transmission from wildlife, to domestic animals, to humans. Then we put three strategies into the basic model to control the emerging zoonoses. Three strategies are corresponding to control measures of isolation, slaughter or similar in wildlife, domestic animals and humans respectively. We analyzed the effects of these three strategies on control reproductive numbers and equilibriums and we took avian influenza epidemic in China as an example to show the impacts of the strategies on emerging zoonoses in different areas at beginning.

Keywords: Zoonoses, strategy, isolation, slaughter, species, wildlife, domestic animals, humans.
Mathematics Subject Classification: Primary: 92B05, 92D30; Secondary: 34D20.
Citation: Jing-An Cui, Fangyuan Chen. Effects of isolation and slaughter strategies in different species on emerging zoonoses. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1119-1140. doi: 10.3934/mbe.2017058
References:
[1]

L. J. S. Allen, Mathematical Modeling of Viral Zoonoses in Wildlife, Natural Resource Modeling, 25 (2012), 5-51.  doi: 10.1111/j.1939-7445.2011.00104.x.

[2]

J. Arino, R. Jordan and P. V. D. Driessche, Quarantine in a multi-species epidemic model with spatial dynamics, Mathematical Biosciences, 206 (2007), 46-60.  doi: 10.1016/j.mbs.2005.09.002.

[3]

R. M. Atlas and S. Maloy, One Health: People, Animals, and the Environment, ASM Press, 2014.

[4]

R. G. Bengis, F. A. Leighton and J. R. Fischer, The role of wildlife in emerging and re-emerging zoonoses, Revue Scientifique Et Technique, 23 (2004), 497-511. 

[5]

F. Brauer and C. Chavez, Mathematical Models in Population Biology and Epidemiology 2$^{nd}$ edition, Springer, 2001. doi: 10.1007/978-1-4757-3516-1.

[6]

N. Busquets, J. Segals and L. Crdoba, Experimental infection with H1N1 European swine influenza virus protects pigs from an infection with the 2009 pandemic H1N1 human influenza virus, Veterinary Research, 41 (2010), 571-584.  doi: 10.1051/vetres/2010046.

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China Agricultural Yearbook Editing Committee, ChinaAgriculture Yearbook, China Agriculture Press, China, 2012.

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G. Chowell, Model parameters and outbreak control for SARS, Emerging Infectious Diseases, 10 (2004), 1258-1263.  doi: 10.3201/eid1007.030647.

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B. J. Coburn, B. G. Wagner and S. Blower, Modeling influenza epidemics and pandemics: Insights into the future of swine flu (H1N1) Bmc Medicine, 7 (2009), p30. doi: 10.1186/1741-7015-7-30.

[10]

B. J. Coburn, C. Cosne and S. Ruan, Emergence and dynamics of influenza super-strains, Bmc Public Health, 11 (2011), 597-615.  doi: 10.1186/1471-2458-11-S1-S6.

[11]

R. W. Compans and M. B. A. Oldstone, Influenza Pathogenesis and Control -Volume I, Current Topics in Microbiology & Immunology, 2014. doi: 10.1007/978-3-319-11155-1.

[12]

M. R. Conover, Human Diseases from Wildlife, Boca Raton : CRC Press, Taylor & Francis Group, 2014. doi: 10.1201/b17428.

[13]

M. Derouich and A. Boutayeb, An avian influenza mathematical model, Applied Mathematical Sciences, 2 (2008), 1749-1760. 

[14]

K. Dietz, W. H. Wernsdorfer and I. Mcgregor, Mathematical Models for Transmission and Control of Malaria, Malaria, 1988.

[15]

A. Dobson, Population dynamics of pathogens with multiple host species, American Naturalist, 164 (2004), 64-78.  doi: 10.1086/424681.

[16]

P. Van Den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[17]

X. Fang, The Role of Mammals in Epidemiology, Acta Theriologica Sinica, 2 (1981), 219-224. 

[18]

Z. Feng, Final and peak epidemic sizes for SEIR models with quarantine and isolation, Mathematical Biosciences & Engineering, 4 (2007), 675-686.  doi: 10.3934/mbe.2007.4.675.

[19]

Z. Feng, Applications of Epidemiological Models to Public Health Policymaking, World Scientific, 2014. doi: 10.1142/8884.

[20]

A. Fritsche, R. Engel and D. Buhl, Mycobacterium bovis tuberculosis: From animal to man and back, International Journal of Tuberculosis & Lung Disease the Official Journal of the International Union Against Tuberculosis & Lung Disease, 8 (2004), 903-904. 

[21]

S. Iwami, Y. Takeuchi and X. Liu, Avian-Chuman influenza epidemic model, Mathematical Biosciences, 207 (2007), 1-25.  doi: 10.1016/j.mbs.2006.08.001.

[22]

A. M. Kilpatrick and S. E. Randolph, Drivers, dynamics, and control of emerging vector-borne zoonotic diseases, Lancet, 380 (2012), 1946-1955.  doi: 10.1016/S0140-6736(12)61151-9.

[23]

J. Lee, J. Kim and H. D. Kwon, Optimal control of an influenza model with seasonal forcing and age-dependent transmission rates, Journal of Theoretical Biology, 317 (2013), 310-320.  doi: 10.1016/j.jtbi.2012.10.032.

[24]

J. S. Mackenzie, One Health: The Human-Animal-Environment Interfaces in Emerging Infectious Diseases, Springer, Berlin, 2013.

[25]

A. Mubayi, A cost-based comparison of quarantine strategies for new emerging diseases, Mathematical Biosciences & Engineering, 7 (2010), 687-717.  doi: 10.3934/mbe.2010.7.687.

[26]

R. A. Saenz, H. W. Hethcote and G. C. Gray, Confined animal feeding operations as amplifiers of influenza, Vector Borne & Zoonotic Diseases, 6 (2006), 338-346.  doi: 10.1089/vbz.2006.6.338.

[27]

P. M. Sharp and B. H. Hahn, Cross-species transmission and recombination of 'AIDS' viruses, Philosophical Transactions of the Royal Society B Biological Sciences, 349 (1995), 41-47.  doi: 10.1098/rstb.1995.0089.

[28]

A. Sing, Zoonoses -Infections Affecting Humans and Animals, Springer Netherlands, Berlin, 2015. doi: 10.1007/978-94-017-9457-2.

[29]

S. Towers and Z. Feng, Pandemic H1N1 influenza: predicting the course of a pandemic and assessing the efficacy of the planned vaccination programme in the United States, European Communicable Disease Bulletin, 14 (2009), 6-8. 

[30]

Q. Xian, L. Cui and Y. Jiao, Antigenic and genetic characterization of a European avian-like H1N1 swine influenza virus from a boy in China in 2011, Archives of Virology, 158 (2013), 39-53. 

[31]

W. D. Zhang, Optimized strategy for the control and prevention of newly emerging influenza revealed by the spread dynamics model, Plos One, 91 (2014), 5-51. 

[32]

J. Zhang, Z. Jin and G. Q. Sun, Modeling seasonal rabies epidemics in China, Bulletin of Mathematical Biology, 74 (2012), 1226-1251.  doi: 10.1007/s11538-012-9720-6.

show all references

References:
[1]

L. J. S. Allen, Mathematical Modeling of Viral Zoonoses in Wildlife, Natural Resource Modeling, 25 (2012), 5-51.  doi: 10.1111/j.1939-7445.2011.00104.x.

[2]

J. Arino, R. Jordan and P. V. D. Driessche, Quarantine in a multi-species epidemic model with spatial dynamics, Mathematical Biosciences, 206 (2007), 46-60.  doi: 10.1016/j.mbs.2005.09.002.

[3]

R. M. Atlas and S. Maloy, One Health: People, Animals, and the Environment, ASM Press, 2014.

[4]

R. G. Bengis, F. A. Leighton and J. R. Fischer, The role of wildlife in emerging and re-emerging zoonoses, Revue Scientifique Et Technique, 23 (2004), 497-511. 

[5]

F. Brauer and C. Chavez, Mathematical Models in Population Biology and Epidemiology 2$^{nd}$ edition, Springer, 2001. doi: 10.1007/978-1-4757-3516-1.

[6]

N. Busquets, J. Segals and L. Crdoba, Experimental infection with H1N1 European swine influenza virus protects pigs from an infection with the 2009 pandemic H1N1 human influenza virus, Veterinary Research, 41 (2010), 571-584.  doi: 10.1051/vetres/2010046.

[7]

China Agricultural Yearbook Editing Committee, ChinaAgriculture Yearbook, China Agriculture Press, China, 2012.

[8]

G. Chowell, Model parameters and outbreak control for SARS, Emerging Infectious Diseases, 10 (2004), 1258-1263.  doi: 10.3201/eid1007.030647.

[9]

B. J. Coburn, B. G. Wagner and S. Blower, Modeling influenza epidemics and pandemics: Insights into the future of swine flu (H1N1) Bmc Medicine, 7 (2009), p30. doi: 10.1186/1741-7015-7-30.

[10]

B. J. Coburn, C. Cosne and S. Ruan, Emergence and dynamics of influenza super-strains, Bmc Public Health, 11 (2011), 597-615.  doi: 10.1186/1471-2458-11-S1-S6.

[11]

R. W. Compans and M. B. A. Oldstone, Influenza Pathogenesis and Control -Volume I, Current Topics in Microbiology & Immunology, 2014. doi: 10.1007/978-3-319-11155-1.

[12]

M. R. Conover, Human Diseases from Wildlife, Boca Raton : CRC Press, Taylor & Francis Group, 2014. doi: 10.1201/b17428.

[13]

M. Derouich and A. Boutayeb, An avian influenza mathematical model, Applied Mathematical Sciences, 2 (2008), 1749-1760. 

[14]

K. Dietz, W. H. Wernsdorfer and I. Mcgregor, Mathematical Models for Transmission and Control of Malaria, Malaria, 1988.

[15]

A. Dobson, Population dynamics of pathogens with multiple host species, American Naturalist, 164 (2004), 64-78.  doi: 10.1086/424681.

[16]

P. Van Den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[17]

X. Fang, The Role of Mammals in Epidemiology, Acta Theriologica Sinica, 2 (1981), 219-224. 

[18]

Z. Feng, Final and peak epidemic sizes for SEIR models with quarantine and isolation, Mathematical Biosciences & Engineering, 4 (2007), 675-686.  doi: 10.3934/mbe.2007.4.675.

[19]

Z. Feng, Applications of Epidemiological Models to Public Health Policymaking, World Scientific, 2014. doi: 10.1142/8884.

[20]

A. Fritsche, R. Engel and D. Buhl, Mycobacterium bovis tuberculosis: From animal to man and back, International Journal of Tuberculosis & Lung Disease the Official Journal of the International Union Against Tuberculosis & Lung Disease, 8 (2004), 903-904. 

[21]

S. Iwami, Y. Takeuchi and X. Liu, Avian-Chuman influenza epidemic model, Mathematical Biosciences, 207 (2007), 1-25.  doi: 10.1016/j.mbs.2006.08.001.

[22]

A. M. Kilpatrick and S. E. Randolph, Drivers, dynamics, and control of emerging vector-borne zoonotic diseases, Lancet, 380 (2012), 1946-1955.  doi: 10.1016/S0140-6736(12)61151-9.

[23]

J. Lee, J. Kim and H. D. Kwon, Optimal control of an influenza model with seasonal forcing and age-dependent transmission rates, Journal of Theoretical Biology, 317 (2013), 310-320.  doi: 10.1016/j.jtbi.2012.10.032.

[24]

J. S. Mackenzie, One Health: The Human-Animal-Environment Interfaces in Emerging Infectious Diseases, Springer, Berlin, 2013.

[25]

A. Mubayi, A cost-based comparison of quarantine strategies for new emerging diseases, Mathematical Biosciences & Engineering, 7 (2010), 687-717.  doi: 10.3934/mbe.2010.7.687.

[26]

R. A. Saenz, H. W. Hethcote and G. C. Gray, Confined animal feeding operations as amplifiers of influenza, Vector Borne & Zoonotic Diseases, 6 (2006), 338-346.  doi: 10.1089/vbz.2006.6.338.

[27]

P. M. Sharp and B. H. Hahn, Cross-species transmission and recombination of 'AIDS' viruses, Philosophical Transactions of the Royal Society B Biological Sciences, 349 (1995), 41-47.  doi: 10.1098/rstb.1995.0089.

[28]

A. Sing, Zoonoses -Infections Affecting Humans and Animals, Springer Netherlands, Berlin, 2015. doi: 10.1007/978-94-017-9457-2.

[29]

S. Towers and Z. Feng, Pandemic H1N1 influenza: predicting the course of a pandemic and assessing the efficacy of the planned vaccination programme in the United States, European Communicable Disease Bulletin, 14 (2009), 6-8. 

[30]

Q. Xian, L. Cui and Y. Jiao, Antigenic and genetic characterization of a European avian-like H1N1 swine influenza virus from a boy in China in 2011, Archives of Virology, 158 (2013), 39-53. 

[31]

W. D. Zhang, Optimized strategy for the control and prevention of newly emerging influenza revealed by the spread dynamics model, Plos One, 91 (2014), 5-51. 

[32]

J. Zhang, Z. Jin and G. Q. Sun, Modeling seasonal rabies epidemics in China, Bulletin of Mathematical Biology, 74 (2012), 1226-1251.  doi: 10.1007/s11538-012-9720-6.

Figure 1.  Emerging zoonotic pathogen transmission from wildlife, to domestic animals, to humans
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Figure 2.  Avian influenza prevalence in wildlife, domestic animals and humans with high risk group: low risk group=1:9 in a, 1:3 in b, 1:1 in c, 3:1 in d 9:1 in e
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Figure 3.  Incidence rate on epidemic equilibrium change in different proportion of high risk group in humans
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Figure 4.  The effect of $\delta$ on $R_{1(WW)}$ in a. $R_{1(WW)}=1$, when $\delta =0.142*10^3$. The effect of $\Delta_I$ on $R_{2(DD)}$ in b. $R_{2(DD)}=1$, when $\Delta_I$=0.258. The effect of $\delta$ on $R_{3(H)}$ in c. $R_{3(H)}$=1, when $\delta$=0.066
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Figure 5.  Phase portrait of $S_H$ and $I_H$ in system (6) with no strategy:${\varepsilon _W} = 0 $, ${\varepsilon _D} = 0,$ ${\varepsilon _H} = 0$. Strategy 1: when $I_{LH} + I_{HH}< I_{WC}$, $\varepsilon _W = 0$; when $I_{LH} + I_{HH}\geq I_{WC}$, $\varepsilon _W = 1$. $\varepsilon _D = 0,\varepsilon _H = 0$. Strategy 2: when $I_{LH} + I_{HH}< I_{DC}$, $\varepsilon _D = 0$; when $I_{LH} + I_{HH}\geq I_{DC}$, $\varepsilon _D = 1$. $\varepsilon _W = 0,\varepsilon _H = 0$. Strategy 3: when $I_{LH} + I_{HH}< I_{HC}$, $\varepsilon _H = 0$; when $I_{LH} + I_{HH}\geq I_{HC}$, $\varepsilon _H = 1$. $\varepsilon _W = 0,\varepsilon _D = 0$. ($\delta$=0.1, $\theta_D$=0.1, $\theta_H=0.1, \Delta_I=1, \Theta_H=0.1, \sigma=0.01$ and $\rho$=0.001; high risk group: low risk group=1:9 in a, 9:1 in b; ${I_{DC}} = {I_{WC}} = {I_{HC}} = 15$, at $26^{th}$ day in a, $17^{th}$ day in b)
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Figure 6.  The effects of $\delta, \theta_D, \theta_H, \Delta_I, \Theta_H, \sigma$ and $\rho$ on the number of infected humans ($I_H$) in the first 90 days ($\delta$ in a, $\theta_D$ in b, $\theta_H$ in c, $\Delta_I$ in d, $\Theta_H$ in e, $\sigma$ in f and $\rho$ in g) (high risk group: low risk group=1:9)
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Table 1.  Impact of different strategies on reproductive numbers
Strategiesno strategyStrategy 1Strategy 2Strategy 3
Reproductive number in wildlife$ {R_{0(WW)}} =\frac{R_A}{R_B}$ $ {R_{1(WW)}} = \frac{R_A}{R_C}$$ {R_{2(WW)}} =$ ${R_{0(WW)}}$${R_{3(WW)}} =$${R_{0(WW)}}$
Reproductive number in domestic animals ${R_{0(DD)}} = \frac{R_D}{R_E}$${R_{1(DD)}} = $${R_{0(DD)}}$${R_{2(DD)}} = \frac{R_D}{R_F}$ ${R_{3(DD)}} = $${R_{0(DD)}}$
Reproductive number in humans ${R_{0(H)}} = \frac{R_G}{R_H}$${R_{1(H)}} = {R_{0(H)}}$${R_{2(H)}} ={R_{0(H)}}$ ${R_{3(H)}} = \frac{R_G}{R_I}$
$R_A={{A_W}{\beta _{WW}}}$, $R_B={{\mu _W}({\mu _W} + {\gamma _W} + {\alpha _W})}$,
$R_C={({\mu _W} + \delta )({\mu _W} + {\gamma _W} + {\alpha _W} + \delta )}$,
$R_D={{A_D}{\beta _{DD}}}$, $R_E={{\mu _D}({\mu _D} + {\gamma _D} + {\alpha _D})}$,
$R_F={{\mu _D}({\mu _D} + {\gamma _D} + {\alpha _D} + {\Delta _I})}$, $R_G={({A_{HH}} + {A_{LH}}){\beta _{HH}}}$,
$R_H={{\mu _H}({\mu _H} + {\gamma _H} + {\alpha _H})}$, $R_I={{\mu _H}({\mu _H} + {\gamma _H} + {\alpha _H} + \sigma )}. $
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Strategiesno strategyStrategy 1Strategy 2Strategy 3
Reproductive number in wildlife$ {R_{0(WW)}} =\frac{R_A}{R_B}$ $ {R_{1(WW)}} = \frac{R_A}{R_C}$$ {R_{2(WW)}} =$ ${R_{0(WW)}}$${R_{3(WW)}} =$${R_{0(WW)}}$
Reproductive number in domestic animals ${R_{0(DD)}} = \frac{R_D}{R_E}$${R_{1(DD)}} = $${R_{0(DD)}}$${R_{2(DD)}} = \frac{R_D}{R_F}$ ${R_{3(DD)}} = $${R_{0(DD)}}$
Reproductive number in humans ${R_{0(H)}} = \frac{R_G}{R_H}$${R_{1(H)}} = {R_{0(H)}}$${R_{2(H)}} ={R_{0(H)}}$ ${R_{3(H)}} = \frac{R_G}{R_I}$
$R_A={{A_W}{\beta _{WW}}}$, $R_B={{\mu _W}({\mu _W} + {\gamma _W} + {\alpha _W})}$,
$R_C={({\mu _W} + \delta )({\mu _W} + {\gamma _W} + {\alpha _W} + \delta )}$,
$R_D={{A_D}{\beta _{DD}}}$, $R_E={{\mu _D}({\mu _D} + {\gamma _D} + {\alpha _D})}$,
$R_F={{\mu _D}({\mu _D} + {\gamma _D} + {\alpha _D} + {\Delta _I})}$, $R_G={({A_{HH}} + {A_{LH}}){\beta _{HH}}}$,
$R_H={{\mu _H}({\mu _H} + {\gamma _H} + {\alpha _H})}$, $R_I={{\mu _H}({\mu _H} + {\gamma _H} + {\alpha _H} + \sigma )}. $
Table 2.  Parameter definitions and their values for avian influenza in China
ParameterDefinitionsValuesSources
$A_W$birth or immigration rate of wild aquatic birds0.137 birds/dayEst.
$\mu_W$natural mortality rate of wild aquatic birds0.000137/day[33]
$\gamma_W$recovery rate of wild aquatic birds0.25/dayEst.
$\alpha_W$disease-induced mortality rate of wild aquatic birds0.0025/dayEst.
$A_D$birth or immigration rate of domestic birds48.72 birds/day[33]
$\mu_D$natural mortality rate of domestic birds0.0058/day[33]
$\gamma_D$recovery rate of domestic birds0.25/day[26]
$\alpha_D$disease-induced mortality rate of domestic birds0.0025/dayEst.
$A_{HH}+A_{LH}$birth or immigration rate of humans0.07people/day[23]
$\mu_H$natural mortality rate of humans0.000035/day[23]
$\gamma_H$recovery rate of humans0.33/day[26, 31]
$\gamma_{H1}$remove rate from isolation compartment to susceptible compartment.0.5/dayEst.
$\gamma_{H2}$remove rate from isolation compartment to recovery individual compartment.0.5/dayEst.
$\alpha_H$disease-induced mortality rate of humans0.0033/dayEst.
$R_{0(WW)}$basic reproductive number of wild aquatic birds2Est.
$R_{0(DD)}$basic reproductive number of domestic birds2Est.
$R_{0(H)}$basic reproductive number of humans1.2[26]
$\beta_{WD}$per capita incidence rate from wild aquatic birds to domestic birds$6.15*10^{-6}$Est.
$\beta_{WH}$per capita incidence rate from wild aquatic birds to humans$2*10^{-5}$Est.
$\beta_{DH}$per capita incidence rate from domestic birds to humans$2*10^{-5}$Est.
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ParameterDefinitionsValuesSources
$A_W$birth or immigration rate of wild aquatic birds0.137 birds/dayEst.
$\mu_W$natural mortality rate of wild aquatic birds0.000137/day[33]
$\gamma_W$recovery rate of wild aquatic birds0.25/dayEst.
$\alpha_W$disease-induced mortality rate of wild aquatic birds0.0025/dayEst.
$A_D$birth or immigration rate of domestic birds48.72 birds/day[33]
$\mu_D$natural mortality rate of domestic birds0.0058/day[33]
$\gamma_D$recovery rate of domestic birds0.25/day[26]
$\alpha_D$disease-induced mortality rate of domestic birds0.0025/dayEst.
$A_{HH}+A_{LH}$birth or immigration rate of humans0.07people/day[23]
$\mu_H$natural mortality rate of humans0.000035/day[23]
$\gamma_H$recovery rate of humans0.33/day[26, 31]
$\gamma_{H1}$remove rate from isolation compartment to susceptible compartment.0.5/dayEst.
$\gamma_{H2}$remove rate from isolation compartment to recovery individual compartment.0.5/dayEst.
$\alpha_H$disease-induced mortality rate of humans0.0033/dayEst.
$R_{0(WW)}$basic reproductive number of wild aquatic birds2Est.
$R_{0(DD)}$basic reproductive number of domestic birds2Est.
$R_{0(H)}$basic reproductive number of humans1.2[26]
$\beta_{WD}$per capita incidence rate from wild aquatic birds to domestic birds$6.15*10^{-6}$Est.
$\beta_{WH}$per capita incidence rate from wild aquatic birds to humans$2*10^{-5}$Est.
$\beta_{DH}$per capita incidence rate from domestic birds to humans$2*10^{-5}$Est.
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