June  2020, 25(6): 2307-2330. doi: 10.3934/dcdsb.2020036

Dynamics of a metapopulation epidemic model with localized culling

1. 

Risk Analysis and Genomic Epidemiology Unit, Istituto Zooprofilattico Sperimentale della Lombardia e dell'Emilia Romagna, Via dei Mercati 13, 43126 Parma, Italy

2. 

Department of Mathematical, Physical and Computer Sciences, Università di Parma, Parco Area delle Scienze 53/A, 43124 Parma, Italy

3. 

Department of Electronics, Information and Bioengineering (DEIB), Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy

* Corresponding author: Maria Groppi

Received  July 2019 Published  February 2020

A two–patches metapopulation mathematical model, describing the dynamics of Susceptibles and Infected in wildlife diseases, is presented. The two patches are identical in absence of control, and culling activities are performed in only one of them. Firstly, the dynamics of the system in absence of control is investigated. Then, two types of localized culling strategies (proactive end reactive) are considered. The proactive control is modeled by a constant culling effort, and for the ensuing model the disease free equilibrium is characterized and existence of the endemic equilibrium is discussed in terms of a suitable control reproduction number. The localized reactive control is modeled by a piecewise constant culling effort function, that introduces an extra–mortality when the number of infected individuals in the patch overcomes a given threshold. The reactive control is then analytically and numerically investigated in the frame of Filippov systems.

We find that localized culling may be ineffective in controlling diseases in wild populations when the infection affects host fecundity in addition to host mortality, even leading to unexpected increases in the number of infected individuals in the nearby areas.

Citation: Luca Bolzoni, Rossella Della Marca, Maria Groppi, Alessandra Gragnani. Dynamics of a metapopulation epidemic model with localized culling. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2307-2330. doi: 10.3934/dcdsb.2020036
References:
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R. M. AndersonH. C. JacksonR. M. May and A. M. Smith, Population dynamics of fox rabies in Europe, Nature, 289 (1981), 765-771.  doi: 10.1038/289765a0.  Google Scholar

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L. Bolzoni and G. A. De Leo, Unexpected consequences of culling on the eradication of wildlife diseases: The role of virulence evolution, Amer. Naturalist, 181 (2013), 301-313.  doi: 10.1086/669154.  Google Scholar

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L. BolzoniG. A. De LeoM. Gatto and A. P. Dobson, Body-size scaling in an SEI model of wildlife diseases, Theoret. Population Biol., 73 (2008), 374-382.  doi: 10.1016/j.tpb.2007.12.003.  Google Scholar

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L. Bolzoni, L. Real and G. De Leo, Transmission heterogeneity and control strategies for infectious disease emergence, PLoS ONE, 2 (2007), e747, 5pp. doi: 10.1371/journal.pone.0000747.  Google Scholar

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L. BolzoniV. TessoniM. Groppi and G. A. De Leo, React or wait: Which optimal culling strategy to control infectious diseases in wildlife, J. Math. Biol., 69 (2014), 1001-1025.  doi: 10.1007/s00285-013-0726-y.  Google Scholar

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L. Bolzoni, E. Bonacini, R. Della Marca and M. Groppi, Optimal control of epidemic size and duration with limited resources, Math. Biosci., 315 (2019), 108232, 12pp. doi: 10.1016/j.mbs.2019.108232.  Google Scholar

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S. P. CarterR. J. DelahayG. C. SmithD. W. MacdonaldP. RiordanT. R. EtheringtonE. R. PimleyN. J. Walker and C. L. Cheeseman, Culling–induced social perturbation in Eurasian badgers Meles meles and the management of TB in cattle: An analysis of a critical problem in applied ecology, Proc. Roy. Soc. London B: Biol. Sci., 274 (2007), 2769-2777.  doi: 10.1098/rspb.2007.0998.  Google Scholar

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R. Casagrandi and M. Gatto, A persistence criterion for metapopulations, Theoret. Population Biol., 61 (2002), 115-125.  doi: 10.1006/tpbi.2001.1558.  Google Scholar

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M. Choisy and P. Rohani, Harvesting can increase severity of wildlife disease epidemics, Proc. Roy. Soc. B, 273 (2006), 2025-2034.  doi: 10.1098/rspb.2006.3554.  Google Scholar

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N. S. ChongB. Dionne and R. J. Smith, An avian-only Filippov model incorporating culling of both susceptible and infected birds in combating avian influenza, J. Math. Biol., 73 (2016), 751-784.  doi: 10.1007/s00285-016-0971-y.  Google Scholar

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N. S. Chong and R. J. Smith, Modeling avian influenza using Filippov systems to determine culling of infected birds and quarantine, Nonlinear Anal. Real World Appl., 24 (2015), 196-218.  doi: 10.1016/j.nonrwa.2015.02.007.  Google Scholar

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S. Cleaveland, K. Laurenson and T. Mlengeya, Impacts of wildlife infections on human and livestock health with special reference to Tanzania: Implications for protected area management, in Conservation and Development Interventions at the Wildlife/Livestock Interface: Implications for Wildlife, Livestock and Human Health, IUCN, Gland, 2005,147–151. Google Scholar

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M. J. CoyneG. Smith and F. E. McAllister, Mathematical model for the population biology of rabies in raccoons in the mid–Atlantic states, Amer. J. Veterinary Res., 50 (1989), 2148-2154.   Google Scholar

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F. DercoleA. GragnaniY. A. Kuznetsov and S. Rinaldi, Numerical sliding bifurcation analysis: An application to a relay control system, IEEE Trans. Circuits Systems I Fund. Theory Appl., 50 (2003), 1058-1063.  doi: 10.1109/TCSI.2003.815214.  Google Scholar

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E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede and X. Wang, Auto97: Continuation and bifurcation software for ordinary differential equaitions (with HomCont), Concordia University, Montreal, Canada, 1997. Google Scholar

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K. Dong-Hyun, Structural factors of the Middle East respiratory syndrome coronavirus outbreak as a public health crisis in Korea and future response strategies, J. Preventive Med. Public Health, 48 (2015), 265-270.   Google Scholar

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C. A. DonnellyR. WoodroffeD. R. CoxF. J. BourneC. CheesemanR. S. Clifton-HadleyG. WeiG. GettinbyP. GilksH. JenkinsW. T. JohnstonA. M. Le FevreJ. P. McInerney and W. I. Morrison, Positive and negative effects of widespread badger culling on tuberculosis in cattle, Nature, 439 (2006), 843-846.  doi: 10.1038/nature04454.  Google Scholar

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C. A. DonnellyR. WoodroffeD. R. CoxJ. BourneG. GettinbyA. M. Le FevreJ. P. McInerney and W. I. Morrison, Impact of localized badger culling on tuberculosis incidence in British cattle, Nature, 426 (2003), 834-837.  doi: 10.1038/nature02192.  Google Scholar

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A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

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L. G. Frank and R. Woddroffe, Behaviour of carnivores in exploited and controlled populations, in Carnivore Conservation, Cambridge University Press, Cambridge, 2001,419-442. Google Scholar

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P. Glendinning and M. R. Jeffrey, Grazing-sliding bifurcations, border collision maps and the curse of dimensionality for piecewise smooth bifurcation theory, Nonlinearity, 28 (2015), 263-283.  doi: 10.1088/0951-7715/28/1/263.  Google Scholar

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S. J. HoganM. E. HomerM. R. Jeffrey and R. Szalai, Piecewise smooth dynamical systems theory: The case of the missing boundary equilibrium bifurcations, J. Nonlinear Sci., 26 (2016), 1161-1173.  doi: 10.1007/s00332-016-9301-1.  Google Scholar

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Y. A. KuznetsovS. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2157-2188.  doi: 10.1142/S0218127403007874.  Google Scholar

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S. LachishH. MccallumD. MannC. E. Pukk and M. E. Jones, Evaluation of selective culling of infected individuals to control Tasmanian devil facial tumor disease, Conservation Biol., 24 (2010), 841-851.  doi: 10.1111/j.1523-1739.2009.01429.x.  Google Scholar

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M. K. MortersO. RestifK. HampsonS. CleavelandJ. L. N. Wood and A. J. K. Conlan, Evidence–based control of canine rabies: A critical review of population density reduction, J. Animal Ecology, 82 (2013), 6-14.  doi: 10.1111/j.1365-2656.2012.02033.x.  Google Scholar

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P. T. Piiroinen and Y. A. Kuznetsov, An event-driven method to simulate Filippov systems with accurate computing of sliding motions, ACM Trans. Math. Software, 34 (2008), 13, 24pp. doi: 10.1145/1356052.1356054.  Google Scholar

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L. C. PopeR. K. ButlinG. J. Wilson and R. Woodroffe, Genetic evidence that culling increases badger movement: Implications for the spread of bovine tuberculosis, Molecular Ecology, 16 (2007), 4919-4929.  doi: 10.1111/j.1365-294X.2007.03553.x.  Google Scholar

[40]

M. J. SmithA. WhiteJ. A. Sherratt and S. Telfer, Disease effects on reproduction can cause population cycles in seasonal environments, J. Animal Ecology, 77 (2008), 378-389.  doi: 10.1111/j.1365-2656.2007.01328.x.  Google Scholar

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M. I. TosaE. M. Schauber and C. K. Nielsen, Localized removal affects white–tailed deer space use and contacts, J. Wildlife Management, 81 (2017), 26-37.  doi: 10.1002/jwmg.21176.  Google Scholar

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F. A. M. TuyttensD. W. MacdonaldL. M. RogersC. L. Cheeseman and A. W. Roddam, Comparative study on the consequences of culling badgers (Meles meles) on biometrics, population dynamics and movement, J. Animal Ecology, 69 (2000), 567-580.  doi: 10.1046/j.1365-2656.2000.00419.x.  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

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F. Vial and C. A. Donnelly, Localized reactive badger culling increases risk of bovine tuberculosis in nearby cattle herds, Biol. Lett., 8 (2012), 50-53.  doi: 10.1098/rsbl.2011.0554.  Google Scholar

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show all references

References:
[1]

R. M. Anderson and R. M. May, Population biology of infectious diseases: Part Ⅰ, Nature, 280 (1979), 361-367.  doi: 10.1038/280361a0.  Google Scholar

[2]

R. M. AndersonH. C. JacksonR. M. May and A. M. Smith, Population dynamics of fox rabies in Europe, Nature, 289 (1981), 765-771.  doi: 10.1038/289765a0.  Google Scholar

[3]

J. Bielby, C. A. Donnelly, L. C. Pope, T. Burke and R. Woodroffe, Badger responses to small–scale culling may compromise targeted control of bovine tuberculosis, Proceedings of the National Academy of Sciences, 111, 2014, 9193–9198. Available from: http://www.pnas.org/content/111/25/9193. doi: 10.1073/pnas.1401503111.  Google Scholar

[4]

J. BielbyF. VialR. Woodroffe and C. A. Donnelly, Localised badger culling increases risk of herd breakdown on nearby, not focal, land, PLoS ONE, 11 (2016), 1-9.  doi: 10.1371/journal.pone.0164618.  Google Scholar

[5]

L. Bolzoni and G. A. De Leo, Unexpected consequences of culling on the eradication of wildlife diseases: The role of virulence evolution, Amer. Naturalist, 181 (2013), 301-313.  doi: 10.1086/669154.  Google Scholar

[6]

L. BolzoniG. A. De LeoM. Gatto and A. P. Dobson, Body-size scaling in an SEI model of wildlife diseases, Theoret. Population Biol., 73 (2008), 374-382.  doi: 10.1016/j.tpb.2007.12.003.  Google Scholar

[7]

L. Bolzoni, L. Real and G. De Leo, Transmission heterogeneity and control strategies for infectious disease emergence, PLoS ONE, 2 (2007), e747, 5pp. doi: 10.1371/journal.pone.0000747.  Google Scholar

[8]

L. BolzoniV. TessoniM. Groppi and G. A. De Leo, React or wait: Which optimal culling strategy to control infectious diseases in wildlife, J. Math. Biol., 69 (2014), 1001-1025.  doi: 10.1007/s00285-013-0726-y.  Google Scholar

[9]

L. Bolzoni, E. Bonacini, R. Della Marca and M. Groppi, Optimal control of epidemic size and duration with limited resources, Math. Biosci., 315 (2019), 108232, 12pp. doi: 10.1016/j.mbs.2019.108232.  Google Scholar

[10]

L. BolzoniE. BonaciniC. Soresina and M. Groppi, Time–optimal control strategies in SIR epidemic models, Math. Biosci., 292 (2017), 86-96.  doi: 10.1016/j.mbs.2017.07.011.  Google Scholar

[11]

M. H. Bond, Host life-history strategy explains pathogen-induced sterility, Amer. Naturalist, 168 (2006), 281-293.  doi: 10.1086/506922.  Google Scholar

[12]

M. S. BoyceA. R. E. Sinclair and G. C. White, Seasonal compensation of predation and harvesting, Oikos, 87 (1999), 419-426.  doi: 10.2307/3546808.  Google Scholar

[13]

S. P. CarterR. J. DelahayG. C. SmithD. W. MacdonaldP. RiordanT. R. EtheringtonE. R. PimleyN. J. Walker and C. L. Cheeseman, Culling–induced social perturbation in Eurasian badgers Meles meles and the management of TB in cattle: An analysis of a critical problem in applied ecology, Proc. Roy. Soc. London B: Biol. Sci., 274 (2007), 2769-2777.  doi: 10.1098/rspb.2007.0998.  Google Scholar

[14]

R. Casagrandi and M. Gatto, A persistence criterion for metapopulations, Theoret. Population Biol., 61 (2002), 115-125.  doi: 10.1006/tpbi.2001.1558.  Google Scholar

[15]

M. Choisy and P. Rohani, Harvesting can increase severity of wildlife disease epidemics, Proc. Roy. Soc. B, 273 (2006), 2025-2034.  doi: 10.1098/rspb.2006.3554.  Google Scholar

[16]

N. S. ChongB. Dionne and R. J. Smith, An avian-only Filippov model incorporating culling of both susceptible and infected birds in combating avian influenza, J. Math. Biol., 73 (2016), 751-784.  doi: 10.1007/s00285-016-0971-y.  Google Scholar

[17]

N. S. Chong and R. J. Smith, Modeling avian influenza using Filippov systems to determine culling of infected birds and quarantine, Nonlinear Anal. Real World Appl., 24 (2015), 196-218.  doi: 10.1016/j.nonrwa.2015.02.007.  Google Scholar

[18]

S. Cleaveland, K. Laurenson and T. Mlengeya, Impacts of wildlife infections on human and livestock health with special reference to Tanzania: Implications for protected area management, in Conservation and Development Interventions at the Wildlife/Livestock Interface: Implications for Wildlife, Livestock and Human Health, IUCN, Gland, 2005,147–151. Google Scholar

[19]

M. J. CoyneG. Smith and F. E. McAllister, Mathematical model for the population biology of rabies in raccoons in the mid–Atlantic states, Amer. J. Veterinary Res., 50 (1989), 2148-2154.   Google Scholar

[20]

F. DercoleA. GragnaniY. A. Kuznetsov and S. Rinaldi, Numerical sliding bifurcation analysis: An application to a relay control system, IEEE Trans. Circuits Systems I Fund. Theory Appl., 50 (2003), 1058-1063.  doi: 10.1109/TCSI.2003.815214.  Google Scholar

[21]

F. Dercole and Y. A. Kuznetsov, Slidecont: An Auto97 driver for sliding bifurcation analysis of Filippov systems, ACM Trans. Math. Software, 31 (2005), 95-119.  doi: 10.1145/1055531.1055536.  Google Scholar

[22]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.  Google Scholar

[23]

O. Diekmann and M. Kretzschmar, Patterns in the effects of infectious diseases on population growth, J. Math. Biol., 29 (1991), 539-570.  doi: 10.1007/BF00164051.  Google Scholar

[24]

E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede and X. Wang, Auto97: Continuation and bifurcation software for ordinary differential equaitions (with HomCont), Concordia University, Montreal, Canada, 1997. Google Scholar

[25]

K. Dong-Hyun, Structural factors of the Middle East respiratory syndrome coronavirus outbreak as a public health crisis in Korea and future response strategies, J. Preventive Med. Public Health, 48 (2015), 265-270.   Google Scholar

[26]

C. A. DonnellyR. WoodroffeD. R. CoxF. J. BourneC. CheesemanR. S. Clifton-HadleyG. WeiG. GettinbyP. GilksH. JenkinsW. T. JohnstonA. M. Le FevreJ. P. McInerney and W. I. Morrison, Positive and negative effects of widespread badger culling on tuberculosis in cattle, Nature, 439 (2006), 843-846.  doi: 10.1038/nature04454.  Google Scholar

[27]

C. A. DonnellyR. WoodroffeD. R. CoxJ. BourneG. GettinbyA. M. Le FevreJ. P. McInerney and W. I. Morrison, Impact of localized badger culling on tuberculosis incidence in British cattle, Nature, 426 (2003), 834-837.  doi: 10.1038/nature02192.  Google Scholar

[28]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[29]

L. G. Frank and R. Woddroffe, Behaviour of carnivores in exploited and controlled populations, in Carnivore Conservation, Cambridge University Press, Cambridge, 2001,419-442. Google Scholar

[30]

P. Glendinning and M. R. Jeffrey, Grazing-sliding bifurcations, border collision maps and the curse of dimensionality for piecewise smooth bifurcation theory, Nonlinearity, 28 (2015), 263-283.  doi: 10.1088/0951-7715/28/1/263.  Google Scholar

[31]

S. J. HoganM. E. HomerM. R. Jeffrey and R. Szalai, Piecewise smooth dynamical systems theory: The case of the missing boundary equilibrium bifurcations, J. Nonlinear Sci., 26 (2016), 1161-1173.  doi: 10.1007/s00332-016-9301-1.  Google Scholar

[32] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.  doi: 10.1017/CBO9780511810817.  Google Scholar
[33]

K. E. JonesN. G. PatelM. A. LevyA. StoreygardD. BalkJ. L. Gittleman and P. Daszak, Global trends in emerging infectious diseases, Nature, 451 (2008), 990-993.  doi: 10.1038/nature06536.  Google Scholar

[34]

A. J. Kucharski, A. Camacho, S. Flasche, R. E. Glover, W. J. Edmunds and S. Funk, Measuring the impact of Ebola control measures in Sierra Leone Proceedings of the National Academy of Sciences, 112, 2015, 14366–14371. Available from: http://www.pnas.org/content/112/46/14366. doi: 10.1073/pnas.1508814112.  Google Scholar

[35]

Y. A. KuznetsovS. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2157-2188.  doi: 10.1142/S0218127403007874.  Google Scholar

[36]

S. LachishH. MccallumD. MannC. E. Pukk and M. E. Jones, Evaluation of selective culling of infected individuals to control Tasmanian devil facial tumor disease, Conservation Biol., 24 (2010), 841-851.  doi: 10.1111/j.1523-1739.2009.01429.x.  Google Scholar

[37]

M. K. MortersO. RestifK. HampsonS. CleavelandJ. L. N. Wood and A. J. K. Conlan, Evidence–based control of canine rabies: A critical review of population density reduction, J. Animal Ecology, 82 (2013), 6-14.  doi: 10.1111/j.1365-2656.2012.02033.x.  Google Scholar

[38]

P. T. Piiroinen and Y. A. Kuznetsov, An event-driven method to simulate Filippov systems with accurate computing of sliding motions, ACM Trans. Math. Software, 34 (2008), 13, 24pp. doi: 10.1145/1356052.1356054.  Google Scholar

[39]

L. C. PopeR. K. ButlinG. J. Wilson and R. Woodroffe, Genetic evidence that culling increases badger movement: Implications for the spread of bovine tuberculosis, Molecular Ecology, 16 (2007), 4919-4929.  doi: 10.1111/j.1365-294X.2007.03553.x.  Google Scholar

[40]

M. J. SmithA. WhiteJ. A. Sherratt and S. Telfer, Disease effects on reproduction can cause population cycles in seasonal environments, J. Animal Ecology, 77 (2008), 378-389.  doi: 10.1111/j.1365-2656.2007.01328.x.  Google Scholar

[41] S. H. Strogatz, Nonlinear Dynamics and Chaos, Westview Press, Boulder, CO, 2015.   Google Scholar
[42]

M. I. TosaE. M. Schauber and C. K. Nielsen, Localized removal affects white–tailed deer space use and contacts, J. Wildlife Management, 81 (2017), 26-37.  doi: 10.1002/jwmg.21176.  Google Scholar

[43]

F. A. M. TuyttensD. W. MacdonaldL. M. RogersC. L. Cheeseman and A. W. Roddam, Comparative study on the consequences of culling badgers (Meles meles) on biometrics, population dynamics and movement, J. Animal Ecology, 69 (2000), 567-580.  doi: 10.1046/j.1365-2656.2000.00419.x.  Google Scholar

[44]

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

[45]

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Figure 1.  The effects of variation in host dispersal rate ($ D $) and localized proactive culling effort ($ \bar{c} $) on model (1)–(2) behaviors. The curves $ TC_0 $ (i.e., $ \bar{c} = \hat{c} $ in (16)) and $ TC_1 $ (i.e., $ \mathcal{R}_C = 1 $), which represent transcritical bifurcations, delimit three different regions in the parameter space $ \left[D, \bar{c}\right] $ where system (1), in the presence of localized proactive culling (2), converges to host extinction, disease–free equilibrium, or endemic equilibrium. Parameter values for model (1)–(2) have been fixed to: $ r = 0.9 $; $ \mu = 0.2 $; $ K = 600 $; $ \alpha = 0 $; $ \mathcal{R}_0 = 10 $ (or rather $ \beta = 0.01833 $, see (13))
Figure 2.  Curves in the parameter space [$ D $, $ \bar{c} $] separating the regions in which condition $ \hat{I}_2(\bar{c})>\bar{I} $ or condition $ \hat{I}_2(\bar{c})<\bar{I} $ are satisfied for different values of the basic reproduction number of model (1) with $ c = 0 $ ($ \mathcal{R}_0 $, given by (13)). Solid thick curve: $ \mathcal{R}_0 = 10 $; dashed thick curve: $ \mathcal{R}_0 = 5 $; dot–dashed thick curve: $ \mathcal{R}_0 = 2.5 $. The dynamics of infected individuals in patch 2 ($ \hat{I}_2 $) corresponding to the parameter space along the thin dotted line ($ D = 0.1r $) is illustrated in Fig. 3. Unspecified parameters as in Fig. 1
Figure 3.  Relative variation of the number of infected individuals in patch 2 at endemic equilibrium $ E_2^{\bar{c}} $ with respect to the number of infected in the absence of control (i.e., $ \delta\hat{I}_2 = (\hat{I}_2-\bar{I})/\bar{I} $) as a function of the localized proactive culling rate ($ \bar{c} $) for a very infrequent disperser host species ($ D/r = 0.1 $). Solid thick curve: $ \mathcal{R}_0 = 10 $; dashed thick curve: $ \mathcal{R}_0 = 5 $; dot–dashed thick curve: $ \mathcal{R}_0 = 2.5 $. The thin dotted line represents the condition $ \hat{I}_2 = \bar{I} $. Unspecified parameters as in Fig. 1
Figure 4.  Bifurcation diagram of epidemic model (1)–(3) in the parameter space $ [\theta, \bar{c}] $. The curves BN1 and BN2 represent boundary–node bifurcations. A stable equilibrium is the only attractor in regions 1 and 3. A pseudo–equilibrium is the only attractor in region 2. The shaded area represents the parameter combinations for which model (1)–(3) displays $ I_2^{\theta, \bar{c}} > I_2^{0, 0} = \bar{I} $. The relative variations of the number of infected individuals at model (1)–(3) steady–state with respect to the number of infected in the absence of control ($ \delta I_j^{\theta, \bar{c}} $) as a function of parameter $ \bar{c} $, corresponding to the dotted lines a and b, are illustrated in Figs. 5 and 6. Other parameters are: $ r = 0.9 $; $ \mu = 0.2 $; $ K = 600 $; $ \mathcal{R}_0 = 10 $; $ \alpha = 0 $, $ D = 0.1 r $
Figure 5.  Relative variation of the number of infected individuals in patches 1 (panel A) and 2 (panel B) at model (1)–(3) steady–state with respect to the number of infected in the absence of control ($ \delta I_j^{\theta, \bar{c}} $, with $ j = 1, 2 $) as a function of parameter $ \bar{c} $. BN2 represents a boundary–node bifurcation point as in Fig. 4. The dashed line represents the condition $ I_j^{\theta, \bar{c}} = \bar{I} $. Parameter $ \theta = 2.5 $, unspecified parameters as in Fig. 4
Figure 6.  Relative variation of the number of infected individuals in patches 1 (panel A) and 2 (panel B) at model (1)–(3) steady–state with respect to the number of infected in the absence of control ($ \delta I_j^{\theta, \bar{c}} $, with $ j = 1, 2 $) as a function of parameter $ \bar{c} $. BN2 represents a boundary–node bifurcation point as in Fig. 4. The dashed line represents the condition $ I_j^{\theta, \bar{c}} = \bar{I} $. Parameter $ \theta = 15 $, unspecified parameters as in Fig. 4
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