# American Institute of Mathematical Sciences

## Investigating the effects of intervention strategies in a spatio-temporal anthrax model

 1 Department of Science and Mathematics, Abraham Baldwin Agricultural College, Tifton, GA 31793, USA 2 Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA

* Corresponding author: bpantha@abac.edu

Received  February 2019 Revised  July 2019 Published  November 2019

In this paper, we extend our previous work on optimal control applied in an anthrax outbreak in wild animals. We use a system of ordinary differential equation (ODE) and partial differential equations (PDEs) to track the change in susceptible, infected and vaccinated animals as well as the infected carcasses. In addition to the assumption that the infected animals and the infected carcasses are the main source of infection, we consider the animal movement by diffusion and see its effects in disease transmission. Two controls: vaccinating susceptible animals and disposing infected carcasses properly are applied in the model and these controls depend on both space and time. We formulate an optimal control problem to investigate the effect of intervention strategies in our spatio-temporal model in controlling the outbreak at minimum cost. Finally some numerical results for the optimal control problem are presented.

Citation: Buddhi Pantha, Judy Day, Suzanne Lenhart. Investigating the effects of intervention strategies in a spatio-temporal anthrax model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019242
##### References:
 [1] S. Altizer, R. Bartel and B. A. Han, Animal migration and infectious disease risk, Science, 331 (2011), 296-302.  doi: 10.1126/science.1194694.  Google Scholar [2] Animal Diversity Web, https://animaldiversity.org/, Accessed May 2018. Google Scholar [3] J. K. Blackburn, A. Curtis, T. L. Hadfield, B. O'Shea, M. A. Mitchell and M. E. Hugh-Jones, Confirmation of Bacillus anthracis from flesh-eating flies collected during a West Texas Anthrax Season, Journel of Wildlife Disease, 46 (2010), 918-922.  doi: 10.7589/0090-3558-46.3.918.  Google Scholar [4] L. Busch, Bison herd suffers worst anthrax outbreak on record, Northern News Services Online, (2012), http://www.nnsl.com/frames/newspapers/2012-08/aug13\_12bs.html. Google Scholar [5] J. R. Castello, Bovids of World: Antelopes, Gazelles, Cattle, Goats, Sheep and Relatives, Prinston University Press, 2016. doi: 10.1515/9781400880652.  Google Scholar [6] S. Chawla and S. M. Lenhart, Application of optimal control theory to bioremediation, Journal of Computational and Applied Mathematics, 114 (2000), 81-102.  doi: 10.1016/S0377-0427(99)00290-3.  Google Scholar [7] S. Clegg. P. Turnbull, C. Foggin and P. Lindeque, Massive Outbreak of anthrex in wildlife in the Malilangwe Wildlife Reserve, Zimbabwe, The Veterinary Record, 160 (2007), 113-118.   Google Scholar [8] Department of Agriculture Forestry and Fisheries, Republic of South Africa, http://http://gadi.agric.za/articles/Furstenburg\_D, Accessed, 2018. Google Scholar [9] D. C. Dragon and B. T. Elkin, An overview of early Anthrax Outbreaks in Northern canada: Field reports of the Health of Animals Branch, Agriculture canada 1962-71, Arctic, 54 (2001), 1-104.  doi: 10.14430/arctic761.  Google Scholar [10] D. Dragon and R. Rennie, The ecology of anthrax spores: Tough but not invincible, Canadian Veterinary Journal, 36 (1995), 295-301.   Google Scholar [11] P. van den Driessche and J. Watmough, Reproduction number and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Bioscience, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar [12] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.  Google Scholar [13] Experience Zimbabwe, http://www.experiencezimbabwe.com/experience/attractions/malilangwe-wildlife-reserve, Accessed, 2018. Google Scholar [14] A. Fasanella, D. Galante, G. Garofolo and M. Hugh-Jones, Anthrax under valued zoonosis, Veterinary Microbiology, 140 (2010), 318-331.   Google Scholar [15] E. M. Fevre, B. M. de C. Bronsvoort, K. A. Hamilton and S. Cleaveland, Animal movements and the spread of infectious diseases, Trends in Microbiology, 14 (2006), 125-131.  doi: 10.1016/j.tim.2006.01.004.  Google Scholar [16] P. R. Furniss and B. D. Hahn, A mathematical model of an anthrax epizootic in the Kruger National Park, Applied Math Modeling, 5 (1981), 130-136.  doi: 10.1016/0307-904X(81)90034-2.  Google Scholar [17] K. R. Fister, S. Lenhart and J. McNally, Optimizing chemotherapy in an HIV model, Electronic Journal of Differential Equations, 1998 (1998), 12 pp.  Google Scholar [18] A. Friedman and A.-A. Yakubu, Anthrax epizootic and migration: Persistence or extinction, Mathematical Bioscience, 241 (2013), 137-144.  doi: 10.1016/j.mbs.2012.10.004.  Google Scholar [19] W. Hackbusch, A numerical method for solving parabolic equations with opposite orientations, Computing, 20 (1978), 229-240.  doi: 10.1007/BF02251947.  Google Scholar [20] B. D. Hahn and P. R. Furniss, A deterministic model of and anthrax epizootic: Threshold results, Ecological Modelling, 20 (1983), 233-241.  doi: 10.1016/0304-3800(83)90009-1.  Google Scholar [21] L. Hartfield, Bad year for anthrax outbreaks in US livestock, Center for Infectious Disease Research and Policy (CIDRAP), University of Minnesota, (2005), http://www.cidrap.umn.edu/news-perspective/2005/08/bad-year-anthrax-outbreaks-us-livestock. Google Scholar [22] M. E. Hugh-Jones and V. De Vos, Anthrax and wildlife, Scientific and Technical Review of the Office International des Epizooties, 21 (2003), 359-383.  doi: 10.20506/rst.21.2.1336.  Google Scholar [23] M. Kot, Elements of Mathematical Ecology, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511608520.  Google Scholar [24] I. Kracalik, L. Malania, M. Broladze, A. Navdarashvili, P. mnadze, S. J. Rya and J. Blackburn, Changing livestock vaccination policy alters the epidemiology of human anthrax, Georgia, 2000-2013., Vaccine, 35 (2017), 6283-6289.  doi: 10.1016/j.vaccine.2017.09.081.  Google Scholar [25] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, FL, 2007.  Google Scholar [26] C. Loehle, Social and behavioral barriers to pathogen transmission in wild animal populations, Clinical & Translational Immunology, 3 (1995), 1-6.  doi: 10.2172/666220.  Google Scholar [27] D. L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering, 162. Academic Press, Inc., London-New York, 1982.   Google Scholar [28] The MathWorks Inc, Global optimization toolbox user's guide, Release 2015a, 2015. Google Scholar [29] R. Miller Neilan and S. Lenhart, Optimal vaccine distribution in a spatiotemporal epidemic model with an application to rabies and raccoons, Journal of Mathematical Analysis and Applications, 378 (2011), 603-619.  doi: 10.1016/j.jmaa.2010.12.035.  Google Scholar [30] J. S. Nishi, D. C. Dragon, B. T. Elkin, J. Mitchell, T. R. Ellsworth and M. E. Hugh-Jones, Emergency response planning for anthrax outbreaks in bison herds of northern canada, Annals of the New York Academy of Sciences, 969 (2002), 245-250.  doi: 10.1111/j.1749-6632.2002.tb04386.x.  Google Scholar [31] B. Pantha, J. Day and S. Lenhart, Optimal control applied in an anthrax epizootic model, Journal of Biological Systems, 24 (2016), 495-517.  doi: 10.1142/S021833901650025X.  Google Scholar [32] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  doi: 10.1007/978-1-4615-3034-3.  Google Scholar [33] C. M. Saad-Roy, P. van den Driessche and A.-A. Yakubu, A mathematical model of anthrax transmission in animal populations, Bulletin of Mathematical Biology, 79 (2017), 303-324.  doi: 10.1007/s11538-016-0238-1.  Google Scholar [34] A. H. Seydack, C. C. Grant, I. P. Smit, W. J. Vermeulen, J. Baard and N. Zambatis, Large herbivore population performance and climate in a South African semi-arid Savanna, KOEDOE, 54 (2012), a1047. doi: 10.4102/koedoe.v54i1.1047.  Google Scholar [35] S. V. Shadomy and T. L. Smith, Anthrax, Journal of the American Veterinary Medical Association, 233 (2008), 63-72.  doi: 10.2460/javma.233.1.63.  Google Scholar [36] J. Simon, Compact sets in the space $L^p(0, T, B)$", Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar [37] J. Skellam, The Formulation and Interpretation of Mathematical Models of Diffusionary Processes in Population Biology, The Mathematical Theory of the Dynamics of Biological Populations, Academic Press, 1973.   Google Scholar [38] J. Tello and G. Van, The natural history of nyala, Tragelaphus angasi (Mammalia, Bovidae) in Mozambique, Bulletin of the AMNH, Bulletin of American Museum of Natural History, 155 (1975), 6283-6289.   Google Scholar [39] Texas Animal Health Commission, Anthrax confirmed in Eadwards county Deer, (2014), http://www.ttha.com/ttha/news/2014/09/08/anthrax-confirmed-in-edwards-county-deer. Google Scholar [40] P. Turnbill, Anthrax in Animals and Humans, WHO Press, Fourth edition, Geneva, 2008.   Google Scholar [41] V. Vos, The ecology of anthrax in the Kruger National Park, Salisbury Medical Bulletin, 68 (1990), 9-23.   Google Scholar [42] V. Vos, G. Rooyen and J. Kloppers, Anthrax immunizations of free ranging roan antelope hippotragus equinus in the Kruger National Park, KOEDOE, 16 (1973), 11-25.   Google Scholar

show all references

##### References:
 [1] S. Altizer, R. Bartel and B. A. Han, Animal migration and infectious disease risk, Science, 331 (2011), 296-302.  doi: 10.1126/science.1194694.  Google Scholar [2] Animal Diversity Web, https://animaldiversity.org/, Accessed May 2018. Google Scholar [3] J. K. Blackburn, A. Curtis, T. L. Hadfield, B. O'Shea, M. A. Mitchell and M. E. Hugh-Jones, Confirmation of Bacillus anthracis from flesh-eating flies collected during a West Texas Anthrax Season, Journel of Wildlife Disease, 46 (2010), 918-922.  doi: 10.7589/0090-3558-46.3.918.  Google Scholar [4] L. Busch, Bison herd suffers worst anthrax outbreak on record, Northern News Services Online, (2012), http://www.nnsl.com/frames/newspapers/2012-08/aug13\_12bs.html. Google Scholar [5] J. R. Castello, Bovids of World: Antelopes, Gazelles, Cattle, Goats, Sheep and Relatives, Prinston University Press, 2016. doi: 10.1515/9781400880652.  Google Scholar [6] S. Chawla and S. M. Lenhart, Application of optimal control theory to bioremediation, Journal of Computational and Applied Mathematics, 114 (2000), 81-102.  doi: 10.1016/S0377-0427(99)00290-3.  Google Scholar [7] S. Clegg. P. Turnbull, C. Foggin and P. Lindeque, Massive Outbreak of anthrex in wildlife in the Malilangwe Wildlife Reserve, Zimbabwe, The Veterinary Record, 160 (2007), 113-118.   Google Scholar [8] Department of Agriculture Forestry and Fisheries, Republic of South Africa, http://http://gadi.agric.za/articles/Furstenburg\_D, Accessed, 2018. Google Scholar [9] D. C. Dragon and B. T. Elkin, An overview of early Anthrax Outbreaks in Northern canada: Field reports of the Health of Animals Branch, Agriculture canada 1962-71, Arctic, 54 (2001), 1-104.  doi: 10.14430/arctic761.  Google Scholar [10] D. Dragon and R. Rennie, The ecology of anthrax spores: Tough but not invincible, Canadian Veterinary Journal, 36 (1995), 295-301.   Google Scholar [11] P. van den Driessche and J. Watmough, Reproduction number and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Bioscience, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar [12] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.  Google Scholar [13] Experience Zimbabwe, http://www.experiencezimbabwe.com/experience/attractions/malilangwe-wildlife-reserve, Accessed, 2018. Google Scholar [14] A. Fasanella, D. Galante, G. Garofolo and M. Hugh-Jones, Anthrax under valued zoonosis, Veterinary Microbiology, 140 (2010), 318-331.   Google Scholar [15] E. M. Fevre, B. M. de C. Bronsvoort, K. A. Hamilton and S. Cleaveland, Animal movements and the spread of infectious diseases, Trends in Microbiology, 14 (2006), 125-131.  doi: 10.1016/j.tim.2006.01.004.  Google Scholar [16] P. R. Furniss and B. D. Hahn, A mathematical model of an anthrax epizootic in the Kruger National Park, Applied Math Modeling, 5 (1981), 130-136.  doi: 10.1016/0307-904X(81)90034-2.  Google Scholar [17] K. R. Fister, S. Lenhart and J. McNally, Optimizing chemotherapy in an HIV model, Electronic Journal of Differential Equations, 1998 (1998), 12 pp.  Google Scholar [18] A. Friedman and A.-A. Yakubu, Anthrax epizootic and migration: Persistence or extinction, Mathematical Bioscience, 241 (2013), 137-144.  doi: 10.1016/j.mbs.2012.10.004.  Google Scholar [19] W. Hackbusch, A numerical method for solving parabolic equations with opposite orientations, Computing, 20 (1978), 229-240.  doi: 10.1007/BF02251947.  Google Scholar [20] B. D. Hahn and P. R. Furniss, A deterministic model of and anthrax epizootic: Threshold results, Ecological Modelling, 20 (1983), 233-241.  doi: 10.1016/0304-3800(83)90009-1.  Google Scholar [21] L. Hartfield, Bad year for anthrax outbreaks in US livestock, Center for Infectious Disease Research and Policy (CIDRAP), University of Minnesota, (2005), http://www.cidrap.umn.edu/news-perspective/2005/08/bad-year-anthrax-outbreaks-us-livestock. Google Scholar [22] M. E. Hugh-Jones and V. De Vos, Anthrax and wildlife, Scientific and Technical Review of the Office International des Epizooties, 21 (2003), 359-383.  doi: 10.20506/rst.21.2.1336.  Google Scholar [23] M. Kot, Elements of Mathematical Ecology, Cambridge University Press, Cambridge, 2001.  doi: 10.1017/CBO9780511608520.  Google Scholar [24] I. Kracalik, L. Malania, M. Broladze, A. Navdarashvili, P. mnadze, S. J. Rya and J. Blackburn, Changing livestock vaccination policy alters the epidemiology of human anthrax, Georgia, 2000-2013., Vaccine, 35 (2017), 6283-6289.  doi: 10.1016/j.vaccine.2017.09.081.  Google Scholar [25] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, FL, 2007.  Google Scholar [26] C. Loehle, Social and behavioral barriers to pathogen transmission in wild animal populations, Clinical & Translational Immunology, 3 (1995), 1-6.  doi: 10.2172/666220.  Google Scholar [27] D. L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering, 162. Academic Press, Inc., London-New York, 1982.   Google Scholar [28] The MathWorks Inc, Global optimization toolbox user's guide, Release 2015a, 2015. Google Scholar [29] R. Miller Neilan and S. Lenhart, Optimal vaccine distribution in a spatiotemporal epidemic model with an application to rabies and raccoons, Journal of Mathematical Analysis and Applications, 378 (2011), 603-619.  doi: 10.1016/j.jmaa.2010.12.035.  Google Scholar [30] J. S. Nishi, D. C. Dragon, B. T. Elkin, J. Mitchell, T. R. Ellsworth and M. E. Hugh-Jones, Emergency response planning for anthrax outbreaks in bison herds of northern canada, Annals of the New York Academy of Sciences, 969 (2002), 245-250.  doi: 10.1111/j.1749-6632.2002.tb04386.x.  Google Scholar [31] B. Pantha, J. Day and S. Lenhart, Optimal control applied in an anthrax epizootic model, Journal of Biological Systems, 24 (2016), 495-517.  doi: 10.1142/S021833901650025X.  Google Scholar [32] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.  doi: 10.1007/978-1-4615-3034-3.  Google Scholar [33] C. M. Saad-Roy, P. van den Driessche and A.-A. Yakubu, A mathematical model of anthrax transmission in animal populations, Bulletin of Mathematical Biology, 79 (2017), 303-324.  doi: 10.1007/s11538-016-0238-1.  Google Scholar [34] A. H. Seydack, C. C. Grant, I. P. Smit, W. J. Vermeulen, J. Baard and N. Zambatis, Large herbivore population performance and climate in a South African semi-arid Savanna, KOEDOE, 54 (2012), a1047. doi: 10.4102/koedoe.v54i1.1047.  Google Scholar [35] S. V. Shadomy and T. L. Smith, Anthrax, Journal of the American Veterinary Medical Association, 233 (2008), 63-72.  doi: 10.2460/javma.233.1.63.  Google Scholar [36] J. Simon, Compact sets in the space $L^p(0, T, B)$", Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar [37] J. Skellam, The Formulation and Interpretation of Mathematical Models of Diffusionary Processes in Population Biology, The Mathematical Theory of the Dynamics of Biological Populations, Academic Press, 1973.   Google Scholar [38] J. Tello and G. Van, The natural history of nyala, Tragelaphus angasi (Mammalia, Bovidae) in Mozambique, Bulletin of the AMNH, Bulletin of American Museum of Natural History, 155 (1975), 6283-6289.   Google Scholar [39] Texas Animal Health Commission, Anthrax confirmed in Eadwards county Deer, (2014), http://www.ttha.com/ttha/news/2014/09/08/anthrax-confirmed-in-edwards-county-deer. Google Scholar [40] P. Turnbill, Anthrax in Animals and Humans, WHO Press, Fourth edition, Geneva, 2008.   Google Scholar [41] V. Vos, The ecology of anthrax in the Kruger National Park, Salisbury Medical Bulletin, 68 (1990), 9-23.   Google Scholar [42] V. Vos, G. Rooyen and J. Kloppers, Anthrax immunizations of free ranging roan antelope hippotragus equinus in the Kruger National Park, KOEDOE, 16 (1973), 11-25.   Google Scholar
Simulation results for model (1)-(4) without control $u_1 = u_2 = 0$. The initial population of susceptible and infected animals are considered to be uniformly distributed in $1\le x\le 34$ and $27\le x\le 31$ respectively while only one initial carcass is considered near an end of the domain, $29\le x\le 30$. The figures in the first row show the plots for susceptible (left) and infected (right) animals; and the figure in the second row represents the carcasses
Simulation results for model (1)-(4) with optimal rates of vaccination and optimal carcass disposal rates $0\le u_1(x,t)\le 0.027,\; \; \text{and}\; \; 0\le u_2(x,t)\le 0.5.$. The initial population of susceptible and infected animals are considered to be uniformly distributed in $1\le x\le 34$ and $27\le x\le 31$ respectively while only one initial carcass is considered near an end of the domain, $29\le x\le30$. The two plots in the first row represent the concentrations of susceptible(left) and infected (right) animals. The plots in the second row represents the concentrations of the infected carcasses(left) and the vaccinated animals(right). The last row represents the vaccination (left) and carcass disposal(right) rates
The model parameters, their description, values and units
 Parm. Description Values Units $r$ Intrinsic growth rate of healthy animals $5.052\times 10^{-4}$ day$^{-1}$ $\gamma$ Disease induced death rate of infecteds $\frac{1}{7.5}$ day$^{-1}$ $\alpha$ Carcass feeding rate by scavengers $0$ animal$^{-1}$ day$^{-1}$ $K$ Carrying capacity of animals 2000 animal $p$ Carcass decay rate $0.02816$ day$^{-1}$ $d$ Diffusion rate of healthy animals $0.12$ $km^2$ day$^{-1}$ $d_1$ Diffusion rate of infected animals $0.024$ $km^2$ day$^{-1}$ $\theta_c$ Disease transmission rate from carcasses $1.65\times 10^{-3}$ carcass$^{-1}$ day$^{-1}$ $\theta_i$ Disease transmission rate from infected animals $2.05\times 10^{-2}$ animal$^{-1}$ day$^{-1}$
 Parm. Description Values Units $r$ Intrinsic growth rate of healthy animals $5.052\times 10^{-4}$ day$^{-1}$ $\gamma$ Disease induced death rate of infecteds $\frac{1}{7.5}$ day$^{-1}$ $\alpha$ Carcass feeding rate by scavengers $0$ animal$^{-1}$ day$^{-1}$ $K$ Carrying capacity of animals 2000 animal $p$ Carcass decay rate $0.02816$ day$^{-1}$ $d$ Diffusion rate of healthy animals $0.12$ $km^2$ day$^{-1}$ $d_1$ Diffusion rate of infected animals $0.024$ $km^2$ day$^{-1}$ $\theta_c$ Disease transmission rate from carcasses $1.65\times 10^{-3}$ carcass$^{-1}$ day$^{-1}$ $\theta_i$ Disease transmission rate from infected animals $2.05\times 10^{-2}$ animal$^{-1}$ day$^{-1}$
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