# American Institute of Mathematical Sciences

2012, 9(3): 461-485. doi: 10.3934/mbe.2012.9.461

## Stochastic models for competing species with a shared pathogen

 1 Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, United States 2 Department of Mathematics, Oregon State University, Corvallis, OR 97331-4605, United States

Received  December 2011 Revised  March 2012 Published  July 2012

The presence of a pathogen among multiple competing species has important ecological implications. For example, a pathogen may change the competitive outcome, resulting in replacement of a native species by a non-native species. Alternately, if a pathogen becomes established, there may be a drastic reduction in species numbers. Stochastic variability in the birth, death and pathogen transmission processes plays an important role in determining the success of species or pathogen invasion. We investigate these phenomena while studying the dynamics of deterministic and stochastic models for $n$ competing species with a shared pathogen. The deterministic model is a system of ordinary differential equations for $n$ competing species in which a single shared pathogen is transmitted among the $n$ species. There is no immunity from infection, individuals either die or recover and become immediately susceptible, an SIS disease model. Analytical results about pathogen persistence or extinction are summarized for the deterministic model for two and three species and new results about stability of the infection-free state and invasion by one species of a system of $n-1$ species are obtained. New stochastic models are derived in the form of continuous-time Markov chains and stochastic differential equations. Branching process theory is applied to the continuous-time Markov chain model to estimate probabilities for pathogen extinction or species invasion. Finally, numerical simulations are conducted to explore the effect of disease on two-species competition, to illustrate some of the analytical results and to highlight some of the differences in the stochastic and deterministic models.
Citation: Linda J. S. Allen, Vrushali A. Bokil. Stochastic models for competing species with a shared pathogen. Mathematical Biosciences & Engineering, 2012, 9 (3) : 461-485. doi: 10.3934/mbe.2012.9.461
##### References:
 [1] E. Allen, "Modeling With Itô Stochastic Differential Equations,'' Mathematical Modelling: Theory and Applications, 22, Springer, Dordrecht, The Netherlands, 2007. [2] E. J. Allen, L. J. S. Allen, A. Arciniega and P. Greenwood, Construction of equivalent stochastic differential equation models, Stoch. Anal. Appl., 26 (2008), 274-297. [3] L. J. S. Allen, "An Introduction to Mathematical Biology,'' Prentice Hall, Upper Saddle River, NJ, 2007. [4] L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology,'' 2nd edition, CRC Press, Boca Raton, FL, 2011. [5] L. J. S. Allen and N. Kirupaharan, Asymptotic dynamics of deterministic and stochastic epidemic models with multiple pathogens, Int. J. Numer. Anal. Modeling, 2 (2005), 329-344. [6] R. M. Anderson and R. M. May, The invasion, persistence and spread of infectious diseases with animal and plant communities, Phil. Trans. R. Soc. Lond. B, 314 (1986), 533-570. doi: 10.1098/rstb.1986.0072. [7] N. T. J. Bailey, "The Elements of Stochastic Processes with Applications to the Natural Sciences,'' Reprint of the 1964 original, Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1990. [8] M. S. Bartlett, The relevance of stochastic models for large-scale epidemiological phenomena, Appl. Statist., 13 (1965), 2-8. [9] M. Begon, R. G. Bowers, N. Kadianakis and D. E. Hodgkinson, Disease and community structure: The importance of host self-regulation in a host-host-pathogen model, Am. Nat., 139 (1992), 1131-1150. doi: 10.1086/285379. [10] V. A. Bokil and M.-R. Leung, An analysis of the coexistence of three competing species with a shared pathogen, Technical Report ORST-MATH-11-02, Oregon State Univ., 2011. Citation URL: http://ir.library.oregonstate.edu/xmlui/handle/1957/13738/. [11] E. T. Borer, P. R. Hosseini, E. W. Seabloom and A. P. Dobson, Pathogen-induced reversal of native dominance in a grassland community, Proc. Natl. Acad. Sci. U. S. A., 104 (2007), 5473-5478. [12] R. G. Bowers and J. Turner, Community structure and the interplay between interspecific infection and competition, J. Theor. Biol., 187 (1997), 95-109. doi: 10.1006/jtbi.1997.0418. [13] S. K. Collinge and C. Ray, "Disease Ecology: Community Structure and Pathogen Dynamics," Oxford Univ. Press, Oxford, 2006. [14] A. Dobson, Population dynamics of pathogens with multiple host species, Am. Nat., 164 (2004), S64-S78. doi: 10.1086/424681. [15] R. Durrett, Mutual invadability implies coexistence in spatial models, Mem. Am. Math. Soc., 156 (2002), viii+118 pp. [16] R. Durrett, Special invited paper: Coexistence in stochastic spatial models, Ann. Appl. Probab., 19 (2009), 477-496. doi: 10.1214/08-AAP590. [17] R. Durrett and C. Neuhauser, Coexistence results for some competition models, Ann. Appl. Probab., 7 (1997), 10-45. [18] L. Gilbert, R. Norman, K. M. Laurenson, H. W. Reid and P. J. Hudson, Disease persistence and apparent competition in a three-host community: An empirical and analytical study of large-scale, wild populations, J. Anim. Ecol., 70 (2001), 1053-1061. doi: 10.1046/j.0021-8790.2001.00558.x. [19] D. T. Gillespie, "Markov Processes: An Introduction for Physical Scientists,'' Academic Press, Inc., Boston, MA, 1992. [20] J. V. Greenman and P. J. Hudson, Infected coexistence instability with and without density-dependent regulation, J. Theor. Biol., 185 (1997), 345-356. doi: 10.1006/jtbi.1996.0309. [21] M. Griffiths and D. Greenhalgh, The probability of extinction in a bovine respiratory syncytial virus epidemic model, Math. Biosci., 231 (2011), 144-158. doi: 10.1016/j.mbs.2011.02.011. [22] B. A. Han, "The Effects of an Emerging Pathogen on Amphibian Host Behaviors and Interactions," Ph.D thesis, Oregon State Univ., Corvallis, OR, 2009. [23] L. Han, Z. Ma and T. Shi, An SIRS epidemic model of two competitive species, Math. Comput. Model., 37 (2003), 87-108. doi: 10.1016/S0895-7177(03)80008-0. [24] L. Han and A. Pugliese, Epidemics in two competing species, Nonlinear Anal. Real World Appl., 10 (2009), 723-744. doi: 10.1016/j.nonrwa.2007.11.005. [25] T. E. Harris, "The Theory of Branching Processes,'' Die Grundlehren der Mathematischen Wissenschaften, Bd. 119, Springer-Verlag, Berlin, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1963. [26] M. J. Hatcher, J. T. A. Dick and A. M. Dunn, How parasites affect interactions between competitors and predators, Ecol. Lett., 9 (2006), 1253-1271. doi: 10.1111/j.1461-0248.2006.00964.x. [27] D. J. Higham, Modeling and simulating chemical reactions, SIAM Rev., 50 (2008), 347-368. doi: 10.1137/060666457. [28] R. D. Holt and A. P. Dobson, Chapter 2: Extending the principles of community ecology to address the epidemiology of host-pathogen systems, in "Disease Ecology: Community Structure and Pathogen Dynamics" (eds. S. K. Collinge and C. Ray), Oxford Univ. Press, Oxford, (2006), 2-27. [29] R. D. Holt and J. Pickering, Infectious disease and species coexistence: A model of Lotka-Volterra form, Am. Nat., 126 (1985), 196-211. doi: 10.1086/284409. [30] P. Hudson and J. Greenman, Competition mediated by parasites: Biological and theoretical progress, Trends Ecol. Evol., 13 (1998), 387-390. [31] P. Jagers, "Branching Processes with Biological Applications,'' Wiley Series in Probability and Mathematical Statistics Applied Probability and Statistics, Wiley-Interscience [John Wiley & Sons], London-New York-Sydney, 1975. [32] S. T. Karlin and H. M. Taylor, "A First Course in Stochastic Processes,'' 2nd edition, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. [33] J. M. Kiesecker and A. R. Blaustein, Pathogen reverses competition between larval amphibians, Ecology, 80 (1999), 2442-2448. [34] P. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,'' Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992. [35] N. Lanchier and C. Neuhauser, A spatially explicit model for competition among specialists and generalists in a heterogeneous environment, Ann. Appl. Probab., 16 (2006), 1385-1410. doi: 10.1214/105051606000000394. [36] N. Lanchier and C. Neuhauser, Stochastic spatial models of host-pathogen and host-mutualist interactions. I, Ann. Appl. Probab., 16 (2006), 448-474. doi: 10.1214/105051605000000782. [37] C. A. Manore, "Non-Spatial and Spatial Models for Multi-Host Pathogen Spread in Competing Species: Applications to Barley Yellow Dwarf Virus and Rinderpest," Ph.D thesis, Oregon State Univ., Corvallis, OR, 2012. [38] R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, Special issue on mathematics and the social and biological sciences, SIAM J. Appl. Math., 29 (1975), 243-253. doi: 10.1137/0129022. [39] H. McCallum, N. Barlow and J. Hone, How should pathogen transmission be modelled?, Trends Ecol. Evol., 16 (2001), 295-300. doi: 10.1016/S0169-5347(01)02144-9. [40] R. K. McCormack, "Multi-Host Multi-Patch Mathematical Epidemic Models for Disease Emergence with Applications to Hantavirus in Wild Rodents," Ph.D thesis, Texas Tech Univ., Lubbock, TX, 2006. [41] R. K. McCormack and L. J. S. Allen, Stochastic SIS and SIR multihost epidemic models, in "Differential & Difference Eqns. Appl.," Hindawi Publ. Corp., New York, (2006), 775-785. [42] R. K. McCormack and L. J. S. Allen, Disease emergence in multi-host epidemic models, Math. Med. Biol., 24 (2007), 17-34. doi: 10.1093/imammb/dql021. [43] C. J. Mode, "Multitype Branching Processes. Theory and Applications,'' Modern Analytic and Computational Methods in Science and Mathematics, No. 34 , American Elsevier Publishing Co., Inc., New York, 1971. [44] S. M. Moore, C. A. Manore, V. A. Bokil, E. T. Borer and P. R. Hosseini, Spatiotemporal model of barley and cereal yellow dwarf virus transmission dynamics with seasonality and plant competition, Bull. Math. Biol., 73 (2011), 2707-2730. [45] C. Neuhauser and S. W. Pacala, An explicitly spatial version of the Lotka-Volterra model with interspecific competition, Ann. Appl. Probab., 9 (1999), 1226-1259. [46] R. Norman, R. G. Bowers, M. Begon and P. J. Hudson, Persistence of tick-borne virus in the presence of multiple host species: Tick reservoirs and parasite mediated competition, J. Theor. Biol., 200 (1999), 111-118. doi: 10.1006/jtbi.1999.0982. [47] J. M. Ortega, "Matrix Theory. A Second Course," The University Series in Mathematics, Plenum Press, New York, 1987. [48] S. Pénisson, "Conditional Limit Theorems for Multitype Branching Processes and Illustration in Epidemiological Risk Analysis," Ph.D thesis, Institut für Mathematik der Unversität Potsdam, Germany, 2010. [49] R. A. Saenz and H. W. Hethcote, Competing species models with an infectious disease, Math. Biosci. Eng., 3 (2006), 219-235. [50] D. M. Tompkins, R. A. H. Draycott and P. J. Hudson, Field evidence for apparent competition mediated via the shared parasites of two gamebird species, Ecol. Lett., 3 (2000), 10-14. [51] D. M. Tompkins, A. R. White and M. Boots, Ecological replacement of native red squirrels by invasive greys driven by disease, Ecol. Lett., 6 (2003), 189-196. [52] 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. [53] P. van den Driessche and M. L. Zeeman, Disease induced oscillations between two competing species, SIAM J. Appl. Dyn. Sys., 3 (2004), 601-619. [54] E. Venturino, The effects of diseases on competing species, Math. Biosci., 174 (2001), 111-131. doi: 10.1016/S0025-5564(01)00081-5. [55] P. Whittle, The outcome of a stochastic epidemic: A note on Bailey's paper, Biometrika, 42 (1955), 116-122. doi: 10.2307/2333427. [56] E. C. Zeeman and M. L. Zeeman, From local to global behavior in competitive Lotka-Volterra systems, Trans. Am. Math. Soc., 355 (2003), 713-734. doi: 10.1090/S0002-9947-02-03103-3.

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##### References:
 [1] E. Allen, "Modeling With Itô Stochastic Differential Equations,'' Mathematical Modelling: Theory and Applications, 22, Springer, Dordrecht, The Netherlands, 2007. [2] E. J. Allen, L. J. S. Allen, A. Arciniega and P. Greenwood, Construction of equivalent stochastic differential equation models, Stoch. Anal. Appl., 26 (2008), 274-297. [3] L. J. S. Allen, "An Introduction to Mathematical Biology,'' Prentice Hall, Upper Saddle River, NJ, 2007. [4] L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology,'' 2nd edition, CRC Press, Boca Raton, FL, 2011. [5] L. J. S. Allen and N. Kirupaharan, Asymptotic dynamics of deterministic and stochastic epidemic models with multiple pathogens, Int. J. Numer. Anal. Modeling, 2 (2005), 329-344. [6] R. M. Anderson and R. M. May, The invasion, persistence and spread of infectious diseases with animal and plant communities, Phil. Trans. R. Soc. Lond. B, 314 (1986), 533-570. doi: 10.1098/rstb.1986.0072. [7] N. T. J. Bailey, "The Elements of Stochastic Processes with Applications to the Natural Sciences,'' Reprint of the 1964 original, Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1990. [8] M. S. Bartlett, The relevance of stochastic models for large-scale epidemiological phenomena, Appl. Statist., 13 (1965), 2-8. [9] M. Begon, R. G. Bowers, N. Kadianakis and D. E. Hodgkinson, Disease and community structure: The importance of host self-regulation in a host-host-pathogen model, Am. Nat., 139 (1992), 1131-1150. doi: 10.1086/285379. [10] V. A. Bokil and M.-R. Leung, An analysis of the coexistence of three competing species with a shared pathogen, Technical Report ORST-MATH-11-02, Oregon State Univ., 2011. Citation URL: http://ir.library.oregonstate.edu/xmlui/handle/1957/13738/. [11] E. T. Borer, P. R. Hosseini, E. W. Seabloom and A. P. Dobson, Pathogen-induced reversal of native dominance in a grassland community, Proc. Natl. Acad. Sci. U. S. A., 104 (2007), 5473-5478. [12] R. G. Bowers and J. Turner, Community structure and the interplay between interspecific infection and competition, J. Theor. Biol., 187 (1997), 95-109. doi: 10.1006/jtbi.1997.0418. [13] S. K. Collinge and C. Ray, "Disease Ecology: Community Structure and Pathogen Dynamics," Oxford Univ. Press, Oxford, 2006. [14] A. Dobson, Population dynamics of pathogens with multiple host species, Am. Nat., 164 (2004), S64-S78. doi: 10.1086/424681. [15] R. Durrett, Mutual invadability implies coexistence in spatial models, Mem. Am. Math. Soc., 156 (2002), viii+118 pp. [16] R. Durrett, Special invited paper: Coexistence in stochastic spatial models, Ann. Appl. Probab., 19 (2009), 477-496. doi: 10.1214/08-AAP590. [17] R. Durrett and C. Neuhauser, Coexistence results for some competition models, Ann. Appl. Probab., 7 (1997), 10-45. [18] L. Gilbert, R. Norman, K. M. Laurenson, H. W. Reid and P. J. Hudson, Disease persistence and apparent competition in a three-host community: An empirical and analytical study of large-scale, wild populations, J. Anim. Ecol., 70 (2001), 1053-1061. doi: 10.1046/j.0021-8790.2001.00558.x. [19] D. T. Gillespie, "Markov Processes: An Introduction for Physical Scientists,'' Academic Press, Inc., Boston, MA, 1992. [20] J. V. Greenman and P. J. Hudson, Infected coexistence instability with and without density-dependent regulation, J. Theor. Biol., 185 (1997), 345-356. doi: 10.1006/jtbi.1996.0309. [21] M. Griffiths and D. Greenhalgh, The probability of extinction in a bovine respiratory syncytial virus epidemic model, Math. Biosci., 231 (2011), 144-158. doi: 10.1016/j.mbs.2011.02.011. [22] B. A. Han, "The Effects of an Emerging Pathogen on Amphibian Host Behaviors and Interactions," Ph.D thesis, Oregon State Univ., Corvallis, OR, 2009. [23] L. Han, Z. Ma and T. Shi, An SIRS epidemic model of two competitive species, Math. Comput. Model., 37 (2003), 87-108. doi: 10.1016/S0895-7177(03)80008-0. [24] L. Han and A. Pugliese, Epidemics in two competing species, Nonlinear Anal. Real World Appl., 10 (2009), 723-744. doi: 10.1016/j.nonrwa.2007.11.005. [25] T. E. Harris, "The Theory of Branching Processes,'' Die Grundlehren der Mathematischen Wissenschaften, Bd. 119, Springer-Verlag, Berlin, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1963. [26] M. J. Hatcher, J. T. A. Dick and A. M. Dunn, How parasites affect interactions between competitors and predators, Ecol. Lett., 9 (2006), 1253-1271. doi: 10.1111/j.1461-0248.2006.00964.x. [27] D. J. Higham, Modeling and simulating chemical reactions, SIAM Rev., 50 (2008), 347-368. doi: 10.1137/060666457. [28] R. D. Holt and A. P. Dobson, Chapter 2: Extending the principles of community ecology to address the epidemiology of host-pathogen systems, in "Disease Ecology: Community Structure and Pathogen Dynamics" (eds. S. K. Collinge and C. Ray), Oxford Univ. Press, Oxford, (2006), 2-27. [29] R. D. Holt and J. Pickering, Infectious disease and species coexistence: A model of Lotka-Volterra form, Am. Nat., 126 (1985), 196-211. doi: 10.1086/284409. [30] P. Hudson and J. Greenman, Competition mediated by parasites: Biological and theoretical progress, Trends Ecol. Evol., 13 (1998), 387-390. [31] P. Jagers, "Branching Processes with Biological Applications,'' Wiley Series in Probability and Mathematical Statistics Applied Probability and Statistics, Wiley-Interscience [John Wiley & Sons], London-New York-Sydney, 1975. [32] S. T. Karlin and H. M. Taylor, "A First Course in Stochastic Processes,'' 2nd edition, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. [33] J. M. Kiesecker and A. R. Blaustein, Pathogen reverses competition between larval amphibians, Ecology, 80 (1999), 2442-2448. [34] P. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,'' Applications of Mathematics (New York), 23, Springer-Verlag, Berlin, 1992. [35] N. Lanchier and C. Neuhauser, A spatially explicit model for competition among specialists and generalists in a heterogeneous environment, Ann. Appl. Probab., 16 (2006), 1385-1410. doi: 10.1214/105051606000000394. [36] N. Lanchier and C. Neuhauser, Stochastic spatial models of host-pathogen and host-mutualist interactions. I, Ann. Appl. Probab., 16 (2006), 448-474. doi: 10.1214/105051605000000782. [37] C. A. Manore, "Non-Spatial and Spatial Models for Multi-Host Pathogen Spread in Competing Species: Applications to Barley Yellow Dwarf Virus and Rinderpest," Ph.D thesis, Oregon State Univ., Corvallis, OR, 2012. [38] R. M. May and W. J. Leonard, Nonlinear aspects of competition between three species, Special issue on mathematics and the social and biological sciences, SIAM J. Appl. Math., 29 (1975), 243-253. doi: 10.1137/0129022. [39] H. McCallum, N. Barlow and J. Hone, How should pathogen transmission be modelled?, Trends Ecol. Evol., 16 (2001), 295-300. doi: 10.1016/S0169-5347(01)02144-9. [40] R. K. McCormack, "Multi-Host Multi-Patch Mathematical Epidemic Models for Disease Emergence with Applications to Hantavirus in Wild Rodents," Ph.D thesis, Texas Tech Univ., Lubbock, TX, 2006. [41] R. K. McCormack and L. J. S. Allen, Stochastic SIS and SIR multihost epidemic models, in "Differential & Difference Eqns. Appl.," Hindawi Publ. Corp., New York, (2006), 775-785. [42] R. K. McCormack and L. J. S. Allen, Disease emergence in multi-host epidemic models, Math. Med. Biol., 24 (2007), 17-34. doi: 10.1093/imammb/dql021. [43] C. J. Mode, "Multitype Branching Processes. Theory and Applications,'' Modern Analytic and Computational Methods in Science and Mathematics, No. 34 , American Elsevier Publishing Co., Inc., New York, 1971. [44] S. M. Moore, C. A. Manore, V. A. Bokil, E. T. Borer and P. R. Hosseini, Spatiotemporal model of barley and cereal yellow dwarf virus transmission dynamics with seasonality and plant competition, Bull. Math. Biol., 73 (2011), 2707-2730. [45] C. Neuhauser and S. W. Pacala, An explicitly spatial version of the Lotka-Volterra model with interspecific competition, Ann. Appl. Probab., 9 (1999), 1226-1259. [46] R. Norman, R. G. Bowers, M. Begon and P. J. Hudson, Persistence of tick-borne virus in the presence of multiple host species: Tick reservoirs and parasite mediated competition, J. Theor. Biol., 200 (1999), 111-118. doi: 10.1006/jtbi.1999.0982. [47] J. M. Ortega, "Matrix Theory. A Second Course," The University Series in Mathematics, Plenum Press, New York, 1987. [48] S. Pénisson, "Conditional Limit Theorems for Multitype Branching Processes and Illustration in Epidemiological Risk Analysis," Ph.D thesis, Institut für Mathematik der Unversität Potsdam, Germany, 2010. [49] R. A. Saenz and H. W. Hethcote, Competing species models with an infectious disease, Math. Biosci. Eng., 3 (2006), 219-235. [50] D. M. Tompkins, R. A. H. Draycott and P. J. Hudson, Field evidence for apparent competition mediated via the shared parasites of two gamebird species, Ecol. Lett., 3 (2000), 10-14. [51] D. M. Tompkins, A. R. White and M. Boots, Ecological replacement of native red squirrels by invasive greys driven by disease, Ecol. Lett., 6 (2003), 189-196. [52] 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. [53] P. van den Driessche and M. L. Zeeman, Disease induced oscillations between two competing species, SIAM J. Appl. Dyn. Sys., 3 (2004), 601-619. [54] E. Venturino, The effects of diseases on competing species, Math. Biosci., 174 (2001), 111-131. doi: 10.1016/S0025-5564(01)00081-5. [55] P. Whittle, The outcome of a stochastic epidemic: A note on Bailey's paper, Biometrika, 42 (1955), 116-122. doi: 10.2307/2333427. [56] E. C. Zeeman and M. L. Zeeman, From local to global behavior in competitive Lotka-Volterra systems, Trans. Am. Math. Soc., 355 (2003), 713-734. doi: 10.1090/S0002-9947-02-03103-3.
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