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A comparison of computational efficiencies of stochastic algorithms in terms of two infection models
Stochastic models for competing species with a shared pathogen
1.  Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 794091042, United States 
2.  Department of Mathematics, Oregon State University, Corvallis, OR 973314605, United States 
References:
[1] 
E. Allen, "Modeling With Itô Stochastic Differential Equations,'', Mathematical Modelling: Theory and Applications, 22 (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. 
[3] 
L. J. S. Allen, "An Introduction to Mathematical Biology,'', Prentice Hall, (2007). 
[4] 
L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology,'', 2^{nd} edition, (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. 
[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. 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, (1964). 
[8] 
M. S. Bartlett, The relevance of stochastic models for largescale epidemiological phenomena,, Appl. Statist., 13 (1965), 2. 
[9] 
M. Begon, R. G. Bowers, N. Kadianakis and D. E. Hodgkinson, Disease and community structure: The importance of host selfregulation in a hosthostpathogen model,, Am. Nat., 139 (1992), 1131. 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 ORSTMATH1102, (2011), 11. 
[11] 
E. T. Borer, P. R. Hosseini, E. W. Seabloom and A. P. Dobson, Pathogeninduced reversal of native dominance in a grassland community,, Proc. Natl. Acad. Sci. U. S. A., 104 (2007), 5473. 
[12] 
R. G. Bowers and J. Turner, Community structure and the interplay between interspecific infection and competition,, J. Theor. Biol., 187 (1997), 95. doi: 10.1006/jtbi.1997.0418. 
[13] 
S. K. Collinge and C. Ray, "Disease Ecology: Community Structure and Pathogen Dynamics,", Oxford Univ. Press, (2006). 
[14] 
A. Dobson, Population dynamics of pathogens with multiple host species,, Am. Nat., 164 (2004). doi: 10.1086/424681. 
[15] 
R. Durrett, Mutual invadability implies coexistence in spatial models,, Mem. Am. Math. Soc., 156 (2002). 
[16] 
R. Durrett, Special invited paper: Coexistence in stochastic spatial models,, Ann. Appl. Probab., 19 (2009), 477. doi: 10.1214/08AAP590. 
[17] 
R. Durrett and C. Neuhauser, Coexistence results for some competition models,, Ann. Appl. Probab., 7 (1997), 10. 
[18] 
L. Gilbert, R. Norman, K. M. Laurenson, H. W. Reid and P. J. Hudson, Disease persistence and apparent competition in a threehost community: An empirical and analytical study of largescale, wild populations,, J. Anim. Ecol., 70 (2001), 1053. doi: 10.1046/j.00218790.2001.00558.x. 
[19] 
D. T. Gillespie, "Markov Processes: An Introduction for Physical Scientists,'', Academic Press, (1992). 
[20] 
J. V. Greenman and P. J. Hudson, Infected coexistence instability with and without densitydependent regulation,, J. Theor. Biol., 185 (1997), 345. 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. 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, (2009). 
[23] 
L. Han, Z. Ma and T. Shi, An SIRS epidemic model of two competitive species,, Math. Comput. Model., 37 (2003), 87. doi: 10.1016/S08957177(03)800080. 
[24] 
L. Han and A. Pugliese, Epidemics in two competing species,, Nonlinear Anal. Real World Appl., 10 (2009), 723. doi: 10.1016/j.nonrwa.2007.11.005. 
[25] 
T. E. Harris, "The Theory of Branching Processes,'', Die Grundlehren der Mathematischen Wissenschaften, (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. doi: 10.1111/j.14610248.2006.00964.x. 
[27] 
D. J. Higham, Modeling and simulating chemical reactions,, SIAM Rev., 50 (2008), 347. 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 hostpathogen systems, in "Disease Ecology: Community Structure and Pathogen Dynamics" (eds. S. K. Collinge and C. Ray),, Oxford Univ. Press, (2006), 2. 
[29] 
R. D. Holt and J. Pickering, Infectious disease and species coexistence: A model of LotkaVolterra form,, Am. Nat., 126 (1985), 196. doi: 10.1086/284409. 
[30] 
P. Hudson and J. Greenman, Competition mediated by parasites: Biological and theoretical progress,, Trends Ecol. Evol., 13 (1998), 387. 
[31] 
P. Jagers, "Branching Processes with Biological Applications,'', Wiley Series in Probability and Mathematical Statistics Applied Probability and Statistics, (1975). 
[32] 
S. T. Karlin and H. M. Taylor, "A First Course in Stochastic Processes,'', 2^{nd} edition, (1975). 
[33] 
J. M. Kiesecker and A. R. Blaustein, Pathogen reverses competition between larval amphibians,, Ecology, 80 (1999), 2442. 
[34] 
P. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,'', Applications of Mathematics (New York), 23 (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. doi: 10.1214/105051606000000394. 
[36] 
N. Lanchier and C. Neuhauser, Stochastic spatial models of hostpathogen and hostmutualist interactions. I,, Ann. Appl. Probab., 16 (2006), 448. doi: 10.1214/105051605000000782. 
[37] 
C. A. Manore, "NonSpatial and Spatial Models for MultiHost Pathogen Spread in Competing Species: Applications to Barley Yellow Dwarf Virus and Rinderpest,", Ph.D thesis, (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, 29 (1975), 243. doi: 10.1137/0129022. 
[39] 
H. McCallum, N. Barlow and J. Hone, How should pathogen transmission be modelled?,, Trends Ecol. Evol., 16 (2001), 295. doi: 10.1016/S01695347(01)021449. 
[40] 
R. K. McCormack, "MultiHost MultiPatch Mathematical Epidemic Models for Disease Emergence with Applications to Hantavirus in Wild Rodents,", Ph.D thesis, (2006). 
[41] 
R. K. McCormack and L. J. S. Allen, Stochastic SIS and SIR multihost epidemic models,, in, (2006), 775. 
[42] 
R. K. McCormack and L. J. S. Allen, Disease emergence in multihost epidemic models,, Math. Med. Biol., 24 (2007), 17. doi: 10.1093/imammb/dql021. 
[43] 
C. J. Mode, "Multitype Branching Processes. Theory and Applications,'', Modern Analytic and Computational Methods in Science and Mathematics, (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. 
[45] 
C. Neuhauser and S. W. Pacala, An explicitly spatial version of the LotkaVolterra model with interspecific competition,, Ann. Appl. Probab., 9 (1999), 1226. 
[46] 
R. Norman, R. G. Bowers, M. Begon and P. J. Hudson, Persistence of tickborne virus in the presence of multiple host species: Tick reservoirs and parasite mediated competition,, J. Theor. Biol., 200 (1999), 111. doi: 10.1006/jtbi.1999.0982. 
[47] 
J. M. Ortega, "Matrix Theory. A Second Course,", The University Series in Mathematics, (1987). 
[48] 
S. Pénisson, "Conditional Limit Theorems for Multitype Branching Processes and Illustration in Epidemiological Risk Analysis,", Ph.D thesis, (2010). 
[49] 
R. A. Saenz and H. W. Hethcote, Competing species models with an infectious disease,, Math. Biosci. Eng., 3 (2006), 219. 
[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. 
[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. 
[52] 
P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. 
[53] 
P. van den Driessche and M. L. Zeeman, Disease induced oscillations between two competing species,, SIAM J. Appl. Dyn. Sys., 3 (2004), 601. 
[54] 
E. Venturino, The effects of diseases on competing species,, Math. Biosci., 174 (2001), 111. doi: 10.1016/S00255564(01)000815. 
[55] 
P. Whittle, The outcome of a stochastic epidemic: A note on Bailey's paper,, Biometrika, 42 (1955), 116. doi: 10.2307/2333427. 
[56] 
E. C. Zeeman and M. L. Zeeman, From local to global behavior in competitive LotkaVolterra systems,, Trans. Am. Math. Soc., 355 (2003), 713. doi: 10.1090/S0002994702031033. 
show all references
References:
[1] 
E. Allen, "Modeling With Itô Stochastic Differential Equations,'', Mathematical Modelling: Theory and Applications, 22 (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. 
[3] 
L. J. S. Allen, "An Introduction to Mathematical Biology,'', Prentice Hall, (2007). 
[4] 
L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology,'', 2^{nd} edition, (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. 
[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. 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, (1964). 
[8] 
M. S. Bartlett, The relevance of stochastic models for largescale epidemiological phenomena,, Appl. Statist., 13 (1965), 2. 
[9] 
M. Begon, R. G. Bowers, N. Kadianakis and D. E. Hodgkinson, Disease and community structure: The importance of host selfregulation in a hosthostpathogen model,, Am. Nat., 139 (1992), 1131. 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 ORSTMATH1102, (2011), 11. 
[11] 
E. T. Borer, P. R. Hosseini, E. W. Seabloom and A. P. Dobson, Pathogeninduced reversal of native dominance in a grassland community,, Proc. Natl. Acad. Sci. U. S. A., 104 (2007), 5473. 
[12] 
R. G. Bowers and J. Turner, Community structure and the interplay between interspecific infection and competition,, J. Theor. Biol., 187 (1997), 95. doi: 10.1006/jtbi.1997.0418. 
[13] 
S. K. Collinge and C. Ray, "Disease Ecology: Community Structure and Pathogen Dynamics,", Oxford Univ. Press, (2006). 
[14] 
A. Dobson, Population dynamics of pathogens with multiple host species,, Am. Nat., 164 (2004). doi: 10.1086/424681. 
[15] 
R. Durrett, Mutual invadability implies coexistence in spatial models,, Mem. Am. Math. Soc., 156 (2002). 
[16] 
R. Durrett, Special invited paper: Coexistence in stochastic spatial models,, Ann. Appl. Probab., 19 (2009), 477. doi: 10.1214/08AAP590. 
[17] 
R. Durrett and C. Neuhauser, Coexistence results for some competition models,, Ann. Appl. Probab., 7 (1997), 10. 
[18] 
L. Gilbert, R. Norman, K. M. Laurenson, H. W. Reid and P. J. Hudson, Disease persistence and apparent competition in a threehost community: An empirical and analytical study of largescale, wild populations,, J. Anim. Ecol., 70 (2001), 1053. doi: 10.1046/j.00218790.2001.00558.x. 
[19] 
D. T. Gillespie, "Markov Processes: An Introduction for Physical Scientists,'', Academic Press, (1992). 
[20] 
J. V. Greenman and P. J. Hudson, Infected coexistence instability with and without densitydependent regulation,, J. Theor. Biol., 185 (1997), 345. 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. 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, (2009). 
[23] 
L. Han, Z. Ma and T. Shi, An SIRS epidemic model of two competitive species,, Math. Comput. Model., 37 (2003), 87. doi: 10.1016/S08957177(03)800080. 
[24] 
L. Han and A. Pugliese, Epidemics in two competing species,, Nonlinear Anal. Real World Appl., 10 (2009), 723. doi: 10.1016/j.nonrwa.2007.11.005. 
[25] 
T. E. Harris, "The Theory of Branching Processes,'', Die Grundlehren der Mathematischen Wissenschaften, (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. doi: 10.1111/j.14610248.2006.00964.x. 
[27] 
D. J. Higham, Modeling and simulating chemical reactions,, SIAM Rev., 50 (2008), 347. 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 hostpathogen systems, in "Disease Ecology: Community Structure and Pathogen Dynamics" (eds. S. K. Collinge and C. Ray),, Oxford Univ. Press, (2006), 2. 
[29] 
R. D. Holt and J. Pickering, Infectious disease and species coexistence: A model of LotkaVolterra form,, Am. Nat., 126 (1985), 196. doi: 10.1086/284409. 
[30] 
P. Hudson and J. Greenman, Competition mediated by parasites: Biological and theoretical progress,, Trends Ecol. Evol., 13 (1998), 387. 
[31] 
P. Jagers, "Branching Processes with Biological Applications,'', Wiley Series in Probability and Mathematical Statistics Applied Probability and Statistics, (1975). 
[32] 
S. T. Karlin and H. M. Taylor, "A First Course in Stochastic Processes,'', 2^{nd} edition, (1975). 
[33] 
J. M. Kiesecker and A. R. Blaustein, Pathogen reverses competition between larval amphibians,, Ecology, 80 (1999), 2442. 
[34] 
P. Kloeden and E. Platen, "Numerical Solution of Stochastic Differential Equations,'', Applications of Mathematics (New York), 23 (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. doi: 10.1214/105051606000000394. 
[36] 
N. Lanchier and C. Neuhauser, Stochastic spatial models of hostpathogen and hostmutualist interactions. I,, Ann. Appl. Probab., 16 (2006), 448. doi: 10.1214/105051605000000782. 
[37] 
C. A. Manore, "NonSpatial and Spatial Models for MultiHost Pathogen Spread in Competing Species: Applications to Barley Yellow Dwarf Virus and Rinderpest,", Ph.D thesis, (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, 29 (1975), 243. doi: 10.1137/0129022. 
[39] 
H. McCallum, N. Barlow and J. Hone, How should pathogen transmission be modelled?,, Trends Ecol. Evol., 16 (2001), 295. doi: 10.1016/S01695347(01)021449. 
[40] 
R. K. McCormack, "MultiHost MultiPatch Mathematical Epidemic Models for Disease Emergence with Applications to Hantavirus in Wild Rodents,", Ph.D thesis, (2006). 
[41] 
R. K. McCormack and L. J. S. Allen, Stochastic SIS and SIR multihost epidemic models,, in, (2006), 775. 
[42] 
R. K. McCormack and L. J. S. Allen, Disease emergence in multihost epidemic models,, Math. Med. Biol., 24 (2007), 17. doi: 10.1093/imammb/dql021. 
[43] 
C. J. Mode, "Multitype Branching Processes. Theory and Applications,'', Modern Analytic and Computational Methods in Science and Mathematics, (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. 
[45] 
C. Neuhauser and S. W. Pacala, An explicitly spatial version of the LotkaVolterra model with interspecific competition,, Ann. Appl. Probab., 9 (1999), 1226. 
[46] 
R. Norman, R. G. Bowers, M. Begon and P. J. Hudson, Persistence of tickborne virus in the presence of multiple host species: Tick reservoirs and parasite mediated competition,, J. Theor. Biol., 200 (1999), 111. doi: 10.1006/jtbi.1999.0982. 
[47] 
J. M. Ortega, "Matrix Theory. A Second Course,", The University Series in Mathematics, (1987). 
[48] 
S. Pénisson, "Conditional Limit Theorems for Multitype Branching Processes and Illustration in Epidemiological Risk Analysis,", Ph.D thesis, (2010). 
[49] 
R. A. Saenz and H. W. Hethcote, Competing species models with an infectious disease,, Math. Biosci. Eng., 3 (2006), 219. 
[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. 
[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. 
[52] 
P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. 
[53] 
P. van den Driessche and M. L. Zeeman, Disease induced oscillations between two competing species,, SIAM J. Appl. Dyn. Sys., 3 (2004), 601. 
[54] 
E. Venturino, The effects of diseases on competing species,, Math. Biosci., 174 (2001), 111. doi: 10.1016/S00255564(01)000815. 
[55] 
P. Whittle, The outcome of a stochastic epidemic: A note on Bailey's paper,, Biometrika, 42 (1955), 116. doi: 10.2307/2333427. 
[56] 
E. C. Zeeman and M. L. Zeeman, From local to global behavior in competitive LotkaVolterra systems,, Trans. Am. Math. Soc., 355 (2003), 713. doi: 10.1090/S0002994702031033. 
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