Local approximation of Markov chains in time and space

References

• L.J.S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Pearson, Upper Saddle River, NJ, 2003.
   Google Scholar
• L.J.S. Allen and A.M. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. Biosci. 163(1) (2000), pp. 1–33.
   PubMed Web of Science ®Google Scholar
• L.J.S. Allen and G.E. Lahodny, Extinction thresholds in deterministic and stochastic epidemic models, J. Biol. Dyn. 6(2) (2012), pp. 590–611.
   PubMed Web of Science ®Google Scholar
• L.J.S. Allen and E.J. Schwartz, Free-virus and cell-to-cell transmission in models of equine infectious anemia virus infection, Math. Biosci. 270 (2015), pp. 237–248.
   PubMed Web of Science ®Google Scholar
• L.J.S. Allen and P. van den Driessche, Relations between deterministic and stochastic thresholds for disease extinction in continuous- and discrete-time infectious disease models, Math. Biosci. 243(1) (2013), pp. 99–108.
   PubMed Web of Science ®Google Scholar
• J. Arino, Diseases in metapopulations, in Modeling and Dynamics of Infectious Diseases, Vol. 11, Z. Ma, Y. Zhou, and J. Wu, eds., World Scientific, New Jersey, 2009, pp. 65–123.
   Google Scholar
• J. Arino, Spatio-temporal spread of infectious pathogens of humans, Infect. Dis. Model. 2(2) (2017), pp. 218–228.
   PubMedGoogle Scholar
• J. Arino and S. Portet, Epidemiological implications of mobility between a large urban centre and smaller satellite cities, J. Math. Biol. 71(5) (2015), pp. 1243–1265.
   PubMed Web of Science ®Google Scholar
• J. Arino and P. van den Driessche, A multi-city epidemic model, Math. Popul. Stud. 10(3) (2003), pp. 175–193.
   Google Scholar
• F. Arrigoni and A. Pugliese, Limits of a multi-patch SIS epidemic model, J. Math. Biol. 45(5) (2002), pp. 419–440.
   PubMed Web of Science ®Google Scholar
• N.T.J. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications, 2nd ed., Hafner, New York, 1975.
   Google Scholar
• F.G. Ball and P. Donnelly, Strong approximations for epidemic models, Stoch. Proc. Appl. 55(1) (1995), pp. 1–21.
   Web of Science ®Google Scholar
• F. Ball, B. Tom, and L. Owen, Stochastic multitype epidemics in a community of households: Estimation and form of optimal vaccination schemes, Math. Biosci. 191(1) (2004), pp. 19–40.
   PubMed Web of Science ®Google Scholar
• M. Behzad and G. Chartrand, Introduction to the Theory of Graphs, Allyn and Bacon, Inc., Boston, 1971.
   Google Scholar
• M. Bóna, A walk through combinatorics: An introduction to enumeration and graph theory, World Scientific, New Jersey, 2013.
   Google Scholar
• C.J. Burke and M. Rosenblatt, A Markovian function of a Markov chain, Ann. Math. Statist. 29(4) (1958), pp. 1112–1122.
   Google Scholar
• A. Dey and A. Mukherjea, Collapsing of non-homogeneous Markov chains, Statist. Probab. Lett. 84(1) (2014), pp. 140–148.
   Google Scholar
• T. Dhirasakdanon, H.R. Thieme, and P. Van Den Driessche, A sharp threshold for disease persistence in host metapopulations, J. Biol. Dyn. 1(4) (2007), pp. 363–378.
   PubMedGoogle Scholar
• O. Diekmann, J. Heesterbeek, and J. Metz, On the definition and computation of the basic reproduction ratio R0 in models of infectious disease in heterogeneous populations, J. Math. Biol. 28 (1990), pp. 365–382.
   PubMed Web of Science ®Google Scholar
• R. Durrett, Essentials of Stochastic Processes, 2nd ed., Springer-Verlag, New York, 2012.
   Google Scholar
• K. Falk, E. Namork, E. Rimstad, S. Mjaaland, and B.H. Dannevig, Characterization of infectious salmon anemia virus, an orthomyxo-like virus isolated from Atlantic salmon (Salmo salar L.), J. Virol. 71(12) (1997), pp. 9016–23.
   PubMed Web of Science ®Google Scholar
• I.I. Gikhman and A.V. Skorokhod, Introduction to the Theory of Random Processes, Dover, Mineola, NY, 1996.
   Google Scholar
• S.A. Gourley, H.R. Thieme, and P. van den Driessche, Stability and persistence in a model for bluetongue dynamics, SIAM J. Appl. Math. 71(4) (2011), pp. 1280–1306.
   Web of Science ®Google Scholar
• A. Gray, D. Greenhalgh, L. Hu, X. Mao, and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math. 71(3) (2011), pp. 876–902.
   Web of Science ®Google Scholar
• J. Hachigian, Collapsed Markov chains and the Chapman-Kolmogorov equation, Ann. Math. Statist. 34(1) (1963), pp. 233–237.
   Google Scholar
• T.E. Harris, The Theory of Branching Processes, Springer-Verlag, Berlin, 1963.
   Google Scholar
• A. Iggidr, G. Sallet, and B. Tsanou, Global stability analysis of a metapopulation SIS epidemic model, Math. Popul. Stud. 19(3) (2012), pp. 115–129.
   Web of Science ®Google Scholar
• M. Jesse, R. Mazzucco, U. Dieckmann, H. Heesterbeek, and J.A.J. Metz, Invasion and persistence of infectious agents in fragmented host populations, PLoS ONE 6(9) (2011).
   Web of Science ®Google Scholar
• S. Karlin and H.M. Taylor, A First Course in Stochastic Processes, 2nd ed., Academic Press, New York, 1975.
   Google Scholar
• M.J. Keeling, Metapopulation moments: Coupling, stochasticity and persistence, J. Anim. Ecol. 69(5) (2000), pp. 725–736.
   PubMed Web of Science ®Google Scholar
• M.J. Keeling, C.A. Gilligan, and M.J. Keeling, Metapopulation dynamics of bubonic plague, Nature 407(6806) (2000), pp. 903–906.
   PubMed Web of Science ®Google Scholar
• M.J. Keeling and J.V. Ross, On methods for studying stochastic disease dynamics, J. R. Soc. Interface 5(19) (2008), pp. 171–181.
   PubMed Web of Science ®Google Scholar
• J.G. Kemeny and J.L. Snell, Finite Markov Chains, Springer-Verlag, New York, 1976.
   Google Scholar
• W.O. Kermack and A.G. McKendrick, Contributions to the mathematical theory of epidemics – I, Proc. R. Soc. A 115(772) (1927), pp. 700–721.
   Google Scholar
• A. Khanafer and B. Tamer, An optimal control problem over infected networks, Proceedings of the International Conference of Control, Dynamic Systems, and Robotics, Ottawa, Ontario, Canada, 2014.
   Google Scholar
• A.L. Krause, L. Kurowski, K. Yawar, and R.A. Van Gorder, Stochastic epidemic metapopulation models on networks: SIS dynamics and control strategies, J. Theor. Biol. 449 (2018), pp. 35–52.
   PubMed Web of Science ®Google Scholar
• G.E. Lahodny and L.J.S. Allen, Probability of a disease outbreak in stochastic multipatch epidemic models, Bull. Math. Biol. 75(7) (2013), pp. 1157–1180.
   PubMed Web of Science ®Google Scholar
• E. Milliken, The probability of extinction of infectious salmon anemia virus in one and two patches, Bull. Math. Biol. 79 (2017), pp. 2887–2904.
   PubMed Web of Science ®Google Scholar
• E. Milliken and S.S. Pilyugin, A model of infectious salmon anemia virus with viral diffusion between wild and farmed patches, DCDS-B 21(6) (2016), pp. 1869–1893.
   Web of Science ®Google Scholar
• J.R. Norris, Markov Chains, Cambridge University Press, New York, 1997.
   Google Scholar
• M.A. Pinsky and S. Karlin, Introduction to Stochastic Modeling, Academic Press, Boston, 2011.
   Google Scholar
• M. Roseblatt, Functions of a Markov process that are Markovian, J. Math. Mech. 8(4) (1959), pp. 585–596.
   Google Scholar
• M. Rosenblatt, Random Processes, Oxford University Press, New York, 1962.
   Google Scholar
• R. Ross, An application of the theory of probabilities to the study of a priori pathometry. Part I, Proc. R. Soc. A 92(638) (1916), pp. 204–230.
   Google Scholar
• P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), pp. 29–48.
   PubMed Web of Science ®Google Scholar
• S. Vike, S. Nylund, and A. Nylund, ISA virus in Chile: Evidence of vertical transmission, Arch. Virol. 154(1) (2009), pp. 1–8.
   PubMed Web of Science ®Google Scholar
• W. Wang and X.-Q. Zhao, An epidemic model in a patchy environment, Math. Biosci. 190(1) (2004), pp. 97–112.
   PubMed Web of Science ®Google Scholar
• H.W. Watson and F. Galton, On the probability of extinction of families, J. Anthropol. Inst. GB Ireland 4 (1875), pp. 138–144.
   Google Scholar
• D.B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, NJ, 1996.
   Google Scholar
• R.W. West and J.R. Thompson, Models for the simple epidemic, Math. Biosci. 141 (1997), pp. 29–39.
   PubMed Web of Science ®Google Scholar
• P. Whittle, The outcome of a stochastic epidemic – a note on Bailey's paper, Biometrika 42 (1955), pp. 116–122.
   Web of Science ®Google Scholar