# American Institute of Mathematical Sciences

June  2015, 20(4): 989-1013. doi: 10.3934/dcdsb.2015.20.989

## Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment

 1 Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275 2 College of Mathematics and Information Science, Wenzhou University, Wenzhou, 325035

Received  April 2014 Revised  July 2014 Published  February 2015

In this paper, we explore a parasite-host epidemiological model incorporating demographic and epidemiological processes in a spatially heterogeneous environment in which the individuals are subject to a random movement. We show the global stability of the extinction equilibrium in three different cases, and prove the existence, uniqueness and the global stability of the disease--free equilibrium. When the death rate in the model becomes a constant, we give the existence of the endemic equilibrium and the global stability of the endemic equilibrium in a special case. Furthermore, we perform a series of numerical simulations to display the effects of the movement of hosts and the heterogeneous environment on the disease dynamics. Our analytical and numerical results reveal that the disease extinction/outbreak can be ignited by both individual mobility and the environmental heterogeneity.
Citation: Yongli Cai, Weiming Wang. Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 989-1013. doi: 10.3934/dcdsb.2015.20.989
##### References:
 [1] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM Journal on Applied Mathematics, 67 (2007), 1283-1309. doi: 10.1137/060672522. [2] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete and Continuous Dynamical Systems, 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1. [3] M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Mathematical Biosciences, 189 (2004), 75-96. doi: 10.1016/j.mbs.2004.01.003. [4] R. M. Anderson, R. M. May and B. Anderson, Infectious Diseases of Humans: Dynamics and Control, Wiley Online Library, 1992. [5] S. Anita and V. Capasso, A stabilizability problem for a reaction-diffusion system modelling a class of spatially structured epidemic systems, Nonlinear Analysis: Real World Applications, 3 (2002), 453-464. doi: 10.1016/S1468-1218(01)00025-6. [6] C. Bain, Applied mathematical ecology, Journal of Epidemiology and Community Health, 44 (1990), p254. doi: 10.1136/jech.44.3.254-b. [7] F. Berezovsky, G. Karev, B. Song and C. Castillo-Chavez, A simple epidemic model with surprising dynamics, Mathematical Biosciences and Engineering, 2 (2005), 133-152. [8] E. Beretta and Y. Takeuchi, Global stability of an SIR epidemic model with time delays, Journal of Mathematical Biology, 33 (1995), 250-260. doi: 10.1007/BF00169563. [9] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, John Wiley & Sons, 2003. doi: 10.1002/0470871296. [10] V. Capasso, Mathematical Structures of Epidemic Systems, Springer, 1993. doi: 10.1007/978-3-540-70514-7. [11] V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Mathematical Biosciences, 42 (1978), 43-61. doi: 10.1016/0025-5564(78)90006-8. [12] C. Castillo-Chavez and A.-A. Yakubu, Dispersal, disease and life-history evolution, Mathematical Biosciences, 173 (2001), 35-53. doi: 10.1016/S0025-5564(01)00065-7. [13] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. [14] D. Ebert, M. Lipsitch and K. L. Mangin, The effect of parasites on host population density and extinction: experimental epidemiology with daphnia and six microparasites, The American Naturalist, 156 (2000), 459-477. [15] W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model of the spread of feline leukemia virus (FeLV) through a highly heterogeneous spatial domain, SIAM Journal on Mathematical Analysis, 33 (2001), 570-588. doi: 10.1137/S0036141000371757. [16] W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A reaction-diffusion system modeling direct and indirect transmission of diseases, Discrete and Continuous Dynamical Systems-B, 4 (2004), 893-910. doi: 10.3934/dcdsb.2004.4.893. [17] W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model for indirectly transmitted diseases, Mathematical Biosciences, 206 (2007), 233-248. doi: 10.1016/j.mbs.2005.07.005. [18] B. Fred and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology(Second Edition), Springer, 2012. doi: 10.1007/978-1-4614-1686-9. [19] D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840. Springer-Verlag Berlin, 1981. [20] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [21] H. W. Hethcote and S. A. Levin, Periodicity in epidemiological models, In Applied Mathematical Ecology, Springer, (1989), 193-211. doi: 10.1007/978-3-642-61317-3_8. [22] W. Huang, M. Han and K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Mathematical Biosciences and Engineering, 7 (2010), 51-66. doi: 10.3934/mbe.2010.7.51. [23] T. W. Hwang and Y. Kuang, Deterministic extinction effect of parasites on host populations, Journal of Mathematical Biology, 46 (2003), 17-30. doi: 10.1007/s00285-002-0165-7. [24] T. W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model, Mathematical Biosciences and Engineering, 2 (2005), 743-751. doi: 10.3934/mbe.2005.2.743. [25] Y. Kang and C. Castillo-Chavez, A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness, Discrete and Continuous Dynamical Systems-B, 19 (2014), 89-130. [26] Y. Kang, S. K. Sasmal, A. R. Bhowmick and J. Chattopadhyay, Dynamics of a predator-prey system with prey subject to Allee effects and disease, Mathematical Biosciences and Engineering, 11 (2014), 877-918. doi: 10.3934/mbe.2014.11.877. [27] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics-I, Bulletin of Mathematical Biology, 53 (1991), 33-55. [28] K. I. Kim, Z. Lin and Q. Zhang, An SIR epidemic model with free boundary, Nonlinear Analysis: Real World Applications, 14 (2013), 1992-2001. doi: 10.1016/j.nonrwa.2013.02.003. [29] A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models, Mathematical Medicine and Biology, 22 (2005), 113-128. doi: 10.1093/imammb/dqi001. [30] X. Z. Li, W. S. Li and M. Ghosh, Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment, Applied Mathematics and Computation, 210 (2009), 141-150. doi: 10.1016/j.amc.2008.12.085. [31] W. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of sirs epidemiological models, Journal of Mathematical Biology, 23 (1986), 187-204. doi: 10.1007/BF00276956. [32] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, CRC Press, 2010. [33] J. Ma and Z. Ma, Epidemic threshold conditions for seasonally forced seir models, Mathematical Biosciences and Engineering, 3 (2006), 161-172. doi: 10.3934/mbe.2006.3.161. [34] Z. Ma, Y. Zhou and J. Wu, Modeling and Dynamics of Infectious Diseases, Higher Education Press, 2009. doi: 10.1142/7223. [35] C. Neuhauser, Mathematical challenges in spatial ecology, Notices of the AMS, 48 (2001), 1304-1314. [36] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York: Springer, 1983. doi: 10.1007/978-1-4612-5561-1. [37] R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. part I, Journal of Differential Equations, 247 (2009), 1096-1119. doi: 10.1016/j.jde.2009.05.002. [38] R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Analysis: Theory, Methods and Applications, 71 (2009), 239-247. doi: 10.1016/j.na.2008.10.043. [39] R. Peng and F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement, Physica D: Nonlinear Phenomena, 259 (2013), 8-25. doi: 10.1016/j.physd.2013.05.006. [40] R. Peng and X. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471. doi: 10.1088/0951-7715/25/5/1451. [41] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis, 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9. [42] M. Robinson, N. I. Stilianakis and Y. Drossinos, Spatial dynamics of airborne infectious diseases, Journal of Theoretical Biology, 297 (2012), 116-126. doi: 10.1016/j.jtbi.2011.12.015. [43] R. Ross, The Prevention of Malaria(2nd ed.), Murray, London, 1911. [44] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, In Mathematics for Life Science and Medicine, Springer, (2007), 97-122. [45] S. Ruan and W. D. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of Differential Equations, 188 (2003), 135-163. doi: 10.1016/S0022-0396(02)00089-X. [46] J. Shi, Persistence and bifurcation of degenerate solutions, Journal of Functional Analysis, 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483. [47] H. L. Smith, Subharmonic bifurcation in an SIR epidemic model, Journal of Mathematical Biology, 17 (1983), 163-177. doi: 10.1007/BF00305757. [48] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995. [49] N. Tuncer and M. Martcheva, Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion, Journal of Biological Dynamics, 6 (2012), 406-439. doi: 10.1080/17513758.2011.614697. [50] V. Volpert and S. Petrovskii, Reaction-diffusion waves in biology, Physics of life reviews, 6 (2009), 267-310. doi: 10.1016/j.plrev.2009.10.002. [51] W. D. Wang, Epidemic models with nonlinear infection forces, Mathematical Biosciences and Engineering, 3 (2006), 267-279. doi: 10.3934/mbe.2006.3.267. [52] W. D. Wang and X. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM Journal on Applied Dynamical Systems, 11 (2012), 1652-1673. doi: 10.1137/120872942. [53] D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical Biosciences, 208 (2007), 419-429. doi: 10.1016/j.mbs.2006.09.025.

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##### References:
 [1] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic patch model, SIAM Journal on Applied Mathematics, 67 (2007), 1283-1309. doi: 10.1137/060672522. [2] L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete and Continuous Dynamical Systems, 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1. [3] M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Mathematical Biosciences, 189 (2004), 75-96. doi: 10.1016/j.mbs.2004.01.003. [4] R. M. Anderson, R. M. May and B. Anderson, Infectious Diseases of Humans: Dynamics and Control, Wiley Online Library, 1992. [5] S. Anita and V. Capasso, A stabilizability problem for a reaction-diffusion system modelling a class of spatially structured epidemic systems, Nonlinear Analysis: Real World Applications, 3 (2002), 453-464. doi: 10.1016/S1468-1218(01)00025-6. [6] C. Bain, Applied mathematical ecology, Journal of Epidemiology and Community Health, 44 (1990), p254. doi: 10.1136/jech.44.3.254-b. [7] F. Berezovsky, G. Karev, B. Song and C. Castillo-Chavez, A simple epidemic model with surprising dynamics, Mathematical Biosciences and Engineering, 2 (2005), 133-152. [8] E. Beretta and Y. Takeuchi, Global stability of an SIR epidemic model with time delays, Journal of Mathematical Biology, 33 (1995), 250-260. doi: 10.1007/BF00169563. [9] R. S. Cantrell and C. Cosner, Spatial Ecology Via Reaction-Diffusion Equations, John Wiley & Sons, 2003. doi: 10.1002/0470871296. [10] V. Capasso, Mathematical Structures of Epidemic Systems, Springer, 1993. doi: 10.1007/978-3-540-70514-7. [11] V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Mathematical Biosciences, 42 (1978), 43-61. doi: 10.1016/0025-5564(78)90006-8. [12] C. Castillo-Chavez and A.-A. Yakubu, Dispersal, disease and life-history evolution, Mathematical Biosciences, 173 (2001), 35-53. doi: 10.1016/S0025-5564(01)00065-7. [13] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, Journal of Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. [14] D. Ebert, M. Lipsitch and K. L. Mangin, The effect of parasites on host population density and extinction: experimental epidemiology with daphnia and six microparasites, The American Naturalist, 156 (2000), 459-477. [15] W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model of the spread of feline leukemia virus (FeLV) through a highly heterogeneous spatial domain, SIAM Journal on Mathematical Analysis, 33 (2001), 570-588. doi: 10.1137/S0036141000371757. [16] W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A reaction-diffusion system modeling direct and indirect transmission of diseases, Discrete and Continuous Dynamical Systems-B, 4 (2004), 893-910. doi: 10.3934/dcdsb.2004.4.893. [17] W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model for indirectly transmitted diseases, Mathematical Biosciences, 206 (2007), 233-248. doi: 10.1016/j.mbs.2005.07.005. [18] B. Fred and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology(Second Edition), Springer, 2012. doi: 10.1007/978-1-4614-1686-9. [19] D. Henry, Geometric Theory of Semilinear Parabolic Equations, volume 840. Springer-Verlag Berlin, 1981. [20] H. W. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [21] H. W. Hethcote and S. A. Levin, Periodicity in epidemiological models, In Applied Mathematical Ecology, Springer, (1989), 193-211. doi: 10.1007/978-3-642-61317-3_8. [22] W. Huang, M. Han and K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission, Mathematical Biosciences and Engineering, 7 (2010), 51-66. doi: 10.3934/mbe.2010.7.51. [23] T. W. Hwang and Y. Kuang, Deterministic extinction effect of parasites on host populations, Journal of Mathematical Biology, 46 (2003), 17-30. doi: 10.1007/s00285-002-0165-7. [24] T. W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model, Mathematical Biosciences and Engineering, 2 (2005), 743-751. doi: 10.3934/mbe.2005.2.743. [25] Y. Kang and C. Castillo-Chavez, A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness, Discrete and Continuous Dynamical Systems-B, 19 (2014), 89-130. [26] Y. Kang, S. K. Sasmal, A. R. Bhowmick and J. Chattopadhyay, Dynamics of a predator-prey system with prey subject to Allee effects and disease, Mathematical Biosciences and Engineering, 11 (2014), 877-918. doi: 10.3934/mbe.2014.11.877. [27] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics-I, Bulletin of Mathematical Biology, 53 (1991), 33-55. [28] K. I. Kim, Z. Lin and Q. Zhang, An SIR epidemic model with free boundary, Nonlinear Analysis: Real World Applications, 14 (2013), 1992-2001. doi: 10.1016/j.nonrwa.2013.02.003. [29] A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models, Mathematical Medicine and Biology, 22 (2005), 113-128. doi: 10.1093/imammb/dqi001. [30] X. Z. Li, W. S. Li and M. Ghosh, Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment, Applied Mathematics and Computation, 210 (2009), 141-150. doi: 10.1016/j.amc.2008.12.085. [31] W. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of sirs epidemiological models, Journal of Mathematical Biology, 23 (1986), 187-204. doi: 10.1007/BF00276956. [32] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, CRC Press, 2010. [33] J. Ma and Z. Ma, Epidemic threshold conditions for seasonally forced seir models, Mathematical Biosciences and Engineering, 3 (2006), 161-172. doi: 10.3934/mbe.2006.3.161. [34] Z. Ma, Y. Zhou and J. Wu, Modeling and Dynamics of Infectious Diseases, Higher Education Press, 2009. doi: 10.1142/7223. [35] C. Neuhauser, Mathematical challenges in spatial ecology, Notices of the AMS, 48 (2001), 1304-1314. [36] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, New York: Springer, 1983. doi: 10.1007/978-1-4612-5561-1. [37] R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. part I, Journal of Differential Equations, 247 (2009), 1096-1119. doi: 10.1016/j.jde.2009.05.002. [38] R. Peng and S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model, Nonlinear Analysis: Theory, Methods and Applications, 71 (2009), 239-247. doi: 10.1016/j.na.2008.10.043. [39] R. Peng and F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: Effects of epidemic risk and population movement, Physica D: Nonlinear Phenomena, 259 (2013), 8-25. doi: 10.1016/j.physd.2013.05.006. [40] R. Peng and X. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471. doi: 10.1088/0951-7715/25/5/1451. [41] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, Journal of Functional Analysis, 7 (1971), 487-513. doi: 10.1016/0022-1236(71)90030-9. [42] M. Robinson, N. I. Stilianakis and Y. Drossinos, Spatial dynamics of airborne infectious diseases, Journal of Theoretical Biology, 297 (2012), 116-126. doi: 10.1016/j.jtbi.2011.12.015. [43] R. Ross, The Prevention of Malaria(2nd ed.), Murray, London, 1911. [44] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, In Mathematics for Life Science and Medicine, Springer, (2007), 97-122. [45] S. Ruan and W. D. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of Differential Equations, 188 (2003), 135-163. doi: 10.1016/S0022-0396(02)00089-X. [46] J. Shi, Persistence and bifurcation of degenerate solutions, Journal of Functional Analysis, 169 (1999), 494-531. doi: 10.1006/jfan.1999.3483. [47] H. L. Smith, Subharmonic bifurcation in an SIR epidemic model, Journal of Mathematical Biology, 17 (1983), 163-177. doi: 10.1007/BF00305757. [48] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, 41. American Mathematical Society, Providence, RI, 1995. [49] N. Tuncer and M. Martcheva, Analytical and numerical approaches to coexistence of strains in a two-strain SIS model with diffusion, Journal of Biological Dynamics, 6 (2012), 406-439. doi: 10.1080/17513758.2011.614697. [50] V. Volpert and S. Petrovskii, Reaction-diffusion waves in biology, Physics of life reviews, 6 (2009), 267-310. doi: 10.1016/j.plrev.2009.10.002. [51] W. D. Wang, Epidemic models with nonlinear infection forces, Mathematical Biosciences and Engineering, 3 (2006), 267-279. doi: 10.3934/mbe.2006.3.267. [52] W. D. Wang and X. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM Journal on Applied Dynamical Systems, 11 (2012), 1652-1673. doi: 10.1137/120872942. [53] D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical Biosciences, 208 (2007), 419-429. doi: 10.1016/j.mbs.2006.09.025.
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