• Previous Article
    Bacteria--phagocyte dynamics, axiomatic modelling and mass-action kinetics
  • MBE Home
  • This Issue
  • Next Article
    Regulation of modular Cyclin and CDK feedback loops by an E2F transcription oscillator in the mammalian cell cycle
2011, 8(2): 463-473. doi: 10.3934/mbe.2011.8.463

Preliminary analysis of an agent-based model for a tick-borne disease

1. 

Department of Biological Sciences, Old Dominion University, Norfolk, VA 23529, United States

Received  March 2010 Revised  September 2010 Published  April 2011

Ticks have a unique life history including a distinct set of life stages and a single blood meal per life stage. This makes tick-host interactions more complex from a mathematical perspective. In addition, any model of these interactions must involve a significant degree of stochasticity on the individual tick level. In an attempt to quantify these relationships, I have developed an individual-based model of the interactions between ticks and their hosts as well as the transmission of tick-borne disease between the two populations. The results from this model are compared with those from previously published differential equation based population models. The findings show that the agent-based model produces significantly lower prevalence of disease in both the ticks and their hosts than what is predicted by a similar differential equation model.
Citation: Holly Gaff. Preliminary analysis of an agent-based model for a tick-borne disease. Mathematical Biosciences & Engineering, 2011, 8 (2) : 463-473. doi: 10.3934/mbe.2011.8.463
References:
[1]

Centers for Disease Control and Prevention, Summary of Notifiable Diseases - United States, 2006,, MMWR, 55 (2008), 1.   Google Scholar

[2]

D. E. Sonenshine and T. N. Mather, "Ecological Dynamics of Tick-Borne Zoonoses,", Oxford University Press, (1994).   Google Scholar

[3]

H. Gaff and L. J. Gross, Analysis of a tick-borne disease model with varying population sizes in various habitats,, Bulletin of Mathematical Biology, 69 (2007), 265.  doi: 10.1007/s11538-006-9125-5.  Google Scholar

[4]

H. Gaff and E. Schaefer, Metapopulation models in tick-borne disease transmission modelling,, In, (2008).   Google Scholar

[5]

H. Gaff, L. Gross and E. Schaefer, Results from a mathematical model for human monocytic ehrlichiosis,, Proceedings of the 5th Conference on Rickettsiae and Rickettsial diseases, 15 (2008), 1.   Google Scholar

[6]

D. G. Haile and G. A. Mount, Computer simulation of population dynamics of the lone star tick, Amblyomma americanum (Acari: Ixodidae),, Journal of Medical Entomology, 24 (1987), 356.   Google Scholar

[7]

G. A. Mount and D. G. Haile, Computer simulation of population dynamics of the American dog tick (Acari: Ixodidae),, Journal of Medical Entomology, 26 (1989), 60.   Google Scholar

[8]

G. A. Mount, D. G. Haile, R. B. Davey and L. M. Cooksey, Computer simulation of boophilus cattle tick (Acari: Ixodidae) population dynamics,, Journal of Medical Entomology, 28 (1991), 223.   Google Scholar

[9]

G. A. Mount, D. G. Haile, D. R. Barnard and E. Daniels, New version of LSTSIM for computer simulation of Amblyomma americanum (Acari: Ixodidae) population dynamics,, Journal of Medical Entomology, 30 (1993), 843.   Google Scholar

[10]

G. A. Mount, D. G. Haile and E. Daniels, Simulation of blacklegged tick (Acari: Ixodidae) population dynamics and transmission of Borrelia burgdorferi,, Journal of Medical Entomology, 34 (1997), 461.   Google Scholar

[11]

G. A. Mount, D. G. Haile and E. Daniels, Simulation of management strategies for the blacklegged tick (Acari: Ixodidae) and the Lyme disease spirochete, Borrelia burgdorferi,, Journal of Medical Entomology, 90 (1997), 672.   Google Scholar

[12]

S. Sandberg, T. E. Awerbuch and A. Spielman, A comprehensive multiple matrix model representing the life cycle of the tick that transmits the age of Lyme disease,, Journal of Theoretical Biology, 157 (1992), 203.  doi: 10.1016/S0022-5193(05)80621-6.  Google Scholar

[13]

T. E. Awerbuch and S. Sandberg, Trends and oscillations in tick population dynamics,, Journal of Theoretical Biology, 175 (1995), 511.  doi: 10.1006/jtbi.1995.0158.  Google Scholar

[14]

S. Randolph, Epidemiological uses of a population model for the tick Rhipicephalus appendiculatus,, Tropical Medicine and International Health, 4 (1999).  doi: 10.1046/j.1365-3156.1999.00449.x.  Google Scholar

[15]

J. E. Bunnell, S. D. Price, A. Das, T. M. Shields and G. E. Glass, Geographic Information Systems and Spatial Analysis of Adult Ixodes scapularis (Acari: Ixodidae) in the Middle Atlantic Region of the U.S.A.,, Journal of Medical Entomology, 40 (2003), 570.  doi: 10.1603/0022-2585-40.4.570.  Google Scholar

[16]

A. Das, S. R. Lele, G. E. Glass, T. Shields and J. Petz, Modelling a discrete spatial response using generalized linear mixed models: Application to Lyme disease vectors,, International Journal of Geographical Information Science, 16 (2002), 151.  doi: 10.1080/13658810110099134.  Google Scholar

[17]

G. E. Glass, B. S. Schwartz, J. M. Morgan, D. T. Johnson, P. M. Noy and E. Israel, Environmental risk factors for Lyme disease identified with geographic information systems,, American Journal of Public Health, 85 (1995), 944.  doi: 10.2105/AJPH.85.7.944.  Google Scholar

[18]

W. E. Fitzgibbon, M. E. Parrott and G. F. Webb, A diffusive epidemic model for a host-vector system,, In, (1996).   Google Scholar

[19]

J. Radcliffe and L. Rass, The spatial spread and final size of models for the deterministic host-vector epidemic,, Mathematical Biosciences, 70 (1984), 123.  doi: 10.1016/0025-5564(84)90094-4.  Google Scholar

[20]

J. Radcliffe and L. Rass, The rate of spread of infection in models for the deterministic host-vector epidemic,, Mathematical Biosciences, 74 (1985), 257.  doi: 10.1016/0025-5564(85)90059-8.  Google Scholar

[21]

A. R. Giardina, K. A. Schmidt, E. M. Schauber and R. S. Ostfeld, Modeling the role of songbirds and rodents in the ecology of Lyme disease,, Canadian Journal of Zoology, 78 (2000), 2184.  doi: 10.1139/cjz-78-12-2184.  Google Scholar

[22]

K. LoGiudice, R. S. Ostfeld, K. A. Schmidt and F. Keesing, The ecology of infectious disease: Effects of host diversity and community composition on Lyme disease risk,, Proceedings of the National Academies of Science, 100 (2003), 567.   Google Scholar

[23]

M. Ghosh and A. Pugliese, Seasonal population dynamics of ticks, and its influence on infection transmission: A semi- discrete approach,, Bulletin of Mathematical Biology, 66 (2004), 1659.  doi: 10.1016/j.bulm.2004.03.007.  Google Scholar

[24]

W. Ding, Optimal Control on Hybrid ODE Systems with Application to a Tick Disease Model,, Mathematical Biosciences and Engineering, 4 (2007), 633.   Google Scholar

[25]

D. L. DeAngelis and L. J. Gross, "Individual-based Models and Approaches in Ecology: Populations, Communities and Ecosystems,", Taylor and Francis, (1992).   Google Scholar

[26]

D. L. DeAngelis, L. J. Gross, W. F. Wolff, D. M. Fleming, M. P. Nott and E. J. Comiskey, Individual-based models on the landscape: Applications to the Everglades,, in, (2000).   Google Scholar

[27]

S. Eubank, H. Guclu, V. S. A. Kumar, M. V. Marathe, A. Srinivasan, Z. Toroczkai and N. Wang, Modelling disease outbreaks in realistic urban social networks,, Nature, 429 (2004), 180.  doi: 10.1038/nature02541.  Google Scholar

[28]

A. G. Barbour, "Lyme Disease: The Cause, the Cure, the Controversy,", John Hopkins University Press, (1996).   Google Scholar

[29]

F. Des Vignes, M. L. Levin and D. Fish, Comparative vector competence of Dermacentor variabilis and Ixodes scapularis (Acari: Ixodidae) for the agent of human granulocytic ehrlichiosis,, Journal of Medical Entomology, 36 (1999), 182.   Google Scholar

[30]

Dania Richter, Andrew Spielman, Nicholas Komar and Franz-Rainer Matuschka, Competence of American robins as reservoir hosts for Lyme disease spirochetes,, Emerging Infectious Diseases, 6 (2000), 133.  doi: 10.3201/eid0602.000205.  Google Scholar

[31]

V. Grimm, U. Berger, F. Bastiansen, S. Eliassen, V. Ginot, J. Giske, J. Goss-Custard, T. Grand, S. K. Heinz, G. Huse, A. Huth, J. U. Jepsen, C. Jørgensen, W. M. Mooij, B. Müller, G. Peer, C. Piou, S. F. Railsback, A. M. Robbins, M. M. Robbins, E. Rossmanith, N. Rüger, E. Strand, S. Souissi, R. A. Stillman, R. Vabø, U. Visser and D. L. DeAngelis, A standard protocol for describing individual-based and agent-based models,, Ecological Modelling \textbf{198} (2006), 198 (2006), 115.  doi: 10.1016/j.ecolmodel.2006.04.023.  Google Scholar

[32]

A. L. Bauer, C. A. A. Beauchemin and A. S. Perelson, Agent-based modeling of host-pathogen systems: The successes and the challenges,, Information Sciences, 179 (2009), 1379.  doi: 10.1016/j.ins.2008.11.012.  Google Scholar

[33]

V. Grimm and S. F. Railsbeck, "Individual-based Modeling and Ecology,", Princeton University Press, (2005).   Google Scholar

[34]

J. M. Lockhart, W. R. Davidson, J. E. Dawson and D. E. Stallknecht, Temporal association of Amblyomma americanum with the presence of Ehrlichia chaffeensis reactive antibodies in white-tailed deer,, Journal of Wildlife Diseases, 31 (1995), 119.   Google Scholar

[35]

J. E. Dawson, J. E. Childs, K. L. Biggie, C. Moore, D. Stallknecht, J. Shaddock, J. Bouseman, E. Hofmeister and J. G. Olson, White-tailed deer as a potential reservoir of Ehrlichia spp.,, Journal of Wildlife Diseases, 30 (1994), 162.   Google Scholar

[36]

B. E. Anderson, K. G. Sims, J. G. Olson, J. E. Childs, J. F. Piesman, C. M. Happ, G. O. Maupin and B. J. B. Johnson, Amblyomma americanum: A potential vector of human ehrlichiosis,, American Journal of Tropical Medicine and Hygiene, 49 (1993), 239.   Google Scholar

show all references

References:
[1]

Centers for Disease Control and Prevention, Summary of Notifiable Diseases - United States, 2006,, MMWR, 55 (2008), 1.   Google Scholar

[2]

D. E. Sonenshine and T. N. Mather, "Ecological Dynamics of Tick-Borne Zoonoses,", Oxford University Press, (1994).   Google Scholar

[3]

H. Gaff and L. J. Gross, Analysis of a tick-borne disease model with varying population sizes in various habitats,, Bulletin of Mathematical Biology, 69 (2007), 265.  doi: 10.1007/s11538-006-9125-5.  Google Scholar

[4]

H. Gaff and E. Schaefer, Metapopulation models in tick-borne disease transmission modelling,, In, (2008).   Google Scholar

[5]

H. Gaff, L. Gross and E. Schaefer, Results from a mathematical model for human monocytic ehrlichiosis,, Proceedings of the 5th Conference on Rickettsiae and Rickettsial diseases, 15 (2008), 1.   Google Scholar

[6]

D. G. Haile and G. A. Mount, Computer simulation of population dynamics of the lone star tick, Amblyomma americanum (Acari: Ixodidae),, Journal of Medical Entomology, 24 (1987), 356.   Google Scholar

[7]

G. A. Mount and D. G. Haile, Computer simulation of population dynamics of the American dog tick (Acari: Ixodidae),, Journal of Medical Entomology, 26 (1989), 60.   Google Scholar

[8]

G. A. Mount, D. G. Haile, R. B. Davey and L. M. Cooksey, Computer simulation of boophilus cattle tick (Acari: Ixodidae) population dynamics,, Journal of Medical Entomology, 28 (1991), 223.   Google Scholar

[9]

G. A. Mount, D. G. Haile, D. R. Barnard and E. Daniels, New version of LSTSIM for computer simulation of Amblyomma americanum (Acari: Ixodidae) population dynamics,, Journal of Medical Entomology, 30 (1993), 843.   Google Scholar

[10]

G. A. Mount, D. G. Haile and E. Daniels, Simulation of blacklegged tick (Acari: Ixodidae) population dynamics and transmission of Borrelia burgdorferi,, Journal of Medical Entomology, 34 (1997), 461.   Google Scholar

[11]

G. A. Mount, D. G. Haile and E. Daniels, Simulation of management strategies for the blacklegged tick (Acari: Ixodidae) and the Lyme disease spirochete, Borrelia burgdorferi,, Journal of Medical Entomology, 90 (1997), 672.   Google Scholar

[12]

S. Sandberg, T. E. Awerbuch and A. Spielman, A comprehensive multiple matrix model representing the life cycle of the tick that transmits the age of Lyme disease,, Journal of Theoretical Biology, 157 (1992), 203.  doi: 10.1016/S0022-5193(05)80621-6.  Google Scholar

[13]

T. E. Awerbuch and S. Sandberg, Trends and oscillations in tick population dynamics,, Journal of Theoretical Biology, 175 (1995), 511.  doi: 10.1006/jtbi.1995.0158.  Google Scholar

[14]

S. Randolph, Epidemiological uses of a population model for the tick Rhipicephalus appendiculatus,, Tropical Medicine and International Health, 4 (1999).  doi: 10.1046/j.1365-3156.1999.00449.x.  Google Scholar

[15]

J. E. Bunnell, S. D. Price, A. Das, T. M. Shields and G. E. Glass, Geographic Information Systems and Spatial Analysis of Adult Ixodes scapularis (Acari: Ixodidae) in the Middle Atlantic Region of the U.S.A.,, Journal of Medical Entomology, 40 (2003), 570.  doi: 10.1603/0022-2585-40.4.570.  Google Scholar

[16]

A. Das, S. R. Lele, G. E. Glass, T. Shields and J. Petz, Modelling a discrete spatial response using generalized linear mixed models: Application to Lyme disease vectors,, International Journal of Geographical Information Science, 16 (2002), 151.  doi: 10.1080/13658810110099134.  Google Scholar

[17]

G. E. Glass, B. S. Schwartz, J. M. Morgan, D. T. Johnson, P. M. Noy and E. Israel, Environmental risk factors for Lyme disease identified with geographic information systems,, American Journal of Public Health, 85 (1995), 944.  doi: 10.2105/AJPH.85.7.944.  Google Scholar

[18]

W. E. Fitzgibbon, M. E. Parrott and G. F. Webb, A diffusive epidemic model for a host-vector system,, In, (1996).   Google Scholar

[19]

J. Radcliffe and L. Rass, The spatial spread and final size of models for the deterministic host-vector epidemic,, Mathematical Biosciences, 70 (1984), 123.  doi: 10.1016/0025-5564(84)90094-4.  Google Scholar

[20]

J. Radcliffe and L. Rass, The rate of spread of infection in models for the deterministic host-vector epidemic,, Mathematical Biosciences, 74 (1985), 257.  doi: 10.1016/0025-5564(85)90059-8.  Google Scholar

[21]

A. R. Giardina, K. A. Schmidt, E. M. Schauber and R. S. Ostfeld, Modeling the role of songbirds and rodents in the ecology of Lyme disease,, Canadian Journal of Zoology, 78 (2000), 2184.  doi: 10.1139/cjz-78-12-2184.  Google Scholar

[22]

K. LoGiudice, R. S. Ostfeld, K. A. Schmidt and F. Keesing, The ecology of infectious disease: Effects of host diversity and community composition on Lyme disease risk,, Proceedings of the National Academies of Science, 100 (2003), 567.   Google Scholar

[23]

M. Ghosh and A. Pugliese, Seasonal population dynamics of ticks, and its influence on infection transmission: A semi- discrete approach,, Bulletin of Mathematical Biology, 66 (2004), 1659.  doi: 10.1016/j.bulm.2004.03.007.  Google Scholar

[24]

W. Ding, Optimal Control on Hybrid ODE Systems with Application to a Tick Disease Model,, Mathematical Biosciences and Engineering, 4 (2007), 633.   Google Scholar

[25]

D. L. DeAngelis and L. J. Gross, "Individual-based Models and Approaches in Ecology: Populations, Communities and Ecosystems,", Taylor and Francis, (1992).   Google Scholar

[26]

D. L. DeAngelis, L. J. Gross, W. F. Wolff, D. M. Fleming, M. P. Nott and E. J. Comiskey, Individual-based models on the landscape: Applications to the Everglades,, in, (2000).   Google Scholar

[27]

S. Eubank, H. Guclu, V. S. A. Kumar, M. V. Marathe, A. Srinivasan, Z. Toroczkai and N. Wang, Modelling disease outbreaks in realistic urban social networks,, Nature, 429 (2004), 180.  doi: 10.1038/nature02541.  Google Scholar

[28]

A. G. Barbour, "Lyme Disease: The Cause, the Cure, the Controversy,", John Hopkins University Press, (1996).   Google Scholar

[29]

F. Des Vignes, M. L. Levin and D. Fish, Comparative vector competence of Dermacentor variabilis and Ixodes scapularis (Acari: Ixodidae) for the agent of human granulocytic ehrlichiosis,, Journal of Medical Entomology, 36 (1999), 182.   Google Scholar

[30]

Dania Richter, Andrew Spielman, Nicholas Komar and Franz-Rainer Matuschka, Competence of American robins as reservoir hosts for Lyme disease spirochetes,, Emerging Infectious Diseases, 6 (2000), 133.  doi: 10.3201/eid0602.000205.  Google Scholar

[31]

V. Grimm, U. Berger, F. Bastiansen, S. Eliassen, V. Ginot, J. Giske, J. Goss-Custard, T. Grand, S. K. Heinz, G. Huse, A. Huth, J. U. Jepsen, C. Jørgensen, W. M. Mooij, B. Müller, G. Peer, C. Piou, S. F. Railsback, A. M. Robbins, M. M. Robbins, E. Rossmanith, N. Rüger, E. Strand, S. Souissi, R. A. Stillman, R. Vabø, U. Visser and D. L. DeAngelis, A standard protocol for describing individual-based and agent-based models,, Ecological Modelling \textbf{198} (2006), 198 (2006), 115.  doi: 10.1016/j.ecolmodel.2006.04.023.  Google Scholar

[32]

A. L. Bauer, C. A. A. Beauchemin and A. S. Perelson, Agent-based modeling of host-pathogen systems: The successes and the challenges,, Information Sciences, 179 (2009), 1379.  doi: 10.1016/j.ins.2008.11.012.  Google Scholar

[33]

V. Grimm and S. F. Railsbeck, "Individual-based Modeling and Ecology,", Princeton University Press, (2005).   Google Scholar

[34]

J. M. Lockhart, W. R. Davidson, J. E. Dawson and D. E. Stallknecht, Temporal association of Amblyomma americanum with the presence of Ehrlichia chaffeensis reactive antibodies in white-tailed deer,, Journal of Wildlife Diseases, 31 (1995), 119.   Google Scholar

[35]

J. E. Dawson, J. E. Childs, K. L. Biggie, C. Moore, D. Stallknecht, J. Shaddock, J. Bouseman, E. Hofmeister and J. G. Olson, White-tailed deer as a potential reservoir of Ehrlichia spp.,, Journal of Wildlife Diseases, 30 (1994), 162.   Google Scholar

[36]

B. E. Anderson, K. G. Sims, J. G. Olson, J. E. Childs, J. F. Piesman, C. M. Happ, G. O. Maupin and B. J. B. Johnson, Amblyomma americanum: A potential vector of human ehrlichiosis,, American Journal of Tropical Medicine and Hygiene, 49 (1993), 239.   Google Scholar

[1]

Wei-Chieh Chen, Bogdan Kazmierczak. Traveling waves in quadratic autocatalytic systems with complexing agent. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020364

[2]

Onur Şimşek, O. Erhun Kundakcioglu. Cost of fairness in agent scheduling for contact centers. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021001

[3]

Pedro Branco. A post-quantum UC-commitment scheme in the global random oracle model from code-based assumptions. Advances in Mathematics of Communications, 2021, 15 (1) : 113-130. doi: 10.3934/amc.2020046

[4]

Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004

[5]

Shuyang Dai, Fengru Wang, Jerry Zhijian Yang, Cheng Yuan. A comparative study of atomistic-based stress evaluation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020322

[6]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

[7]

Wolfgang Riedl, Robert Baier, Matthias Gerdts. Optimization-based subdivision algorithm for reachable sets. Journal of Computational Dynamics, 2021, 8 (1) : 99-130. doi: 10.3934/jcd.2021005

[8]

Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074

[9]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071

[10]

Yi An, Bo Li, Lei Wang, Chao Zhang, Xiaoli Zhou. Calibration of a 3D laser rangefinder and a camera based on optimization solution. Journal of Industrial & Management Optimization, 2021, 17 (1) : 427-445. doi: 10.3934/jimo.2019119

[11]

Bing Yu, Lei Zhang. Global optimization-based dimer method for finding saddle points. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 741-753. doi: 10.3934/dcdsb.2020139

[12]

Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260

[13]

Håkon Hoel, Gaukhar Shaimerdenova, Raúl Tempone. Multilevel Ensemble Kalman Filtering based on a sample average of independent EnKF estimators. Foundations of Data Science, 2020, 2 (4) : 351-390. doi: 10.3934/fods.2020017

[14]

Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021004

[15]

Liam Burrows, Weihong Guo, Ke Chen, Francesco Torella. Reproducible kernel Hilbert space based global and local image segmentation. Inverse Problems & Imaging, 2021, 15 (1) : 1-25. doi: 10.3934/ipi.2020048

[16]

Hongfei Yang, Xiaofeng Ding, Raymond Chan, Hui Hu, Yaxin Peng, Tieyong Zeng. A new initialization method based on normed statistical spaces in deep networks. Inverse Problems & Imaging, 2021, 15 (1) : 147-158. doi: 10.3934/ipi.2020045

[17]

Karan Khathuria, Joachim Rosenthal, Violetta Weger. Encryption scheme based on expanded Reed-Solomon codes. Advances in Mathematics of Communications, 2021, 15 (2) : 207-218. doi: 10.3934/amc.2020053

[18]

Kai Zhang, Xiaoqi Yang, Song Wang. Solution method for discrete double obstacle problems based on a power penalty approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021018

[19]

Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020032

[20]

Ténan Yeo. Stochastic and deterministic SIS patch model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021012

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]