Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness
Pages: 89 - 130, Issue 1, January 2014

doi:10.3934/dcdsb.2014.19.89      Abstract        References        Full text (8850.3K)                  Related Articles

Yun Kang - Science and Mathematics Faculty, School of Letters and Sciences, Arizona State University, Mesa, AZ 85212, United States (email)
Carlos Castillo-Chávez - Mathematical, Computational & Modeling Science Center, Arizona State University, Tempe, AZ 85287-1904, United States (email)

1 W. C. Allee, The Social Life of Animals, Norton, New York, 1938.
2 L. H. R. Alvarez, Optimal harvesting under stochastic fluctuations and critical depensation, Mathematical Biosciences, 152 (1998), 63-85.       
3 R. M. Anderson and R. M. May, Regulation and stability of host-parasite population interactions I: Regulatory processes; II: Destabilizing processes, J. Anita. Ecol. 47 (1978), 219-247; 249-267.
4 E. Angulo, G. W. Roemer, L. Berec, J. Gascoigne and F. Courchamp, Double Allee effects and extinction in the island fox, Conservation Biology, 21 (2007), 1082-1091.
5 L. Berec, D. S. Boukal and M. Berec, Linking the Allee effect, sexual reproduction, and temperature-dependent sex determination via spatial dynamics, The American Naturalist, 157 (2001), 217-230.
6 D. S. Boukal and L. Berec, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters, Journal of Theoretical Biology, 218 (2002), 375-394.       
7 F. Berezovskaya, G. Karev, B. Song and C. Castillo-Chavez, A simple epidemic model with surprising dynamics, Mathematical Biosciences and Engineering, 2 (2005), 133-152.       
8 F. S. Berezovskaya, B. Song and C. Castillo-Chavez, Role of prey dispersal and refuges on predator-prey dynamics, SIAM J. APPL. MATH., 70 (2010), 1821-1839.       
9 F. Brauer, Backward bifurcations in simple vaccination models, J. Math. Anal. Appl., 298 (2004), 418-431.       
10 F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, 2nd Edition, Texts in Applied Mathematics, 40, Springer-Verlag, 530 pages, 2012.       
11 R. Burrows, H. Hofer and M. L. East, Population dynamics, intervention and survival in African wild dogs Lycaon pictus, Proceedings of the Royal Society B: Biological Sciences, 262 (1995), 235-245.
12 C. Castillo-Chavez, K. Cooke, W. Huang and S. A. Levin, Results on the dynamics for models for the sexual transmission of the human immunodeficiency virus, Applied Math. Letters, 2 (1989), 327-331.
13 C. Castillo-Chavez and A. A. Yakubu, Dispersal,disease and life history evolution, Math. Biosc., 173 (2001), 35-53.       
14 C. Castillo-Chavez and B. Song, Models for the transmission dynamics of fanatic behaviors, in Bioterrorism: Mathematical Modeling Applications to Homeland Security (eds. T. Banks and C. Castillo-Chavez), SIAM Series Frontiers in Applied Mathematics, 28 (2003), 240.       
15 A. Cintron-Arias, C. Castillo-Chavez, L. M. Bettencourt, A. L. Lloyd and H. T. Banks, Estimation of the effective reproductive number from disease outbreak data, Math. Biosc. & Eng., 6 (2009), 261-282.       
16 B. R. Clark and S. H. Faeth, The consequences of larval aggregation in the butterfly Chlosyne lacinia, Ecological Entomology, 22 (1997), 408-415.
17 D. L. Clifford, J. A. K. Mazet, E. J. Dubovi, D. K. Garcelon, T. J. Coonan, P. A. Conrad and L. Munson, Pathogen exposure in endangered island fox Urocyon littoralis populations: implications for conservation management, Biological Conservation, 131 (2006), 230-243.
18 F. Courchamp,T. Clutton-Brock and B. Grenfell, Multipack dynamics and the Allee effect in the African wild dog, Lycaon pictus, Animal Conservation, 3 (2000), 277-285.
19 F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, 2008.
20 J. M. Cushing, Oscillations in age-structured population models with an Allee effect. Oscillations in nonlinear systems: Applications and numerical aspects, J. Comput. Appl. Math., 52 (1994), 71-80.       
21 P. Daszak, L. Berger, A. A. Cunningham, A. D. Hyatt, D. E. Green and R. Speare, Emerging infectious diseases and amphibian population declines, Emerging Infectious Diseases, 5 (1999), 735-748.
22 S. Del Valle, H. W. Hethcote, J. M. Hyman and C. Castillo-Chavez, Effects of behavioral changes in a smallpox attack model, Mathematical Biosciences, 195 (2005), 228-251.       
23 A. Deredec and F. Courchamp, Combined impacts of Allee effects and parasitism, OIKOS, 112 (2006), 667-679.
24 A. Drew, E. J. Allen and L. J. S. Allen, Analysis of climate and geographic factors affecting the presence of chytridiomycosis in Australia, Dis. Aquat. Org., 68 (2006), 245-250.
25 O. Diekmann and M. Kretzshmar, Patterns in the effects of infectious diseases on population growth, Journal of Mathematical Biology, 29 (1991), 539-570.       
26 J. Dushoff, W. Huang and C. Castillo-Chavez, Backwards bifurcations and catastrophe in simple models of fatal diseases, J. Math. Biol., 36 (1998), 227-248.       
27 G. Dwyer, S. A. Levin and L. Buttel, A simulation model of the population dynamics and evolution of myxomatosis, Ecological Monographs, 60 (1990), 423-447.
28 L. Edelstein-Keshet, Mathematical Models in Biology, SIAM, Philadelphia, 2005.       
29 K. E. Emmert and L. J. S. Allen, Population persistence and extinction in a discrete-time stage-structured epidemic model, J. Differ. Eqn Appl., 10 (2004), 1177-1199.       
30 W. F. Fagan, M. A. Lewis, M. G. Neubert and P. Van Den Driessche, Invasion theory and biological control, Ecology Letters, 5 (2002), 148-157.
31 E. P. Fenichel, C. Castillo-Chavez, M. G. Ceddiac, G. Chowell, P. Gonzalez, G. J. Hickling, G. Holloway, R. Horan, B. Morin, C. Perrings, M. Springborn, L. Velazquez and C. Villalobos, Adaptive human behavior in epidemiological models, Proc. Natl. Acad. Sci., 108 (2011), 6306-6311.       
32 Z. Feng, C. Castillo-Chavez and A. Capurro, A model for tb with exogenous re-infection, Journal of Theoretical Population Biology, 57 (2000), 235-247.
33 A. Friedman and A-A. Yakubu, Fatal disease and demographic Allee effect: Population persistence and extinction, Journal of Biological Dynamics, 6 (2012), 495–-508.       
34 J. C. Gascoigne and R. N. Lipcius, Allee effects driven by predation, Journal of Applied Ecology, 41 (2004), 801-810.
35 B. Gonzalez, E. Huerta-Sanchez, A. Ortiz-Nieves, T. Vazquez-Alvarez and C. Kribs-Zaleta, Am I too fat? Bulimia as an epidemic, Journal of Mathematical Psychology, 47 (2003), 515-526.       
36 D. Greenhalgh and M. Griffiths, Dynamic phenomena arising from an extended core group model, Mathematical Biosciences, 221 (2009), 136-149.       
37 Y. Gruntfest, R. Arditi and Y. Dombrovsky, A fragmented population in a varying environment, Journal of Theoretical Biology, 185 (1997), 539-547.
38 J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag. 1983.       
39 S. Gupta, R. M. Anderson and R. M. May, Potential of community-wide chemotherapy or immunotherapy to control the spread of HIV, Nature, 350 (1991), 356-359.
40 K. P. Hadeler and K. Dietz, Nonlinear hyperbolic partial differential equations for the dynamics of parasite populations, Comput. Math. Appl., 9 (1983), 415-430.       
41 K. P. Hadeler and J. Müfiller, The effects of vaccination on sexually transmitted disease in heterosexual populations, In Mathematical Population Dynamics (eds. O. Arino, D. Axelrod, M. Kimmel and M. Langlois), 3d Int. Conf., 1 1992, Wuerz, Winnipeg, {251-278}.
42 K. P. Hadeler and C. Castillo-Chavez, A core group model for disease transmission, Math. Biosci., 128 (1995), 41-55.
43 K. P. Hadeler and P. van den Driessche, Backward bifurcation in epidemic control, Mathematical Biosciences, 146 (1997), 15-35.       
44 C. D. Harvell, C. E. Mitchell, J. R. Ward, S. Altizer, A. P. Dobson, R. S. Ostfeld and M. D. Samuel, Climate warming and disease risks for terrestrial and marine biota, Science, 296 (2002), 2158-2162.
45 H. Hethcote and J. Yorke, Gonorrhea: Transmission Dynamics and Control, Lecture Notes in Biomathematics, 56, Springer-Verlag, Berlin, 1984.       
46 H. W. Hethcote and J. W. van Ark, Epidemiological models for heterogeneous populations: Proportionate mixing, parameter estimation, and immunization programs, Math. Biosci., 84 (1987), 85-118.       
47 F. M. Hilker, M. A. Lewis, H. Seno, M. Langlais and H. Malchow, Pathogens can slow down or reverse invasion fronts of their hosts, Biol. Invasions, 7 (2005), 817-832.
48 F. M. Hilker, M. Langlais and H. Malchow, The Allee Effect and Infectious Diseases: Extinction, Multistability, and the (Dis-)Appearance of Oscillations, The American Naturalist, 173 (2009), 72-88.
49 F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect, Journal of Biological Dynamics, 4 (2010), 86-101.       
50 K. R. Hopper and R. T. Roush, Mate finding, dispersal, number released, and the success of biological control introductions, Ecological Entomology, 18 (1993), 321-331.
51 V. Hutson, A theorem on average Liapunov functions, Monatshefte für Mathematik, 98 (1984), 267-275.       
52 W. Huang, K. L. Cooke and C. Castillo-Chavez, Stability and bifurcation for a multiple group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math., 52 (1992), 835-854.       
53 C. Castillo-Chavez and W. Huang, Age-structured Core Groups and their impact on HIV dynamics, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods and Theory, IMA 126, 261-273, Springer-Verlag, Berlin-Heidelberg-New York. Edited by Carlos Castillo-Chavez with Pauline van den Driessche, Denise Kirschner and Abdul-Aziz Yakubu, 2002.       
54 S. R.-J. Jang and S. L. Diamond, A host-parasitoid interaction with Allee effects on the host, Comp. Math. Appl., 53 (2007), 89-103.       
55 Y. Kang and D. Armbruster, Dispersal effects on a two-patch discrete model for plant-herbivore interactions, Journal of Theoretical Biology, 268 (2011), 84-97.       
56 Y. Kang and N. Lanchier, Expansion or extinction: deterministic and stochastic two-patch models with Allee effects, Journal of Mathematical Biology, 62 (2011), 925-973.       
57 W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. A, 115 (1927), 700-721.
58 C. M. Kribs-Zaleta, Center manifolds and normal forms in epidemic models, in Mathematical Approaches for Emerging and Re-emerging Infectious Diseases: An Introduction (Edited by C. Castillo-Chavez, S. Blower, D. Kirschner, P. van den Driessche and A. A.Yakubu), Springer-Verlag, NewYork, 2001, 269-286.       
59 A. Lajmanovich and J. A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28 (1976), 221-236.       
60 R. Lande, Anthropogenic, ecological and genetic factors in extinction and conservation, Researches on Population Ecology, 40 (1998), 259-269.
61 M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43 (1993), 141-158.
62 V. Padrón and M. C. Trevisan, Effect of aggregating behavior on population recovery on a set of habitat islands, Mathematical Biosciences, 165 (2000), 63-78.       
63 D. Pauly, V. Christensen, S. Guenette, T. J. Pitcher, U. R. Sumaila, C. J. Walters and D. Zeller, Towards sustainability in world fisheries, Nature, 418 (2002), 689-695.
64 L. J. Rachowicz, J.-M. Hero, R. A. Alford, J. W. Taylor, J. A. T. Morgan, V. T. Vredenburg, J. P. Collins and C. J. Briggs, The novel and endemic pathogen hypotheses: Competing explanations for the origin of emerging infectious diseases of wildlife, Conserv. Biol., 19 (2005), 1441-1448.
65 F. Sanchez, X. Wang, C. Castillo-Chavez, P. Gruenewald and D. Gorman, Drinking as an epidemic, a simple mathematical model with recovery and relapse, in Therapist's Guide to Evidence Based Relapse Prevention (Edited by Katie Witkiewitz and G. Alan Marlatt), 2007, 353-368.       
66 P. Scalia-Tomba, The effects of structural behavior change on the spread of HIV in one sex populations, Math. Biosci., 107 (1991), 547-555.
67 K. Sherman and A. M. Duda, Large marine ecosystems: An emerging paradigm for fishery sustainability, Fisheries, 24 (1999), 15-26.
68 L. F. Skerrat, L. Berger, R. Speare, S. Cashins, K. R. McDonald, A. D. Phillott, H. B. Hines and N. Kenyon, Spread of chytridiomycosis has caused the rapid global decline and extinction of frogs, EcoHealth, 4 (2007), 125-134.
69 K. F. Smith, D. F. Sax and K. D. Lafferty, Evidence for the role of infectious disease in species extinction and endangerment, Conservation Biology, 20 (2006), 1349-1357.
70 B. Song, Dynamical Epidemic Models and Their Applications, Phd Dissertation, Cornell University, Ithaca, NY, 2002.       
71 B. Song, M. Garsow-Castillo, K. Rios-Soto, M. Mejran, L. Henso and C. Castillo-Chavez, Raves clubs, and ecstasy: The impact of peer pressure, Journal of Mathematical Biosciences and Engineering, 3 (2006), 249-266.       
72 P. A. Stephens and W. J. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation, Trends in Ecology & Evolution, 14 (1999), 401-405.
73 P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect? Oikos, 87 (1999), 185-190.
74 H. R. Thieme, T. Dhirasakdanon, Z. Han and R. Trevino, Species decline and extinction: Synergy of infectious disease and Allee effect? Journal of Biological Dynamics, 3 (2009), 305-323.       
75 P. van den Driessche and J. Watmough, A simple SIS epidemic model with a backward bifurcation, J. Math. Biol., 40 (2000), 525-540.       
76 X. Wang, Backward Bifurcation in a Mathematical Model for Tuberculosis with Loss of Immunity, Ph.D. Thesis, Purdue University, 2005.       
77 A-A. Yakubu, Allee effects in a discrete-time SIS epidemic model with infected newborns, Journal of Difference Equations and Applications, 13 (2007), 341-356.       

Go to top