2014, 19(1): 89-130. doi: 10.3934/dcdsb.2014.19.89

A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness

1. 

Science and Mathematics Faculty, School of Letters and Sciences, Arizona State University, Mesa, AZ 85212

2. 

Mathematical, Computational & Modeling Science Center, Arizona State University, Tempe, AZ 85287-1904

Received  April 2012 Revised  September 2013 Published  December 2013

The focus here is on the study disease dynamics under the assumption that a critical mass of susceptible individuals is required to guarantee the population's survival. Specifically, the emphasis is on the study of the role of an Allee effect on a Susceptible-Infectious (SI) model where the possibility that susceptible and infected individuals reproduce, with the S-class being the best fit. It is further assumed that infected individuals loose some of their ability to compete for resources, the cost imposed by the disease. These features are set in motion as simple model as possible. They turn out to lead to a rich set of dynamical outcomes. This toy model supports the possibility of multi-stability (hysteresis), saddle node and Hopf bifurcations, and catastrophic events (disease-induced extinction). The analyses provide a full picture of the system under disease-free dynamics including disease-induced extinction and proceed to identify required conditions for disease persistence. We conclude that increases in (i) the maximum birth rate of a species, or (ii) in the relative reproductive ability of infected individuals, or (iii) in the competitive ability of a infected individuals at low density levels, or in (iv) the per-capita death rate (including disease-induced) of infected individuals, can stabilize the system (resulting in disease persistence). We further conclude that increases in (a) the Allee effect threshold, or (b) in disease transmission rates, or in (c) the competitive ability of infected individuals at high density levels, can destabilize the system, possibly leading to the eventual collapse of the population. The results obtained from the analyses of this toy model highlight the significant role that factors like an Allee effect may play on the survival and persistence of animal populations. Scientists involved in biological conservation and pest management or interested in finding sustainability solutions, may find these results of this study compelling enough to suggest additional focused research on the role of disease in the regulation and persistence of animal populations. The risk faced by endangered species may turn out to be a lot higher than initially thought.
Citation: Yun Kang, Carlos Castillo-Chávez. A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 89-130. doi: 10.3934/dcdsb.2014.19.89
References:
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show all references

References:
[1]

W. C. Allee, The Social Life of Animals,, Norton, (1938).

[2]

L. H. R. Alvarez, Optimal harvesting under stochastic fluctuations and critical depensation,, Mathematical Biosciences, 152 (1998), 63. doi: 10.1016/S0025-5564(98)10018-4.

[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), 47 (1978), 219.

[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. doi: 10.1111/j.1523-1739.2007.00721.x.

[5]

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[6]

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[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.

[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. doi: 10.1137/080730603.

[9]

F. Brauer, Backward bifurcations in simple vaccination models,, J. Math. Anal. Appl., 298 (2004), 418. doi: 10.1016/j.jmaa.2004.05.045.

[10]

F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology,, 2nd Edition, (2012). doi: 10.1007/978-1-4614-1686-9.

[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. doi: 10.1098/rspb.1995.0201.

[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. doi: 10.1016/0893-9659(89)90080-3.

[13]

C. Castillo-Chavez and A. A. Yakubu, Dispersal,disease and life history evolution,, Math. Biosc., 173 (2001), 35. doi: 10.1016/S0025-5564(01)00065-7.

[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), 28 (2003).

[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. doi: 10.3934/mbe.2009.6.261.

[16]

B. R. Clark and S. H. Faeth, The consequences of larval aggregation in the butterfly Chlosyne lacinia,, Ecological Entomology, 22 (1997), 408.

[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. doi: 10.1016/j.biocon.2006.04.029.

[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.

[19]

F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation,, Oxford University Press, (2008). doi: 10.1093/acprof:oso/9780198570301.001.0001.

[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. doi: 10.1016/0377-0427(94)90349-2.

[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. doi: 10.3201/eid0506.990601.

[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. doi: 10.1016/j.mbs.2005.03.006.

[23]

A. Deredec and F. Courchamp, Combined impacts of Allee effects and parasitism,, OIKOS, 112 (2006), 667. doi: 10.1111/j.0030-1299.2006.14243.x.

[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. doi: 10.3354/dao068245.

[25]

O. Diekmann and M. Kretzshmar, Patterns in the effects of infectious diseases on population growth,, Journal of Mathematical Biology, 29 (1991), 539. doi: 10.1007/BF00164051.

[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. doi: 10.1007/s002850050099.

[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.

[28]

L. Edelstein-Keshet, Mathematical Models in Biology,, SIAM, (2005). doi: 10.1137/1.9780898719147.

[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. doi: 10.1080/10236190410001654151.

[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. doi: 10.1046/j.1461-0248.2002.0_285.x.

[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. doi: 10.1073/pnas.1011250108.

[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.

[33]

A. Friedman and A-A. Yakubu, Fatal disease and demographic Allee effect: Population persistence and extinction,, Journal of Biological Dynamics, 6 (2012). doi: 10.1080/17513758.2011.630489.

[34]

J. C. Gascoigne and R. N. Lipcius, Allee effects driven by predation,, Journal of Applied Ecology, 41 (2004), 801. doi: 10.1111/j.0021-8901.2004.00944.x.

[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. doi: 10.1016/j.jmp.2003.08.002.

[36]

D. Greenhalgh and M. Griffiths, Dynamic phenomena arising from an extended core group model,, Mathematical Biosciences, 221 (2009), 136. doi: 10.1016/j.mbs.2009.08.003.

[37]

Y. Gruntfest, R. Arditi and Y. Dombrovsky, A fragmented population in a varying environment,, Journal of Theoretical Biology, 185 (1997), 539.

[38]

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