Search Advanced

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 10: 18717-18760. doi: 10.3934/mbe.2023830
Previous Article Next Article
Research article Special Issues

Can infectious diseases eradicate host species? The effect of infection-age structure

  • School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA
  • Dedicated to Stephan Luckhaus on the occasion of his 70th birthday.
  • Received: 29 July 2023 Revised: 18 September 2023 Accepted: 21 September 2023 Published: 09 October 2023

  • It is a fundamental question in mathematical epidemiology whether deadly infectious diseases only lead to a mere decline of their host populations or whether they can cause their complete disappearance. Upper density-dependent incidences do not lead to host extinction in simple, deterministic SI or SIS (susceptible-infectious) epidemic models. Infection-age structure is introduced into SIS models because of the biological accuracy offered by considering arbitrarily distributed infectious periods. In an SIS model with infection-age structure, survival of the susceptible host population is established for incidences that depend on the infection-age density in a general way. This confirms previous host persistence results without infection-age for incidence functions that are not generalizations of frequency-dependent transmission. For certain power incidences, hosts persist if some infected individuals leave the infected class and become susceptible again and the return rate dominates the infection-age dependent infectivity in a sufficient way. The hosts may be driven into extinction by the infectious disease if there is no return into the susceptible class at all.

    Citation: Joan Ponce, Horst R. Thieme. Can infectious diseases eradicate host species? The effect of infection-age structure[J]. Mathematical Biosciences and Engineering, 2023, 20(10): 18717-18760. doi: 10.3934/mbe.2023830

    Related Papers:

  • It is a fundamental question in mathematical epidemiology whether deadly infectious diseases only lead to a mere decline of their host populations or whether they can cause their complete disappearance. Upper density-dependent incidences do not lead to host extinction in simple, deterministic SI or SIS (susceptible-infectious) epidemic models. Infection-age structure is introduced into SIS models because of the biological accuracy offered by considering arbitrarily distributed infectious periods. In an SIS model with infection-age structure, survival of the susceptible host population is established for incidences that depend on the infection-age density in a general way. This confirms previous host persistence results without infection-age for incidence functions that are not generalizations of frequency-dependent transmission. For certain power incidences, hosts persist if some infected individuals leave the infected class and become susceptible again and the return rate dominates the infection-age dependent infectivity in a sufficient way. The hosts may be driven into extinction by the infectious disease if there is no return into the susceptible class at all.


    加载中


    [1] A. P. Farrell, J. P. Collins, A. L. Greer, H. R. Thieme, Do fatal infectious diseases eradicate host species?, J. Math. Biol., 77 (2018), 2103–2164. https://doi.org/10.1007/s00285-018-1249-3 doi: 10.1007/s00285-018-1249-3
    [2] A. P. Farrell, J. P. Collins, A. L. Greer, H. R. Thieme, Times from infection to disease-induced death and their influence on final population sizes after epidemic outbreaks, Bull. Math. Biol., 80 (2018), 1937–1961. https://doi.org/10.1007/s11538-018-0446-y doi: 10.1007/s11538-018-0446-y
    [3] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, in Proceedings of the Royal Society of London. Series A, Containing papers of a mathematical and physical character, 115 (1927), 700–721. https://doi.org/10.1098/rspa.1927.0118
    [4] H. R. Thieme, Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations, Math. Biosci., 111 (1992), 99–130. https://doi.org/10.1016/0025-5564(92)90081-7 doi: 10.1016/0025-5564(92)90081-7
    [5] L. Han, A. Pugliese, Epidemics in two competing species, Nonlinear Anal. Real World Appl., 10 (2009), 723–744. https://doi.org/10.1016/j.nonrwa.2007.11.005 doi: 10.1016/j.nonrwa.2007.11.005
    [6] H. W. Hethcote, W. Wang, Y. Li, Species coexistence and periodicity in host-host-pathogen models, J. Math. Biol., 51 (2005), 629–660. https://doi.org/10.1007/s00285-005-0335-5 doi: 10.1007/s00285-005-0335-5
    [7] R. D. Holt, J. Pickering, Infectious disease and species coexistence: A model of Lotka-Volterra form, Am. Natur., 126 (1985), 196–211. https://doi.org/10.1086/284409 doi: 10.1086/284409
    [8] A. Friedman, A. A. Yakubu, Host demographic Allee effect, fatal disease, and migration: Persistence or extinction, SIAM J. Appl. Math., 72 (2012), 1644–1666. https://doi.org/10.1137/120861382 doi: 10.1137/120861382
    [9] F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect, J. Biol. Dynam., 4 (2010), 86–101. https://doi.org/10.1080/17513750903026429 doi: 10.1080/17513750903026429
    [10] H. R. Thieme, T. Dhirasakdanon, Z. Han, R. Trevino, Species decline and extinction: Synergy of infectious disease and Allee effect?, J. Biol. Dynam., 3 (2009), 305–323. https://doi.org/10.1080/17513750802376313 doi: 10.1080/17513750802376313
    [11] S. Busenberg, K. Cooke, H. R. Thieme, Demographic change and persistence of HIV/AIDS in a heterogeneous population, SIAM J. Appl. Math., 51 (1991), 1030–1052. https://doi.org/10.1137/0151052 doi: 10.1137/0151052
    [12] S. Busenberg, P. van den Driessche, Analysis of a disease transmission model in a population with varying size, J. Math. Biol., 28 (1990), 257–270. https://doi.org/10.1007/BF00178776 doi: 10.1007/BF00178776
    [13] L. Q. Gao, H. W. Hethcote, Disease transmission models with density-dependent demographics, J. Math. Biol., 30 (1992), 717–731. https://doi.org/10.1007/BF00173265 doi: 10.1007/BF00173265
    [14] W. M. Getz, J. Pickering, Epidemic models: Thresholds and population regulation, Am. Natur., 121 (1983), 892–898. https://doi.org/10.1086/284112 doi: 10.1086/284112
    [15] D. Greenhalgh, R. Das, An SIR epidemic model with a contact rate depending on population density, Math. Popul. Dynam. Anal. Heterog., 1 (1995), 79–101.
    [16] K. P. Hadeler, K. Dietz, M. Safan, Case fatality models for epidemics in growing populations, Math. Biosci., 281 (2016), 120–127. https://doi.org/10.1016/j.mbs.2016.09.007 doi: 10.1016/j.mbs.2016.09.007
    [17] T.-W. Hwang, Y. Kuang, Deterministic extinction effect of parasites on host populations, J. Math. Biol., 46 (2003), 17–30. https://doi.org/10.1007/s00285-002-0165-7 doi: 10.1007/s00285-002-0165-7
    [18] T.-W. Hwang, Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model, Math. Biosci. Eng., 2 (2005), 743–751. https://doi.org/10.3934/mbe.2005.2.743 doi: 10.3934/mbe.2005.2.743
    [19] R. M. May, R. M. Anderson, A. R. McLean, Possible demographic consequences of HIV/AIDS epidemics: ii, assuming HIV infection does not necessarily lead to AIDS, in Mathematical Approaches to Problems in Resource Management and Epidemiology, Springer, 1989,220–248. https://doi.org/10.1007/978-3-642-46693-9_16
    [20] J. Mena-Lorca, H. W. Hethcote, Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biol., 30 (1992), 693–716. https://doi.org/10.1007/BF00173264 doi: 10.1007/BF00173264
    [21] J. Zhou, H. W. Hethcote, Population size dependent incidence in models for diseases without immunity, J. Math. Biol., 32 (1994), 809–834. https://doi.org/10.1007/BF00168799 doi: 10.1007/BF00168799
    [22] O. Diekmann, H. Heesterbeek, T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton University Press, Princeton, 2013. https://doi.org/10.23943/princeton/9780691155395.001.0001
    [23] F. De Castro, B. Bolker, Mechanisms of disease-induced extinction, Ecol. Letters, 8 (2005), 117–126. https://doi.org/10.1111/j.1461-0248.2004.00693.x doi: 10.1111/j.1461-0248.2004.00693.x
    [24] K. F. Smith, D. F. Sax, K. D. Lafferty, Evidence for the role of infectious disease in species extinction and endangerment, Conserv. Biol., 20 (2006), 1349–1357. https://doi.org/10.1111/j.1523-1739.2006.00524.x doi: 10.1111/j.1523-1739.2006.00524.x
    [25] A. L. Greer, C. J. Briggs, J. P. Collins, Testing a key assumption of host-pathogen theory: density and disease transmission, Oikos, 117 (2008), 1667–1673. https://doi.org/10.1111/j.1600-0706.2008.16783.x doi: 10.1111/j.1600-0706.2008.16783.x
    [26] B. H. Carlson, H. R. Thieme, Can infectious diseases that do not confer immunity eradicate host species?, J. Biol. Syst. (to appear). https://doi.org/10.1142/S0218339023400016
    [27] A. Taylor, W. Mann, Advanced Calculus, Sec. ed., Xerox College Publishing, Lexington, 1972.
    [28] P. E. Sartwell, The distribution of incubation periods of infectious disease, Am. J. Hyg., 51 (1950), 310–318. https://doi.org/10.1093/oxfordjournals.aje.a119397 doi: 10.1093/oxfordjournals.aje.a119397
    [29] P. E. Sartwell, The incubation period and the dynamics of infectious disease, Am. J. Epidemiol., 83 (1966), 204–216. https://doi.org/10.1093/oxfordjournals.aje.a120576 doi: 10.1093/oxfordjournals.aje.a120576
    [30] Z. Feng, Applications of Epidemiological Models to Public Health Policymaking: The Role of Heterogeneity in Model Predictions, World Scientific, Singapore, 2014. https://doi.org/10.1142/8884
    [31] W. O. Kermack, A. G. McKendrick, Contributions to the mathematical theory of epidemics. ii. The problem of endemicity, in Proceedings of the Royal Society of London. Series A, containing papers of a mathematical and physical character, 138 (1932), 55–83. https://doi.org/10.1098/rspa.1932.0171
    [32] W. O. Kermack, A. G. McKendrick, Contributions to the mathematical theory of epidemics. iii. Further studies of the problem of endemicity, in Proceedings of the Royal Society of London. Series A, Containing papers of a mathematical and physical character, 141 (1933), 94–122. https://doi.org/10.1098/rspa.1933.0106
    [33] H. R. Thieme, C. Castillo-Chavez, How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM J. Appl. Math., 53 (1993), 1447–1479. https://doi.org/10.1137/0153068 doi: 10.1137/0153068
    [34] H. Inaba, Age-structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. https://doi.org/10.1007/978-981-10-0188-8
    [35] M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015. https://doi.org/10.1007/978-1-4899-7612-3
    [36] H. R. Thieme, J. Yang, An endemic model with variable re-infection rate and applications to influenza, Math. Biosci., 180 (2002), 207–235. https://doi.org/10.1016/S0025-5564(02)00102-5 doi: 10.1016/S0025-5564(02)00102-5
    [37] F. Hoppensteadt, Mathematical Theories of Populations: Demographics Genetics and Epidemics, Society for Industrial and Applied Mathematics, Philadelphia, 1975. https://doi.org/10.1137/1.9781611970487
    [38] S. Luckhaus, A. Stevens, A free boundary problem-in time-for the spread of Covid-19, J. Math. Biol., 86 (2023), 17. https://doi.org/10.1007/s00285-023-01881-0 doi: 10.1007/s00285-023-01881-0
    [39] S. Luckhaus, A. Stevens, Kermack and McKendrick models on a two-scale network and connections to the Boltzmann equations, in Mathematics Going Forward: Collected Mathematical Brushstrokes, Springer, 2022,417–427. https://doi.org/10.1007/978-3-031-12244-6_29
    [40] Z. Ma, J. Li, Dynamical Modeling and Analysis of Epidemics, World Scientific, Singapore, 2009.
    [41] F. Abbona, E. Venturino, An eco-epidemic model for infectious keratoconjunctivitis caused by Mycoplasma conjunctivae in domestic and wild herbivores, with possible vaccination strategies, Math. Methods Appl. Sci., 41 (2018), 2269–2280. https://doi.org/10.1002/mma.4209 doi: 10.1002/mma.4209
    [42] T. Dhiraksakdanon, A model of infectious diseases in amphibian populations with ephemeral larval habitat, Dissertation, Arizona State University, 2010.
    [43] O. Diekmann, Thresholds and traveling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109–130. https://doi.org/10.1007/BF02450783 doi: 10.1007/BF02450783
    [44] H. R. Thieme, A model for the spatial spread of an epidemic, J. Math. Biol., 4 (1977), 337–351. https://doi.org/10.1007/BF00275082 doi: 10.1007/BF00275082
    [45] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, NJ, USA, 2003.
    [46] H. L. Smith, H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Soc., 2011. https://doi.org/10.1090/gsm/118
    [47] A. Farrell, Prey-predator-parasite: An ecosystem model with fragile persistence, Arizona State University, 2017.
    [48] J. A. P. Heesterbeek, J. A. Metz, The saturating contact rate in marriage-and epidemic models, J. Math. Biol., 31 (1993), 529–539. https://doi.org/10.1007/BF00173891 doi: 10.1007/BF00173891
    [49] P. Magal, X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251–275. https://doi.org/10.1137/S0036141003439173 doi: 10.1137/S0036141003439173
    [50] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003.
    [51] C. K. Busenberg S, Vertically Transmitted Diseases. Models and Dynamics, Springer, Berlin Heidelberg, 1993. https://doi.org/10.1007/978-3-642-75301-5
    [52] V. Capasso, Mathematical Structures of Epidemic Systems, Springer, Berlin Heidelberg, 1993. https://doi.org/10.1007/978-3-540-70514-7
    [53] D. Breda, O. Diekmann, W. De Graaf, A. Pugliese, R. Vermiglio, On the formulation of epidemic models (an appraisal of Kermack and McKendrick), J. Biol. Dynam., 6 (2012), 103–117. https://doi.org/10.1080/17513758.2012.716454 doi: 10.1080/17513758.2012.716454
    [54] O. Diekmann, The 1927 epidemic model of Kermack and Mckendrick: A success story or a tragicomedy?, SMB Newsletter, 92, 1–4.
    [55] H. R.Thieme, Disease extinction and disease persistence in age-structured epidemic models, Nonlinear Anal. Theory Methods Appl., 47 (2001), 6181–6194. https://doi.org/10.1016/S0362-546X(01)00677-0 doi: 10.1016/S0362-546X(01)00677-0
    [56] E. Venturino, Ecoepidemiology: A more comprehensive view of population interactions, Math. Model. Nat. Phenom., 11 (2016), 49–90. https://doi.org/10.1051/mmnp/201611104 doi: 10.1051/mmnp/201611104
    [57] L. Han, Z. Ma, H. W. Hethcote, Four predator-prey models with infectious diseases, Math. Computer Model., 34 (2001), 849–858. https://doi.org/10.1016/S0895-7177(01)00104-2 doi: 10.1016/S0895-7177(01)00104-2
    [58] O. Diekmann, M. Kretzschmar, Patterns in the effects of infectious diseases on population growth, J. Math. Biol., 29 (1991), 539–570. https://doi.org/10.1007/BF00164051 doi: 10.1007/BF00164051
    [59] T. Dhirasakdanon, H. R. Thieme, Stability of the endemic coexistence equilibrium for one host and two parasites, Math. Model. Nat. Phenom., 5 (2010), 109–138. https://doi.org/10.1051/mmnp/20105606 doi: 10.1051/mmnp/20105606
    [60] L. J. S. Allen, E. J. Allen, Deterministic and stochastic SIR epidemic models with power function transmission and recovery rates, Math. Contin. Discrete Dyn. Syst., 618 (2014), 1–15. https://doi.org/10.1090/conm/618/12329 doi: 10.1090/conm/618/12329
    [61] O. Diekmann, H. Inaba, A systematic procedure for incorporating separable static hetero- geneity into compartmental epidemic models, J. Math. Biol., 86 (2023), 29. https://doi.org/10.1007/s00285-023-01865-0 doi: 10.1007/s00285-023-01865-0
    [62] A. S. Novozhilov, On the spread of epidemics in a closed heterogeneous population, Math. Biosci., 215 (2008), 177–185. https://doi.org/10.1016/j.mbs.2008.07.010 doi: 10.1016/j.mbs.2008.07.010
    [63] N. T. Bailey, The Mathematical Theory of Infectious Diseases and its Applications, Charles Griffin & Company Ltd, 5a Crendon Street, High Wycombe, 1975.
    [64] F. Brauer, C. Castillo-Chavez, Z. Feng, Mathematical Models in Epidemiology, Springer, New York, 2019. https://doi.org/10.1007/978-1-4939-9828-9
    [65] A. Ducrot, Q. Griette, Z. Liu, P. Magal, Differential Equations and Population Dynamics I, Springer, Cham, 2022. https://doi.org/10.1007/978-3-030-98136-5
    [66] X.-Z. Li, J. Yang, d M. Martcheva, Age Structured Epidemic Modeling, vol. 52 makes no sense unless the series is mentioned as well, Springer Nature, Cham, 2020.
    [67] L. Rass, J. Radcliffe, Spatial Deterministic Epidemics, American Mathematical Soc., Providence, 2003. https://doi.org/10.1090/surv/102
    [68] K. P. Hadeler, Topics in Mathematical Biology, Springer, Cham, 2017. https://doi.org/10.1007/978-3-319-65621-2
    [69] M. Iannelli, F. Milner, The Basic Approach to Age-Structured Population Dynamics. Models, Methods and Numerics, Springer, Dordrecht, 2017. https://doi.org/10.1007/978-94-024-1146-1
    [70] J. McDonald, N. Weiss, A Course in Real Analysis, Academic Press, San Diego, 1999.
    [71] A. G. McKendrick, Applications of mathematics to medical problems, Proceed. Edinburgh Math. Soc., 14 (1926), 98–130. https://doi.org/10.1017/S0013091500034428 doi: 10.1017/S0013091500034428
    [72] A. J. Lotka, Relation between birth rates and death rates, Science, 26 (1907), 21–22. https://doi.org/10.1126/science.26.653.21.b doi: 10.1126/science.26.653.21.b
    [73] E. Hewitt and K. Stromberg, Real and Abstract analysis, Springer, Berlin Heidelberg, 1956.
    [74] G. Gripenberg, S.-O. Londen, O. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, 1990. https://doi.org/10.1017/CBO9780511662805
    [75] W. M. Hirsch, H. Hanisch, J.-P. Gabriel, Differential equation models of some parasitic infections: methods for the study of asymptotic behavior, Commun. Pure Appl. Math., 38 (1985), 733–753. https://doi.org/10.1002/cpa.3160380607 doi: 10.1002/cpa.3160380607
    [76] K. Yosida, Functional Analysis, Sec. edition, Springer, Berlin Heidelberg, 1965–1968. https://doi.org/10.1007/978-3-662-25762-3
    [77] H. R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations, J. fur die Reine und Angewandte Mathematik, 1979 (1979), 94–121. https://doi.org/10.1515/crll.1979.306.94 doi: 10.1515/crll.1979.306.94
    [78] H. Amann, Gewöhnliche Differentialgleichungen, De Gruyter, Berlin, 1983.
    [79] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin Heidelberg, 1985. https://doi.org/10.1007/978-3-662-00547-7
    [80] C. D. Aliprantis, K. C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, Springer, Berlin Heidelberg, 1999, 2006. https://doi.org/10.1007/978-3-662-03961-8
    [81] P. Gwiazda, A. Marciniak-Czochra and H. R. Thieme, Measures under the flat norm as ordered normed vector space, Positivity, 22 (2018), 105–138. https://doi.org/10.1007/s11117-017-0503-z doi: 10.1007/s11117-017-0503-z
    [82] C. Düll, P. Gwiazda, A. Marciniak-Czochra, J. Skrzeczkowski, Spaces of Measures and their Applications to Structured Population Models, Cambridge University Press, Cambridge, 2021. https://doi.org/10.1017/9781009004770
    [83] P. Magal, S. Ruan, Theory and applications of abstract semilinear Cauchy problems, Springer, Cham, 2018. https://doi.org/10.1007/978-3-030-01506-0
    [84] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differ. Integral Equat., 3 (1990), 1035–1066. https://doi.org/10.57262/die/1379101977 doi: 10.57262/die/1379101977
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Mathematical Biosciences and Engineering
3.9

Metrics

Article views(984) PDF downloads(44) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Figures and Tables

Tables(1)

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog