×

zbMATH — the first resource for mathematics

Case fatality models for epidemics in growing populations. (English) Zbl 1348.92152
Summary: The asymptotically homogeneous SIR model of H. R. Thieme [ibid. 111, No. 1, 99–130 (1992; Zbl 0782.92018)] for growing populations, with incidence depending in a general way on total population size, is reconsidered with respect to other parameterizations that give clear insight into epidemiological relevant relations and thresholds. One important feature of the present approach is case fatality as opposed to differential mortality. Although case fatality models and differential mortality models are equivalent via a transformation in parameter space, the underlying ideas and the dynamic behaviors are different, e.g. the basic reproduction number depends on differential mortality but not on case fatality. The persistent distributions and exponents of growth of infected solutions are computed and discussed in terms of the parameters. The notion of asymptotically exponentially growing state (as opposed to stationary state or exponential solution) coined by Thieme is interpreted in terms of stability theory. Of some interest are limiting cases of models without recovery where two infected solutions exist.

MSC:
92D30 Epidemiology
92D25 Population dynamics (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Thieme, H., Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations, Math. Biosci., 111, 99-130, (1992) · Zbl 0782.92018
[2] Anderson, R.; May, R., Population biology of infectious diseases: part I, Nature, 280, 361-367, (1979)
[3] Brauer, F., Models for the spread of universally fatal diseases, J. Math. Biol., 28, 451-462, (1990) · Zbl 0718.92021
[4] Busenberg, S.; van den Driessche, P., Analysis of a disease transmission model in a population with varying size, J. Math. Biol., 29, 257-270, (1990) · Zbl 0725.92021
[5] Gao, L.; Hethcote, H., Disease transmission models with density-dependent demographics, J. Math. Biol., 30, 717-731, (1992) · Zbl 0774.92018
[6] Louie, K.; Roberts, M.; Wake, G., The regulation of an age-structured population by a fatal disease, IMA J. Math. Appl. Med. Biol., 11, 229-244, (1994) · Zbl 0812.92019
[7] Martcheva, M.; Castillo-Chavez, C., Diseases with chronic stage in a population with varying size, Math. Biosci., 182, 1-25, (2003) · Zbl 1012.92024
[8] Mena-Lorca, J.; Hethcote, H., Dynamic models of infectious diseases as regulators of population sizes, J. Math. Biol., 30, 693-716, (1992) · Zbl 0748.92012
[9] Zhou, J.; Hethcote, H., Population size dependent incidence in models for diseases without immunity, J. Math. Biol., 32, 809-834, (1994) · Zbl 0823.92027
[10] Diekmann, O.; Heesterbeek, J., Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, (2000), Wiley · Zbl 0997.92505
[11] Andreasen, V., Disease regulation of age-structured host populations, Theor. Popul. Biol., 36, 214-239, (1989) · Zbl 0688.92009
[12] Thieme, H., Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 24, 407-435, (1993) · Zbl 0774.34030
[13] Busenberg, S.; Hadeler, K., Demography and epidemics, Math. Biosci., 101, 63-74, (1990) · Zbl 0751.92012
[14] Safan, M., Spread of Infectious Diseases: Impact on Demography and the Eradication Effort in Models with Backward Bifurcation, (2006), University of Tübingen
[15] Blower, S., Daniel Bernoulli, an attempt at a new analysis of the mortality caused by smallpox and of the advantages of inoculation to prevent it, Rev. Med. Virol., 14, 275-288, (2004)
[16] Dietz, K.; Heesterbeek, J., Daniel bernoulli’s epidemiological model revisited, Math. Biosci., 180, 1-21, (2002) · Zbl 1019.92028
[17] Anderson, R.; Gupta, S.; May, R., Potential of community-wide chemotherapy or immunotherapy to control the spread of HIV-1, Nature, 350, 356-359, (1991)
[18] Day, T., On the evolution of virulence and the relationship between various measures of mortality, Proc. R. Soc. Lond. B, 269, 1317-1323, (2002)
[19] Ma, J.; van den Driessche, P., Case fatality proportion, Bull. Math. Biol., 70, 118-133, (2008) · Zbl 1281.92061
[20] Ghani, A.; Donnelly, C.; Cox, D.; Griffin, J.; Fraser, C.; Lam, T.; Ho, L.; Chan, W.; Anderson, R.; Hedley, A.; Leung, G., Methods for estimating the case fatality ratio for a novel, emerging infectious disease, Am. J. Epidemiol., 162, 479-486, (2005)
[21] Lipsitch, M.; Donnelly, C.; Fraser, C.; Blake, I.; Cori, A.; Dorigatti, I.; Ferguson, N.; Garske, T.; Mills, H.; Riley, S., Potential biases in estimating absolute and relative case fatality risks during outbreaks, PLoS Negl. Trop. Dis, 9, 7, e0003846, (2015)
[22] Hadeler, K., Periodic solutions of homogeneous equations, J. Differ. Equ., 95, 183-202, (1992) · Zbl 0747.34030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.