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A semi-hybrid model for malaria with limited superinfection

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Philippe Picard
Philippe Picard*
Affiliation:
Université de Lyon 1
*
∗ Postal address: Mathématiques Appliquées, Université de Lyon 1, 43 Boulevard du 11 Novembre 1918, F-69622 Villeurbanne Cedex, France. Email address: isfa@univ_lyon1.fr
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Abstract

Modelling malaria with consistency necessitates the introduction of at least two families of interconnected processes. Even in a Markovian context the simplest fully stochastic model is intractable and is usually transformed into a hybrid model, by supposing that these two families are stochastically independent and linked only through two deterministic connections. A model closer to the fully stochastic model is presented here, where one of the two families is subordinated to the other and just a unique deterministic connection is required. For this model a threshold theorem can be proved but the threshold level is not the one obtained in a hybrid model. The difference disappears only when the human population size approaches infinity.

Keywords

Nåsell-Hirsch modelhybrid modelthreshold conditionasymptotic eradicationsaturation levelstwo-state Markov processsubordinated process

MSC classification

Primary: 92D30: Epidemiology
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 30 , Issue 4 , December 1998 , pp. 1027 - 1057
Copyright
Copyright © Applied Probability Trust 1998 

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References

Bailey, N. (1982). The Biomathematics of Malaria. Griffin, London.Google Scholar
Ball, F. and Nåsell, I., (1994). The shape of the size distribution of an epidemic in a finite population. Math. Biosci. 123, 167–181.CrossRefGoogle Scholar
Dietz, K. (1980). Models of vector-borne parasitic diseases. Lecture Notes Biomath. 39, 264–277.CrossRefGoogle Scholar
Gani, J. (1967). A problem of virus populations: attachment and detachment of antibodies. Math. Biosci. 1, 545–554.CrossRefGoogle Scholar
Gani, J. and Srivastava, R. C. (1968). A stochastic model for the attachment and detachment of antibodies to virus. Math. Biosci. 3, 307–322.CrossRefGoogle Scholar
Macdonald, G. (1957). The Epidemiology and Control of Malaria. Oxford University Press, London.Google Scholar
Milligan, P. and Downham, D. (1996). Models of superinfection and acquired immunity to multiple parasite strains. J. Appl. Prob. 33, 915–932.CrossRefGoogle Scholar
Nåsell, I. (1976). A hybrid model of schistosomiasis with snail latency. Theoret. Popn. Biol. 10, 47–69.CrossRefGoogle ScholarPubMed
Nåsell, I. (1985). Hybrid Models of Tropical infections. Springer, Heidelberg.CrossRefGoogle Scholar
Nåsell, I. (1986). On superinfection in malaria. IMA J. Math. Appl. Medicine Biology 3, 211–227.CrossRefGoogle ScholarPubMed
Nåsell, I. (1991). On the quasi-stationary distribution of the Ross malaria model. Math. Biosc. 107, 187–208.CrossRefGoogle ScholarPubMed
Nåsell, I. (1995). The threshold concept in stochastic epidemic and endemic models. In Epidemic Models: Their Structure and Relation to Data, ed. Mollison, D.. Cambridge University Press, Cambridge, pp. 71–83.Google Scholar
Nåsell, I. and Hirsch, W. M. (1972). A mathematical model of some helminthic infections. Comm. Pure Appl. Math., 25, 459–477.CrossRefGoogle Scholar
Picard, Ph. (1980). Applications of martingale theory to some epidemic models. J. Appl. Prob. 17, 583–599.CrossRefGoogle Scholar
Picard, Ph. (1985). On the integral functionals of linear birth death immigration processes. Stat. Decisions, Suppl. 2, 105–109.Google Scholar
Picard, Ph. (1989). Two variants of the first Nå sell–Hirsch model. Math. Biosci. 94, 45–85.CrossRefGoogle Scholar
Ross, R. (1911). The Prevention of Malaria. Murray, London.Google ScholarPubMed
Ross, R. (1916). An application of the theory of probabilities to the study of a priori pathometry, I. Proc. R. Soc. London A 92, 204–230.Google Scholar