The epidemic threshold of vector-borne diseases with seasonality. (English) Zbl 1098.92056

Summary: Cutaneous leishmaniasis is a vector-borne disease transmitted to humans by sandflies. We develop a mathematical model which takes into account the seasonality of the vector population and the distribution of the latent period from infection to symptoms in humans. Parameters are fitted to real data from the province of Chichaoua, Morocco. We also introduce a generalization of the definition of the basic reproduction number \(R_0\) which is adapted to periodic environments. This \(R_0\) is estimated numerically for the epidemic in Chichaoua; \(R_0\simeq 1.94\). The model suggests that the epidemic could be stopped if the vector population were reduced by a factor \((R_0)^2\simeq 3.76\).


92D30 Epidemiology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
Full Text: DOI HAL


[1] Anderson R.M., May R.M. (1991) Infectious Diseases of Humans – Dynamics and Control. Oxford University Press, Oxford
[2] Anita S., Iannelli M., Kim M.Y., Park E.J. (1998) Optimal harvesting for periodic age-dependent population dynamics. SIAM J. Appl. Math. 58, 1648–1666 · Zbl 0935.92030
[3] Ben Salah A., Smaoui H., Mbarki L., Anderson R.M., Ben Ismaïl R. (1994) Développement d’un modèle mathématique de la dynamique de transmission de la leishmaniose canine. Archs. Inst. Pasteur Tunis 71, 431–438
[4] Burattini M.N., Coutinho F.A.B., Lopez L.F., Massad E. (1998) Modelling the dynamics of leishmaniasis considering human, animal host and vector populations. J. Biol. Syst. 6, 337–356
[5] Chaves L.F., Hernandez M.J. (2004) Mathematical modelling of American Cutaneous Leishmaniasis: incidental hosts and threshold conditions for infection persistence. Acta Tropica 92, 245–252
[6] Coale A.J. (1972) The Growth and Structure of Human Populations – a Mathematical Investigation. Princeton University Press, Princeton
[7] Desjeux P. (2004) Leishmaniasis: current situation and new perspectives. Comp. Immunol. Microbiol. Infect. Dis. 27, 305–318
[8] Diekmann O., Heesterbeek J.A.P. (2000) Mathematical Epidemiology of Infectious Diseases – Model Building, Analysis and Interpretation. Wiley, Chichester · Zbl 0997.92505
[9] Feliciangeli M.D. (2004) Natural breeding places of phlebotomine sandflies. Med. Vet. Entomol. 18, 71–80
[10] Guernaoui S., Boumezzough A., Pesson B., Pichon G. (2005) Entomological investigations in Chichaoua: an emerging epidemic focus of cutaneous leishmaniasis in Morocco. J. Med. Entomol. 42, 697–701
[11] Hasibeder G., Dye C., Carpenter J. (1992) Mathematical modelling and theory for estimating the basic reproduction number of canine leishmaniasis. Parasitology 105, 43–53
[12] Heesterbeek J.A.P., Roberts M.G. (1995) Threshold quantities for helminth infections. J. Math. Biol. 33, 415–434 · Zbl 0822.92018
[13] Heesterbeek J.A.P., Roberts M.G. (1995) Threshold quantities for infectious diseases in periodic environments. J. Biol. Syst. 3, 779–787
[14] Jagers P., Nerman O. (1985) Branching processes in periodically varying environment. Ann. Prob. 13, 254–268 · Zbl 0557.60072
[15] Kerr S.F., Grant W.E., Dronen N.O Jr. (1997) A simulation model of the infection cycle of Leishmania mexicana in Neotoma micropus. Ecol. Modell. 98, 187–197
[16] Lotka A.J. (1923) Contribution to the analysis of malaria epidemiology. Am. J. Hygiene 3, 1–121
[17] Kermack W.O., McKendrick A.G. (1927) A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond A 115, 700–721 · JFM 53.0517.01
[18] Ma J., Ma Z. (2006) Epidemic threshold conditions for seasonally forced SEIR models. Math. Biosci. Eng. 3, 161–172 · Zbl 1089.92048
[19] Ministère de la Santé Publique du Maroc Etat d’avancement des programmes de lutte contre les maladies parasitaires. Direction de l’épidémiologie et de lutte contre les maladies, Rabat (2001)
[20] Rabinovich J.E., Feliciangeli M.D. (2004) Parameters of Leishmania Braziliensis transmission by indoor Lutzomyia Ovallesi in Venezuela. Am. J. Trop. Med. Hygiene. 70, 373–382
[21] Ross R. (1911) The Prevention of Malaria. John Murray, London
[22] Thieme H.R. (1984) Renewal theorems for linear periodic Volterra integral equations. J. Integral Equations 7, 253–277 · Zbl 0566.45016
[23] Williams B.G., Dye C. (1997) Infectious disease persistence when transmission varies seasonally. Math. Biosci. 145, 77–88 · Zbl 0896.92024
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.