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On a Reproductive Stage-Structured Model for the Population Dynamics of the Malaria Vector

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Abstract

A reproductive stage-structured deterministic differential equation model for the population dynamics of the human malaria vector is derived and analysed. The model captures the gonotrophic and behavioural life characteristics of the female Anopheles sp. mosquito and takes into consideration the fact that for the purposes of reproduction, the female Anopheles sp. mosquito must visit and bite humans (or animals) to harvest necessary proteins from blood that it needs for the development of its eggs. Focusing on mosquitoes that feed exclusively on humans, our results indicate the existence of a threshold parameter, the vectorial reproduction number, whose size increases with increasing number of gonotrophic cycles, and is also affected by the female mosquito’s birth rate, its attraction and visitation rate to human residences, and its contact rate with humans. A stability analysis of the model indicates that the mosquito can establish itself in the environment if and only if the value of the vectorial reproduction number exceeds unity and that mosquito eradication is possible if the vectorial reproduction number is less than unity, since, then, the trivial steady state which always exist is unique and is globally and asymptotically stable. When a persistent vector population steady state exists, it is locally and asymptotically stable for a range of reproduction numbers, but can also be driven to instability via a Hopf bifurcation as the reproduction number increases further away from unity. The model derivation identifies and characterizes control parameters relating to activities such as human-mosquito contact and the mosquito’s survival chances between blood meals and egg laying. Our results show that the total mosquito population size increases with increasing number of gonotrophic cycles. Therefore understanding the fundamental aspects of the mosquito’s behaviour provides a pathway for the study of human-mosquito contact and mosquito population control. Control of the mosquito population densities would ultimately lead to malaria control.

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Notes

  1. Females Anopheles can survive for up to a month (or more in captivity) but most probably do not live longer than 2–3 weeks in nature Giles and Warrel (2002).

  2. A Metamorphic cycle starts when the female mosquito lays eggs on water (Eggs are laid every two to three days in batches and each batch can contain over one hundred eggs.). After about 2–3 days, the eggs hatch into larvae which live in a aquatic environment. The larvae eventually mature into pupae in about 4–10 days, and the pupae then change into adult free flying adult mosquitoes in about 2–4 days. The duration of the metamorphic cycle, from egg laying to an adult mosquito eclosion, varies between 7 and 20 days depending on the ambient temperature of the swamp and the species of mosquito involved.

  3. The process of blood feeding, egg maturation and oviposition that is repeated several times during the mosquito’s entire reproductive life is referred to as the gonotrophic cycle. The mosquito’s first batch of eggs may require more than one blood meal to mature but subsequently, blood meals and oviposition alternate regularly.

  4. In the context of malaria transmission, it is the questing vectors that are in at least their second or higher reproductive stage that can transmit infection.

  5. After an unsuccessful interaction with the human, the vector may have a chance to try again, but for now, we have assumed for simplicity that the vector is killed in case of an unsuccessful interaction.

  6. The most general form of the initial condition would take the form \(V_{i}(t_{0}) = V_{i}^{0}\), \(W_{i}(t_{0}) = W_{i}^{0}\) and \(U_{i+1}(t_{0}) = U_{i+1}^{0}\) for \(i\in \{0,1,2,3,\ldots ,N-1\}\), where \(t_{0}\) is some initial starting time and \(V_{i}^{0}\), \(W_{i}^{0}\) and \(U_{i}^{0}\) some prescribed constants. However, for simplicity, and without loss of generality, we have taken \(t_{0}=0\), together with the form (16) for \(V_{i}^{0}\), \(W_{i}^{0}\) and \(U_{i}^{0}\).

  7. Recall that, for this model, all new births go to vectors of type \(V_{0}\).

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Acknowledgments

GAN and MYF-N acknowledge support of the Cameroon Ministry of Higher Education through the initiative for the modernization of research in Cameroon’s Higher Education granting scheme for 2013. GAN, MIT-E and CNN acknowledge the support of NIMBioS through the investigative workshop grant that supported a workshop on Malaria Modeling and Control, co-organized by MIT-E et al. in 2011. They also acknowledge support of the NSF grant DMS-1261662 that supported a 2013 School on Stochastic Analysis, Financial, and Actuarial Mathematics with Applications, that enabled all five authors to meet. CNN was supported by a Postdoctoral Fellowship from the National Institute for Mathematical and Biological Synthesis, an Institute sponsored by the National Science Foundation, the U.S. Department of Homeland Security, and the U.S. Department of Agriculture through NSF Award #EF-0832858, with additional support from The University of Tennessee, Knoxville. TTW acknowledges AMMSI through the Department of Mathematics at the University of Buea for support towards his research. All authors acknowledge the constructive comments of the reviewers which greatly improved the clarity of our results.

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Authors and Affiliations

  1. Department of Mathematics, University of Buea, P.O. Box 63, Buea, Cameroon

    Gideon A. Ngwa, Terence T. Wankah & Mary Y. Fomboh-Nforba

  2. National Institute for Mathematical and Biological Synthesis (NIMBioS), The University of Tennessee, Knoxville, TN, 37996-1527, USA

    Calsitus N. Ngonghala

  3. Department of Mathematics, Lehigh University, Bethlehem, PA, 18015, USA

    Miranda I. Teboh-Ewungkem

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  1. Gideon A. Ngwa
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  2. Terence T. Wankah
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Correspondence to Miranda I. Teboh-Ewungkem.

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Ngwa, G.A., Wankah, T.T., Fomboh-Nforba, M.Y. et al. On a Reproductive Stage-Structured Model for the Population Dynamics of the Malaria Vector. Bull Math Biol 76, 2476–2516 (2014). https://doi.org/10.1007/s11538-014-0021-0

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