## Abstract

We derive and study a deterministic compartmental model for malaria transmission with varying human and mosquito populations. Our model considers disease-related deaths, asymptomatic immune humans who are also infectious, as well as mosquito demography, reproduction and feeding habits. Analysis of the model reveals the existence of a backward bifurcation and persistent limit cycles whose period and size is determined by two threshold parameters: the vectorial basic reproduction number \(\fancyscript{R}_{m}\), and the disease basic reproduction number \(\fancyscript{R}_0\), whose size can be reduced by reducing \(\fancyscript{R}_{m}\). We conclude that malaria dynamics are indeed oscillatory when the methodology of explicitly incorporating the mosquito’s demography, feeding and reproductive patterns is considered in modeling the mosquito population dynamics. A sensitivity analysis reveals important control parameters that can affect the magnitudes of \(\fancyscript{R}_{m}\) and \(\fancyscript{R}_0\), threshold quantities to be taken into consideration when designing control strategies. Both \(\fancyscript{R}_{m}\) and the intrinsic period of oscillation are shown to be highly sensitive to the mosquito’s birth constant \(\lambda _{m}\) and the mosquito’s feeding success probability \(p_{w}\). Control of \(\lambda _{m}\) can be achieved by spraying, eliminating breeding sites or moving them away from human habitats, while \(p_{w}\) can be controlled via the use of mosquito repellant and insecticide-treated bed-nets. The disease threshold parameter \(\fancyscript{R}_0\) is shown to be highly sensitive to \(p_{w}\), and the intrinsic period of oscillation is also sensitive to the rate at which reproducing mosquitoes return to breeding sites. A global sensitivity and uncertainty analysis reveals that the ability of the mosquito to reproduce and uncertainties in the estimations of the rates at which exposed humans become infectious and infectious humans recover from malaria are critical in generating uncertainties in the disease classes.

This is a preview of subscription content, access via your institution.

## Notes

- 1.
A common assumption made in the analysis of epidemic models is that the duration of immunity is independent of exposure to infection (Anderson and May 1979; Hethcote et al. 1982; Anderson 1991). However, immunity to malaria is sustained by continuous exposure (Aron 1983), and such acquired immunity may confer protection against severe clinical illness without eliminating chronic or mild infections. Thus, asymptomatic immune malaria carriers can be infective (Aron 1988), which we’ve assumed here, that it is at a lower rate.

## References

World Health Organisation, The World Malaria Report 2010, WHO Press (2011)

Murray CJL, Rosenfeld LC, Lim SS, Andrews KG, Foreman KJ, Haring D, Fullman N, Naghavi M, Lozano R, Lopez AD Global malaria mortality between 1980 and 2010: a systematic analysis, The Lancet 379 (9814) (9814) (2012) 413–431

Chitnis N, Cushing JM, Hyman JM (2006) Bifurcation analysis of a mathematical model for malaria transmission. SIAM J Appl Math 67(1):24–45

Chitnis N, Hyman JM, Cushing JM (2008) Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull Math Biol 70:1272–1296

Ngwa GA, Shu WS (2000) A mathematical model for endemic malaria with variable human and mosquito populations. Mathematical and Computational Modeling 32(7):747–763

Ngwa GA, Ngonghala CN, Wilson NBS (2001) A model for endemic malaria with delay and variable populations. J Cameroon Acad Sci 1(3):169–186

Ngwa GA (2004) Modelling the dynamics of endemic malaria in growing populations. Discret Contin Dyn Syst Ser B 4(4):1173–1202

Teboh-Ewungkem MI, Podder CN, Gumel AB (2010) Mathematical study of the role of gametocytes and an imperfect vaccine on malaria transmission dynamics. Bull Math Biol 72(1):63–93

Teboh-Ewungkem MI Malaria Control (2009) The role of local communities as seen through a mathematical model in a changing population-Cameroon, Chapter 4, pp 103–140, in advances in disease epidemiology, Nova Science Publishers

Teboh-Ewungkem MI, Yuster T (2010) A within-vector mathematical model of plasmodium falciparum and implications of incomplete fertilization on optimal gametocyte sex ratio. J Theor Biol 264:273–286

Teboh-Ewungkem MI, Wang M (2012) Male fecundity and optimal gametocyte sex ratios for Plasmodium falciparum during incomplete fertilization. J Theor Biol 307:183–192

Ngwa GA (2006) On the Population Dynamics of the Malaria Vector. Bulletin of Mathematical Biology 68(8):2161–2189

Nourridine S, Teboh-Ewungkem MI, Ngwa GA (2011) A mathematical model of the population dynamics of disease transmitting vectors with spatial consideration. J Biol Dyn 5(4):335–365

Ngwa GA, Niger AM, Gumel AB (2010) Mathematical assessment of the role of non-linear birth and maturation delay in the population dynamics of the malaria vector. Appl Math Comput 217:3286–3313

Ngonghala CN, Ngwa GA, Teboh-Ewungkem MI (2012) Periodic oscillations and backward bifurcation in a model for the dynamics of malaria transmission. Math Biosci 240(1):45–62

Aron JL (1983) Dynamics of acquired immunity boosted by exposure to infection. Mathematical Biosciences 64:249–253

Mwambi H, Baumgärtner J, Hadeler K (2000) Ticks and tick-borne diseases: a vector-host interaction model for the brown ear tick (rhipicephalus appendiculatus). Stat Methods Med Res 9(3):279–301

Rosa R, Pugliese A (2007) Effects of tick population dynamics and host densities on the persistence of tick-borne infections. Math Biosci 208(1):216–240

Reiner RC, Perkins TA, Barker CM, Niu T, Chaves LF, Ellis AM, George DB, Le Menach A, Pulliam JR, Bisanzio D et al (2013) A systematic review of mathematical models of mosquito-borne pathogen transmission: 1970–2010. Journal of The Royal Society Interface 10(81):20120921

Central Intelligence Agency, Country comparison: birth rate, The world fact book, Available at https://www.cia.gov/library/publications/the-world-factbook/rankorder/2054rank.html. Assessed June 2012

Central Intelligence Agency, Country comparison: Life expectancy at birth, The world fact book, Available at https://www.cia.gov/library/publications/the-world-factbook/rankorder/2102rank.html Assessed June 2012

Prudhomme O’Meara W, Smith DL, McKenzie FE (2006) Potential impact of intermittent preventive treatment (IPT) on spread of drug resistant malaria. PLoS Med 3(5):0633–0642

Klowden M, Briegel H (1994) Mosquito gonotrophic cycle and multiple feeding potential: contrasts between Anopheles and Aedes (Diptera: Culicidae). J Med Entomol 31(4):618–622

Ngonghala CN (2012) Mathematical modeling and analysis of epidemiological and chemical systems, ProQuest, UMI Dissertation Publishing

Pearl R, Reed L (1920) On the rate of growth of the population of the united states since 1970 and its mathematical interpretation. Proceedings of the National academy of Sciences 6:275–288

Verhulst P (1938) Notice sur la loi que la population suit dans son accroissement. Correspondence Mathématique et Physique 10:113–121

Beverton RJH, Holt SJ (1994) On the dynamics of exploited fish populations. Rev Fish Biol Fish 4:259–260

Ricker WE (1954) Stock and recruitment. J Fish Res Board Canada 11:559–623

Maynard-Smith J, Slatkin M (1973) The stability of predator-prey systems. Ecology 54:384–391

Anderson RM, May RM (1979) Population biology of infectious diseases: part I. Nature 280:361–367

Hethcote HW, Stech HW, van den Driessche P (1982) Periodicity and stability in epidemic models: a survey. In: Differential equations and applications in ecology, epidemics, and population problems, Academic Press, San Diego

Anderson RM, May R (1991) Infectious diseases of humans: dynamics and control. Oxford University Press, Oxford

Aron JL (1988) Acquired immunity dependent upon exposure in an SIRS epidemic model. Math Biosci 88:37–47

Lakshmikantham V, Leela S, Martynyuk AA (1989) Stability analysis of nonlinear systems. Marcel Dekker Inc., New York and Basel

LaSalle JP (1976) The stability of dynamical systems. Society for Industrial and Applied Mathematics, Philadelphia

Mutero C, Birley M (1987) Estimation of the survival rate and oviposition cycle of field populations of malaria vectors in kenya. J Appl Ecol 24(3): 853–863

van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180:29–48

Smith DL, McKenzie FE, Snow RW, Hay SI (2007) Revisiting the basic reproductive number for malaria and its implications for malaria control. PLoS Biol 5(3):e42

Dushoff J, Huang W, Castillo-Chavez C (1998) Backwards bifurcations and catastrophe in simple models of fatal diseases. J Math Biol 36(3):227–248

Castillo-Chavez C, Song B (2004) Dynamical models of tuberculosis and their applications. Math Biosci Eng 1(2):361–404

Hadeler K, Van den Driessche P (1997) Backward bifurcation in epidemic control. Math Biosci 146(1):15–35

Moore S, Shrestha S, Tomlinson K, Vuong H (2012) Predicting the effect of climate change on african trypanosomiasis: integrating epidemiology with parasite and vector biology. J Royal Soc Interface 9(70):817–830

Blower S, Hartel D, Dowlatabadi H, Anderson R, May R, Blower S, Hartel D, Dowlatabadi H, Anderson R (1991) Drugs, sex and hiv: a mathematical model for New York City, Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences 331(1260):171–187

Blower S, Dowlatabadi H (1994) Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example. International Statistical Review/Revue Internationale de Statistique. J Appl Ecol 62(2): 229–243

Marino S, Hogue I, Ray C, Kirschner D (2008) A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theor Biol 254(1):178–196

Gouagna L, Ferguson H, Okech B, Killeen G, Kabiru E, Beier J, Githure J, Yan G (2004) Plasmodium falciparum malaria disease manifestations in humans and transmission to anopheles gambiae: a field study in western kenya. Parasitology 128(03):235–243

Baton L, Ranford-Cartwright L et al (2005) Spreading the seeds of million-murdering death: metamorphoses of malaria in the mosquito. Trends Parasitol 21(12):573–580

He D, Earn D (2007) Epidemiological effects of seasonal oscillations in birth rates. Theor Popul Biol 72(2):274–291

Ireland J, Norman R, Greenman J (2004) The effect of seasonal host birth rates on population dynamics: the importance of resonance. Journal of theoretical biology 231(2):229–238

Yang G, Brook B, Bradshaw C (2009) Predicting the timing and magnitude of tropical mosquito population peaks for maximizing control efficiency. PLoS Negl Trop Dis 3(2):e385

Lloyd AL, Zhang J, Root AM (2007) Stochasticity and heterogeneity in host-vector models. J Royal Soc Interface 4(16):851–863

## Acknowledgments

CNN acknowledges the support of the National Institute for Mathematical and Biological Synthesis (NIMBioS), an Institute sponsored by the National Science Foundation (NSF), 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, and NSF Award #OISE-0855380. Additional funding for CNN comes from a Scholar Award in Complex Systems Science to MHB from the James S. McDonnell Foundation. MIT-E acknowledges support of NSF Award #OISE-0855380. GAN acknowledges the grants and support of the Cameroon Ministry of Higher Education through the initiative for the mordenization of research in Cameroon’s Higher Education.

## Author information

### Affiliations

### Corresponding author

## Rights and permissions

## About this article

### Cite this article

Ngonghala, C.N., Teboh-Ewungkem, M.I. & Ngwa, G.A. Persistent oscillations and backward bifurcation in a malaria model with varying human and mosquito populations: implications for control.
*J. Math. Biol.* **70, **1581–1622 (2015). https://doi.org/10.1007/s00285-014-0804-9

Received:

Revised:

Published:

Issue Date:

### Keywords

- Malaria transmission and control
- Mosquito demography
- Backward bifurcation
- Oscillatory dynamics
- Sensitivity analysis
- Reproduction numbers

### Mathematics Subject Classification

- 34A
- 34C
- 37G
- 92B
- 92D