# The Impact of Recruitment on the Dynamics of an Immune-Suppressed Within-Human–Host Model of the *Plasmodium falciparum* Parasite

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## Abstract

A model is developed and used to study within-human malaria parasite dynamics. The model integrates actors involved in the development–progression of parasitemia, gametocytogenesis and mechanisms for immune response activation. Model analyses under immune suppression reveal different dynamical behaviours for different healthy red blood cell (HRBC) generation functions. Existence of a threshold parameter determines conditions for HRBCs depletion. Oscillatory dynamics reminiscent of malaria parasitemia are obtained. A dependence exists on the type of recruitment function used to generate HRBCs, with complexities observed for a more nonlinear function. An upper bound that delimits the size of feasible parasitized steady-state solution exists for a logistic function but not a constant function. The upper bound is completely characterized and is affected by parameters associated with HRBCs recruitment, parasitized red blood cells generation and the release and time-to-release of free merozoites. A stable density size for mature gametocytes, the bridge to invertebrate hosts, is derived.

## Keywords

Within-human–host dynamics Innate and adaptive immune response Parasitemia Gametocytogenesis Global stability Red blood cells Malaria Recruitment## Notes

### Acknowledgements

The first author, WA, acknowledges support from the Department of Mathematics at Lehigh University, MIT-E’s Lehigh University Development fund, and Lehigh University as a whole, for supporting him and making available to him Lehigh’s resources, and for sponsoring as well as hosting him for more than two months as a visiting pre-doctoral scholar, enabling him to make significant progress on the work related to this manuscript and his thesis under the mentorship of MIT-E in conjunction with GAN via SKYPE. WA also acknowledges support from the African Institute for the Mathematical Sciences (AIMS) Cameroon that paid his flight for him to visit Lehigh University as a visiting pre-doctoral scholar, paving the path towards a successful completion of this manuscript and his doctoral dissertation. GAN acknowledges the grants and support of the Cameroon Ministry of Higher Education through the initiative for the modernization of research in Cameroon’s Higher Education. All three authors, WA, MIT-E and GAN acknowledge the support of the NSF -Directorate for Mathematical and Physical Science grant DMS-1544434 that created the opportunity for all three authors, who were present at the grant related activities (on School on Stochastic Analysis, Financial and Actuarial Mathematics with Applications) to meet as a unit for the first time and commence discussions on the manuscript and related project.

## References

- An G, Widness JA, Mock DM, Veng-Pedersen P (2016) A novel physiology-based mathematical model to estimate red blood cell lifespan in different human age groups. AAPS J 18(5):1182–1191Google Scholar
- Anderson RM (1998) Complex dynamic behaviours in the interaction between parasite populations and the host’s immune system. Int J Parasitol 28(4):551–566Google Scholar
- Anderson RM, May RM (1979) Population biology of infectious diseases: Part I. Nature 280:361–367Google Scholar
- Anderson RM, May RM (1991) Infectious diseases of humans: dynamics and control. Oxford University Press, OxfordGoogle Scholar
- Anderson RM, May RM, Gupta S (1989) Non-linear phenomena in hostparasite interactions. Parasitology 99(S1):S59–S79Google Scholar
- Aron JL (1983) Dynamics of acquired immunity boosted by exposure to infection. Math Biosci 64:249–253zbMATHGoogle Scholar
- Aron JL (1988a) Acquired immunity dependent upon exposure in an sirs epidemic model. Math Biosci 88:37–47MathSciNetzbMATHGoogle Scholar
- Aron JL (1988b) Mathematical modelling of immunity to malaria. Math Biosci 90(1):385–396MathSciNetzbMATHGoogle Scholar
- Augustine AD, Hall BF, Leitner WW, Mo AX, Wali Tonu M, Fauci Anthony S (2009) Niaid workshop on immunity to malaria: addressing immunological challenges. Nat Immunol 10(7):673–678Google Scholar
- Baron S (1996) Medical microbiolgy—galveston (tx). University of Texas Medical Branch at GalvestonGoogle Scholar
- Baton LA, Ranford-Cartwright LC (2005) Spreading the seeds of million-murdering death: metamorphoses of malaria in the mosquito. Trends Parasitol 21(12):573–580Google Scholar
- Bianconi E, Piovesan A, Facchin F, Beraudi A, Casadei Raffaella, Frabetti Flavia, Vitale Lorenza, Pelleri Maria Chiara, Tassani Simone, Piva Francesco et al (2013) An estimation of the number of cells in the human body. Ann Hum Biol 40(6):463–471Google Scholar
- Bichara D, Cozic N, Iggidr A (2012) On the estimation of sequestered parasite population in falciparum malaria patients. [Research Report] INRIA, RR-8178:22Google Scholar
- Bousema T, Drakeley C (2011) Epidemiology and infectivity of plasmodium falciparum and plasmodium vivax gametocytes in relation to malaria control and elimination. Clin Microbiol Rev 24(2):377–410Google Scholar
- Bousema T, Sutherland CJ, Churcher TS, Mulder B, Gouagna Louis C, Riley Eleanor M, Targett Geoffrey AT, Drakeley Chris J (2011) Human immune responses that reduce the transmission of plasmodium falciparum in african populations. Int J Parasitol 41(3):293–300Google Scholar
- Brännström BÅ, Sumpter DJT (2005) The role of competition and clustering in population dynamics. Proc R Soc B 272:2065–2072Google Scholar
- Brookhaven National Labortory (BNL) (2017) 56 Facts About Blood and Blood Donation. https://www.bnl.gov/hr/blooddrive/56facts.asp. Accessed April 2017
- Chiyaka C, Garira W, Dube S (2008) Modelling immune response and drug therapy in human malaria infection. Comput Math Method Med 9(2):143–163MathSciNetzbMATHGoogle Scholar
- Cowman AF, Berry D, Baum J (2012) The cellular and molecular basis for malaria parasite invasion of the human red blood cell. J Cell Biol 198(6):961–971Google Scholar
- Cuomo MJ, Noel LB, White DB (2009) Diagnosing medical parasites: a public health officers guide to assisting laboratory and medical officers. Technical report, DTIC DocumentGoogle Scholar
- Dean L, National Center for Biotechnology Information (U.S.) (2005) Blood groups and red cell antigens. NCBIGoogle Scholar
- Eichner M, Diebner HH, Molineaux L, Collins WE, Jeffery GM, Dietz K (2001) Genesis, sequestration and survival of plasmodium falciparum gametocytes: parameter estimates from fitting a model to malariatherapy data. Trans R Soc Trop Med Hyg 95(5):497–501Google Scholar
- Gardiner DL, Trenholme KR (2015) Plasmodium falciparum gametocytes: playing hide and seek. Ann Transl Med 3(4):45Google Scholar
- Ginsburg H, Hoshen MB (2002) Is the development of falciparum malaria in the human host limited by the availability of uninfected erythrocytes? Malar J 1(1):18Google Scholar
- Ginsburg H, Stein WD (1987) New permeability pathways induced by the malarial parasite in the membrane of its host erythrocyte: potential routes for targeting of drugs into infected cells. Biosci Rep 7(6):455–463Google Scholar
- Gottlieb Y, Topaz O, Cohen LA, Yakov LD, Haber Tom, Morgenstern Abigail, Weiss Avital, Berman Karen Chait, Fibach Eitan, Meyron-Holtz Esther G (2012) Physiologically aged red blood cells undergo erythrophagocytosis in vivo but not in vitro. Haematologica 97(7):994–1002Google Scholar
- Gravenor MB, Kwiatkowski D (1998) An analysis of the temperature effects of fever on the intra-host population dynamics of plasmodium falciparum. Parasitology 117(02):97–105Google Scholar
- Gravenor MB, Lloyd AL (1998) Reply to: Models for the in-host dynamics of malaria revisited: errors in some basic models lead to large over-estimates of growth rates. Parasitology 117(05):409–410Google Scholar
- Gurarie D, Karl S, Zimmerman PA, King CH, Pierre Timothy G St, Davis Timothy ME (2012) Mathematical modeling of malaria infection with innate and adaptive immunity in individuals and agent-based communities. PLoS One 7(3):e34040Google Scholar
- Heffernan JM (2011) Mathematical immunology of infectious diseases. Math Popul Stud 18(2):47–54MathSciNetzbMATHGoogle Scholar
- Hellriegel B (1992) Modelling the immune response to malaria with ecological concepts: short-term behaviour against long-term equilibrium. Proc R Soc Lond B Biol Sci 250(1329):249–256Google Scholar
- Hethcote HW, Stech HW, van den Driessche P (1982) Periodicity and stability in epidemic models: a survey. In: Busenberg S, Cooke KL (eds) Differential equations and applications in ecology, epidemics, and population problems. Academic Press, San Diego, pp 65–82Google Scholar
- Hetzel C, Anderson RM (1996) The within-host cellular dynamics of bloodstage malaria: theoretical and experimental studies. Parasitology 113(01):25–38Google Scholar
- Hoffman SL, Crutcher JM (2017) Malaria, Chapter 83. Medical Microbiology, Galveston (TX): University of Texas Medical Branch at Galveston, 4th ed, 1996. Accessed March 2017Google Scholar
- Hollowell JG, Van Assendelft OW, Gunter EW, Lewis BG, Najjar M, Pfeiffer C (2005) Hematological and iron-related analytes-reference data for persons aged 1 year and over: United states, 1988–94. Vital Health Stat Ser 11 Data Natl Health Surv 247(247):1–156Google Scholar
- Hoshen MB, Heinrich R, Stein WD, Ginsburg H (2000) Mathematical modelling of the within-host dynamics of plasmodium falciparum. Parasitology 121(03):227–235Google Scholar
- Iggidr A, Kamgang J-C, Sallet G, Tewa J-J (2006) Global analysis of new malaria intrahost models with a competitive exclusion principle. SIAM J Appl Math 67(1):260–278MathSciNetzbMATHGoogle Scholar
- Ingemar N (1985) Lecture notes in biomathematics. Springer, BerlinGoogle Scholar
- Janeway CA Jr, Travers P, Walport M, Shlomchik MJ (2001) Immunobiology: the immune system in health and disease, 5th edn. Garland Science, New York. Available from: https://www.ncbi.nlm.nih.gov/books/NBK10757/
- Josling GA, Llinás M (2015) Sexual development in plasmodium parasites: knowing when it’s time to commit. Nat Rev Microbiol 13(9):573–587Google Scholar
- Kaushal DC, Carter R, Miller LH, Krishna G (1980) Gametocytogenesis by malaria parasites in continuous culture. Nature 286(5772):490–2Google Scholar
- Kirk K (2001) Membrane transport in the malaria-infected erythrocyte. Physiol Rev 81(2):495–537Google Scholar
- Kiszewski Anthony E (2010) Blocking plasmodium falciparum malaria transmission with drugs: the gametocytocidal and sporontocidal properties of current and prospective antimalarials. Pharmaceuticals 4(1):44–68Google Scholar
- Kuehn A, Pradel G (2010) The coming-out of malaria gametocytes. BioMed Res Int 21(4):683–696Google Scholar
- Landaw SA (1987) Factors that accelerate or retard red blood cell senescence. Blood Cells 14(1):47–67Google Scholar
- Langhorne J (2006) Immunology and immunopathogenesis of malaria. Current topics in microbiology and immunology. Springer, BerlinGoogle Scholar
- Langhorne J, Ndungu FM, Sponaas A-M, Marsh K (2008) Immunity to malaria: more questions than answers. Nat Immunol 9(7):725–732Google Scholar
- Li Y, Ruan S, Xiao D (2011) The within-host dynamics of malaria infection with immune response. Math Biosci Eng 8(4):999–1018MathSciNetzbMATHGoogle Scholar
- McKenzie EF, Bossert WH (1997) The dynamics ofplasmodium falciparumblood-stage infection. J Theor Biol 188(1):127–140Google Scholar
- Mitri C, Thiery I, Bourgouin C, Paul REL (2009) Density-dependent impact of the human malaria parasite plasmodium falciparum gametocyte sex ratio on mosquito infection rates. Proc R Soc Lond B Biol Sci 276(1673):3721–3726Google Scholar
- National Institute of Allergy and Infectious Diseases (NIAID) (2010) The life cycle of the malaria parasite. https://www.cdc.gov/malaria/about/biology/index.html. Accessed Jan 2018
- 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):45–62MathSciNetzbMATHGoogle Scholar
- Ngonghala CN, Teboh-Ewungkem MI, Ngwa GA (2015) Persistent oscillations and backward bifurcation in a malaria model with varying human and mosquito populations: implications for control. J Math Biol 70(7):1581–1622MathSciNetzbMATHGoogle Scholar
- Ngonghala CN, Teboh-Ewungkem MI, Ngwa GA (2016) Observance of period-doubling bifurcation and chaos in an autonomous ode model for malaria with vector demography. Theor Ecol 9(3):337–351Google Scholar
- Ngwa CJ, de Rosa A, Thiago F, Pradel G (2017) The Biology of Malaria Gametocytes, chapter Current Topics in Malaria. InTech, 2016. Accessed March 2017Google Scholar
- Ngwa GA, Teboh-Ewungkem MI (2016) A mathematical model with quarantine states for the dynamics of ebola virus disease in human populations. Comput Math Method Med, Vol 2016, Article ID 9352725, 93 ppGoogle Scholar
- Okrinya A (2015) Mathematical modelling of malaria transmission and pathogenesis. PhD thesis, Loughborough UniversityGoogle Scholar
- Pearl R (1925) The biology of population growth. Alfred A. Knopf, New YorkGoogle Scholar
- Perlmann P, Troye-Blomberg M (2002) Malaria immunology, chemical immunology and allergy. Karger, BaselGoogle Scholar
- Rothman KJ, Greenland S, Lash TL (2008) Modern epidemiology. Lippincott Williams & Wilkins, BaltimoreGoogle Scholar
- Sackmann E (1995) Biological membranes architecture and function. Struct Dyn Membr 1:1–63Google Scholar
- Shemin D, Rittenberg D (1946) The life span of the human red blood cell. J Biol Chem 166(2):627–636Google Scholar
- Sinden RE (1982) Gametocytogenesis of plasmodium falciparum in vitro: an electron microscopic study. Parasitology 84(01):1–11Google Scholar
- Sompayrac LM (2015) How the immune system works. John Wiley & Sons, New YorkGoogle Scholar
- Talman AM, Domarle O, McKenzie FE, Ariey F, Robert Vincent (2004) Gametocytogenesis: the puberty of plasmodium falciparum. Malar J 3(1):24Google Scholar
- Tavares JC (2013) Malaria. Colloquium series on integrated systems physiology: from molecule to function. Biota Publishing, PrincetonGoogle Scholar
- Teboh-Ewungkem MI, Wang M (2012) Male fecundity and optimal gametocyte sex ratios for plasmodium falciparum during incomplete fertilization. J Theor Biol 307:183–192MathSciNetzbMATHGoogle Scholar
- 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(2):273–286MathSciNetzbMATHGoogle Scholar
- Teboh-Ewungkem MI, Yuster T (2016) Evolutionary implications for the determination of gametocyte sex ratios under fecundity variation for the malaria parasite. J Theor Biol 408:260–273MathSciNetzbMATHGoogle Scholar
- 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):63–93MathSciNetzbMATHGoogle Scholar
- Teboh-Ewungkem MI, Ngwa GA, Ngonghala CN (2013) Models and proposals for malaria: a review. Math Popul Stud 20(2):57–81MathSciNetzbMATHGoogle Scholar
- Tewa J-J, Fokouop R, Mewoli B, Bowong S (2012) Mathematical analysis of a general class of ordinary differential equations coming from within-hosts models of malaria with immune effectors. Appl Math Comput 218(14):7347–7361MathSciNetzbMATHGoogle Scholar
- Tumwiine J, Luckhaus S, Mugisha JYT, Luboobi LS (2008) An age-structured mathematical model for the within host dynamics of malaria and the immune system. J Math Model Algor 7(1):79–97MathSciNetzbMATHGoogle Scholar
- Tumwiine J, Mugisha JYT, Luboobi LS (2008) On global stability of the intra-host dynamics of malaria and the immune system. J Math Anal Appl 341(2):855–869MathSciNetzbMATHGoogle Scholar
- Van den Driessche P, Watmough J (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180(1):29–48MathSciNetzbMATHGoogle Scholar
- Verhulst PF (1838) Notice sur la loi que la population suit dans son acroissement. Correspondence Mathématiwue et Physique 10:113–121Google Scholar
- Wahlgren M, Perlmann P (1999) Malaria: molecular and clinical aspects. CRC Press, Boca RatonGoogle Scholar
- Weekley C, Smith DS (2013) Malaria: the clinical basics. Global Health Education Consortium (GHEC)Google Scholar
- WHO (2015) World malaria report 2015. World Health Organisisation BulletineGoogle Scholar
- Willekens FLA, Werre JM, Groenen-Döpp YAM, Roerdinkholder-Stoelwinder B, De Pauw Ben, Bosman Giel JCGM (2008) Erythrocyte vesiculation: a self-protective mechanism? Br J Haematol 141(4):549–556Google Scholar
- Wongsrichanalai C, Barcus MJ, Muth S, Sutamihardja A, Wernsdorfer Walther H (2007) A review of malaria diagnostic tools: microscopy and rapid diagnostic test (rdt). Am J Trop Med Hyg 77(6 Suppl):119–127Google Scholar
- World Health Organization and Center for Disease Control (2010) Basic malaria microscopy: tutor’s guide. World Health OrganizationGoogle Scholar