# American Institute of Mathematical Sciences

2011, 8(4): 999-1018. doi: 10.3934/mbe.2011.8.999

## The Within-Host dynamics of malaria infection with immune response

 1 Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China 2 Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250 3 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240

Received  December 2010 Revised  March 2011 Published  August 2011

Malaria infection is one of the most serious global health problems of our time. In this article the blood-stage dynamics of malaria in an infected host are studied by incorporating red blood cells, malaria parasitemia and immune effectors into a mathematical model with nonlinear bounded Michaelis-Menten-Monod functions describing how immune cells interact with infected red blood cells and merozoites. By a theoretical analysis of this model, we show that there exists a threshold value $R_0$, namely the basic reproduction number, for the malaria infection. The malaria-free equilibrium is global asymptotically stable if $R_0<1$. If $R_0>1$, there exist two kinds of infection equilibria: malaria infection equilibrium (without specific immune response) and positive equilibrium (with specific immune response). Conditions on the existence and stability of both infection equilibria are given. Moreover, it has been showed that the model can undergo Hopf bifurcation at the positive equilibrium and exhibit periodic oscillations. Numerical simulations are also provided to demonstrate these theoretical results.
Citation: Yilong Li, Shigui Ruan, Dongmei Xiao. The Within-Host dynamics of malaria infection with immune response. Mathematical Biosciences & Engineering, 2011, 8 (4) : 999-1018. doi: 10.3934/mbe.2011.8.999
##### References:
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R. Engwerda, Development and regulation of cell-mediated immune responses to the blood stages of malaria: Implications from vaccine research,, Annu. Rev. Immunol., 23 (2005), 69. Google Scholar [13] M. B. Gravenor and A. L. Lloyd, Reply to: Models for the in-host dynamics of malaria revisited: Errors in some basic models lead to large overestimates of growth rates,, Parasitology, 117 (1998), 409. Google Scholar [14] M. B. Gravenor, A. L. Lloyd, P. G. Kremsner, M. A. Missinou, M. English, K. Marsh and D. Kwiatkowski, A model for estimating total parasite load in falciparum malaria patients,, J. Theoret. Biol., 217 (2002), 137. Google Scholar [15] M. B. Gravenor, M. B. Van Hensbroek and D. Kwiatkowski, Estimating sequestered parasite population dynamics in cerebral malaria,, Proc. Natl. Acad. Sci. USA, 95 (1998), 7620. Google Scholar [16] C. Hetzel and R. M. Anderson, The within-host cellular dynamics of bloodstage malaria-theoretical and experimental studies,, Parasitology, 113 (1996), 25. Google Scholar [17] M. B. Hoshen, R. Heinrich, W. D. Stein and H. Ginsburg, Mathematical modeling of the within-host dynamics of Plasmodium falciparum,, Parasitology, 121 (2000), 227. Google Scholar [18] A. Iggidr, J.-C. Kamgang, G. Sallet and J.-J. Tewa, Global analysis of new malaria intrahost models with a competitive exclusion principle,, SIAM J. Appl. Math., 67 (2006), 260. Google Scholar [19] T. Kajiwara and T. Sasaki, A note on the stability analysis of pathogen-immune interaction dynamics,, Discret. Contin. Dynam. Syst. Ser. B, 4 (2004), 615. Google Scholar [20] D. Kwiatkowsti and M. Nowak, Periodic and chaotic host-parasite interactions in human malaria,, Proc. Natl. Acad. Sci. USA, 88 (1991), 5111. Google Scholar [21] J. Langhorne, F. M. Ndungu, A.-M. Sponaas and K. Marsh, Immunity to malaria: More questions than answers,, Nature Immunol., 9 (2008), 725. Google Scholar [22] W. Liu, Nonlinear oscillation in models of immune responses to persistent viruses,, Theoret. Pop. Biol., 52 (1997), 224. Google Scholar [23] L. Malaguarnera and S. Musumeci, The immune response to Plasmodium falciparum malaria,, Lancet Infect. Dis., 2 (2002), 472. Google Scholar [24] G. L. Mandell, J. E. Bennett and R. Dolin, "Principles and Practice of Infectious Diseases,'', Churchill Livingstone, (1995). Google Scholar [25] F. E. McKenzie and H. W. Bossert, An integrated model of Plasmodium falciparum dynamics,, J. Theoret. Biol., 232 (2005), 411. Google Scholar [26] P. G. McQueen and F. E. McKenzie, Age-structured red blood cell susceptibility and the dynamics of malaria infections,, Proc. Natl. Acad. Sci. USA, 101 (2004), 9161. Google Scholar [27] P. G. McQueen and F. E. McKenzie, Host control of malaria infections: Constrains on immune and erythropoeitic response kinetics,, PLoS Comput. Biol., 4 (2008). doi: 10.1371/journal.pcbi.1000149. Google Scholar [28] J. L. Mitchell and T. W. Carr, Oscillations in an intra-host model of plasmodium falciparum malaria due to cross-reactive immune response,, Bull. Math. Biol., 72 (2010), 590. Google Scholar [29] L. Molineaux and K. Dietz, Review of intra-host models of malaria,, Parassitologia, 41 (1999), 221. Google Scholar [30] A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics,, J. Math. Biol., 51 (2005), 247. Google Scholar [31] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses,, Nature, 272 (1996), 74. Google Scholar [32] S. S. Pilyugin and R. Antia, Modeling immune responses with handling time,, Bull. Math. Biol., 62 (2000), 869. Google Scholar [33] S. I. Rapaport, "Introduction to Hematology,'', Lippincott, (1987). Google Scholar [34] I. M. Rouzine and F. E. McKenzie, Link between immune response and parasite synchronization in malaria,, Proc. Natl. Acad. Sci. USA, 100 (2003), 3473. Google Scholar [35] S. Ruan and G. S. K. Wolkowicz, Bifurcation analysis of a chemostat model with a distributed delay,, J. Math. Anal. Appl., 204 (1996), 786. Google Scholar [36] A. Saul, Models for the in-host dynamics of malaria revisited: Errors in some basic models lead to large over-estimates of growth rates,, Parasitology, 117 (1998), 405. Google Scholar [37] J. Stark, C. Chan and A. J. T. George, Oscillations in immune system,, Immunol. Rev., 216 (2007), 213. Google Scholar [38] M. M. Stevenson and E. M. Riley, Innate immunity to malaria,, Nat. Rev. Immunol., 4 (2004), 169. Google Scholar [39] Y. Su, S. Ruan and J. Wei, Periodicity and synchronization in blood-stage malaria infection,, J. Math. Biol., 63 (2011), 557. doi: 10.1007/s00285-010-0381-5. Google Scholar [40] J. Tumwiine, J. Y. T. Mugisha and L. S. Luboobi, On global stability of the intra-host dynamics of malaria and the immune system,, J. Math. Anal. Appl., 341 (2008), 855. Google Scholar [41] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. Google Scholar [42] , WHO, "Malaria,", 2008. Available from: \url{http://www.who.int/malaria/en}., (2008). Google Scholar [43] D. Xiao and H. W. Bossert, An intra-host mathematical model on interaction between HIV and malaria,, Bull. Math. Biol., 72 (2010), 1892. doi: p10.1007/s11538-010-9515-6. Google Scholar

show all references

##### References:
 [1] P. Adda, J. L. Dimi, A. Iggidr, J. C. Kamgang, G. Sallet and J. J. Tewa, General models of host-parasite systems. Global analysis,, Dis. Contin. Dynam. Syst. Ser. B, 8 (2007), 1. Google Scholar [2] Z. Agur, D. Abiri and L. H. T. van der Ploeg, Ordered appearance of antigenic variants of African trypanosomes explained in a mathematical model based on a stochastic switch process and immune-selection against putative switch intermediates,, Proc. Natl. Acad. Sci. USA, 86 (1989), 9626. Google Scholar [3] R. M. Anderson, Complex dynamic behaviors in the interaction between parasite populations and the host's immune system,, Intl. J. Parasitol., 28 (1998), 551. Google Scholar [4] R. M. Anderson, R. M. May and S. Gupta, Non-linear phenomena in host-parasite interactions,, Parasitology, 99 (1989). Google Scholar [5] R. Antia, B. R. Levin and R. M. May, Within-host population dynamics and the evolution and maintenance of microparasite virulence,, Am. Nat., 144 (1994), 457. Google Scholar [6] A. D. Augustine, B. F. Hall, W. W. Leitner, A. X. Mo, T. M. Wali and A. S. Fauci, NIAID workshop on immunity to malaria: Addressing immunological challenges,, Nature Immunol., 10 (2009), 673. Google Scholar [7] C. Chiyaka, W. Garira and S. Dube, Modelling immune response and drug therapy in human malaria infection,, Comput. Math. Meth. Med., 9 (2008), 143. Google Scholar [8] C. Coban, K. J. Ishii, T. Horii and S. Akira, Manipulation of host innate immune responses by the malaria parasite,, TRENDS Microbiol., 15 (2007), 271. Google Scholar [9] J. A. Deans and Cohen, Immunology of malaria,, Annu. Rev. Microbiol., 37 (1983), 25. Google Scholar [10] R. J. De Boer and A. S. Perelson, Towards a general function describing T cell proliferation,, J. Theoret. Biol., 175 (1995), 567. Google Scholar [11] Z. Dong and J.-A. Cui, Dynamical model of vivax malaria intermittence attack in vivo,, Intl. J. Biomath., 2 (2009), 507. Google Scholar [12] M. F. Good, H. Xu, M. Wykes and C. R. Engwerda, Development and regulation of cell-mediated immune responses to the blood stages of malaria: Implications from vaccine research,, Annu. Rev. Immunol., 23 (2005), 69. Google Scholar [13] M. B. Gravenor and A. L. Lloyd, Reply to: Models for the in-host dynamics of malaria revisited: Errors in some basic models lead to large overestimates of growth rates,, Parasitology, 117 (1998), 409. Google Scholar [14] M. B. Gravenor, A. L. Lloyd, P. G. Kremsner, M. A. Missinou, M. English, K. Marsh and D. Kwiatkowski, A model for estimating total parasite load in falciparum malaria patients,, J. Theoret. Biol., 217 (2002), 137. Google Scholar [15] M. B. Gravenor, M. B. Van Hensbroek and D. Kwiatkowski, Estimating sequestered parasite population dynamics in cerebral malaria,, Proc. Natl. Acad. Sci. USA, 95 (1998), 7620. Google Scholar [16] C. Hetzel and R. M. Anderson, The within-host cellular dynamics of bloodstage malaria-theoretical and experimental studies,, Parasitology, 113 (1996), 25. Google Scholar [17] M. B. Hoshen, R. Heinrich, W. D. Stein and H. Ginsburg, Mathematical modeling of the within-host dynamics of Plasmodium falciparum,, Parasitology, 121 (2000), 227. Google Scholar [18] A. Iggidr, J.-C. Kamgang, G. Sallet and J.-J. Tewa, Global analysis of new malaria intrahost models with a competitive exclusion principle,, SIAM J. Appl. Math., 67 (2006), 260. Google Scholar [19] T. Kajiwara and T. Sasaki, A note on the stability analysis of pathogen-immune interaction dynamics,, Discret. Contin. Dynam. Syst. Ser. B, 4 (2004), 615. Google Scholar [20] D. Kwiatkowsti and M. Nowak, Periodic and chaotic host-parasite interactions in human malaria,, Proc. Natl. Acad. Sci. USA, 88 (1991), 5111. Google Scholar [21] J. Langhorne, F. M. Ndungu, A.-M. Sponaas and K. Marsh, Immunity to malaria: More questions than answers,, Nature Immunol., 9 (2008), 725. Google Scholar [22] W. Liu, Nonlinear oscillation in models of immune responses to persistent viruses,, Theoret. Pop. Biol., 52 (1997), 224. Google Scholar [23] L. Malaguarnera and S. Musumeci, The immune response to Plasmodium falciparum malaria,, Lancet Infect. Dis., 2 (2002), 472. Google Scholar [24] G. L. Mandell, J. E. Bennett and R. Dolin, "Principles and Practice of Infectious Diseases,'', Churchill Livingstone, (1995). Google Scholar [25] F. E. McKenzie and H. W. Bossert, An integrated model of Plasmodium falciparum dynamics,, J. Theoret. Biol., 232 (2005), 411. Google Scholar [26] P. G. McQueen and F. E. McKenzie, Age-structured red blood cell susceptibility and the dynamics of malaria infections,, Proc. Natl. Acad. Sci. USA, 101 (2004), 9161. Google Scholar [27] P. G. McQueen and F. E. McKenzie, Host control of malaria infections: Constrains on immune and erythropoeitic response kinetics,, PLoS Comput. Biol., 4 (2008). doi: 10.1371/journal.pcbi.1000149. Google Scholar [28] J. L. Mitchell and T. W. Carr, Oscillations in an intra-host model of plasmodium falciparum malaria due to cross-reactive immune response,, Bull. Math. Biol., 72 (2010), 590. Google Scholar [29] L. Molineaux and K. Dietz, Review of intra-host models of malaria,, Parassitologia, 41 (1999), 221. Google Scholar [30] A. Murase, T. Sasaki and T. Kajiwara, Stability analysis of pathogen-immune interaction dynamics,, J. Math. Biol., 51 (2005), 247. Google Scholar [31] M. A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses,, Nature, 272 (1996), 74. Google Scholar [32] S. S. Pilyugin and R. Antia, Modeling immune responses with handling time,, Bull. Math. Biol., 62 (2000), 869. Google Scholar [33] S. I. Rapaport, "Introduction to Hematology,'', Lippincott, (1987). Google Scholar [34] I. M. Rouzine and F. E. McKenzie, Link between immune response and parasite synchronization in malaria,, Proc. Natl. Acad. Sci. USA, 100 (2003), 3473. Google Scholar [35] S. Ruan and G. S. K. Wolkowicz, Bifurcation analysis of a chemostat model with a distributed delay,, J. Math. Anal. Appl., 204 (1996), 786. Google Scholar [36] A. Saul, Models for the in-host dynamics of malaria revisited: Errors in some basic models lead to large over-estimates of growth rates,, Parasitology, 117 (1998), 405. Google Scholar [37] J. Stark, C. Chan and A. J. T. George, Oscillations in immune system,, Immunol. Rev., 216 (2007), 213. Google Scholar [38] M. M. Stevenson and E. M. Riley, Innate immunity to malaria,, Nat. Rev. Immunol., 4 (2004), 169. Google Scholar [39] Y. Su, S. Ruan and J. Wei, Periodicity and synchronization in blood-stage malaria infection,, J. Math. Biol., 63 (2011), 557. doi: 10.1007/s00285-010-0381-5. Google Scholar [40] J. Tumwiine, J. Y. T. Mugisha and L. S. Luboobi, On global stability of the intra-host dynamics of malaria and the immune system,, J. Math. Anal. Appl., 341 (2008), 855. Google Scholar [41] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. Google Scholar [42] , WHO, "Malaria,", 2008. Available from: \url{http://www.who.int/malaria/en}., (2008). Google Scholar [43] D. Xiao and H. W. Bossert, An intra-host mathematical model on interaction between HIV and malaria,, Bull. Math. Biol., 72 (2010), 1892. doi: p10.1007/s11538-010-9515-6. Google Scholar
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