2014, 11(4): 741-759. doi: 10.3934/mbe.2014.11.741

On the estimation of sequestered infected erythrocytes in Plasmodium falciparum malaria patients

1. 

Inria, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine, UMR 7502, ISGMP Bat. A, Metz, F-57045, France, France, France

Received  January 2013 Revised  December 2013 Published  March 2014

The aim of this paper is to give a method for the estimation of total parasite burden of the patient and the rate of infection in a malaria's intra-host model by using control theory tools. More precisely, we use an auxiliary system, called observer or estimator, whose solutions tend exponentially to those of the original model. This observer uses only the available measurable data, namely, the values of peripheral infected erythrocytes. It provides estimates of the sequestered infected erythrocytes, that cannot be measured by clinical methods. Therefore this method allows to estimate the total parasite burden within a malaria patient. Moreover, our constructed observer does not use the uncertain infection rate parameter $\beta$. In fact, we derive a simple method to estimate this parameter $\beta$. We apply this estimation method using real data that have been collected when malaria was used as therapy for neurosyphilis by the US Public Health Service.
Citation: Derdei Bichara, Nathalie Cozic, Abderrahman Iggidr. On the estimation of sequestered infected erythrocytes in Plasmodium falciparum malaria patients. Mathematical Biosciences & Engineering, 2014, 11 (4) : 741-759. doi: 10.3934/mbe.2014.11.741
References:
[1]

R. M. Anderson, R. M. May and S. Gupta, Non-linear phenomena in host-parasite interactions, Parasitology, 99 (1989), S59-S79. doi: 10.1017/S0031182000083426.

[2]

W. E. Collins and G. M. Jeffery, A retrospective examination of sporozoite- and trophozoite-induced infections with plasmodium falciparum in patients previously infected with heterologous species of plasmodium: Effect on development of parasitologic and clinical immunity, Am. J. Trop. Med. Hyg., 61 (1999), 36-43. doi: 10.4269/tropmed.1999.61-036.

[3]

M. Eichner, H. H. Diebner, L. Molineaux, W. E. Collins, G. M. Jeffery and K. Dietz, Genesis, sequestration and survival of plasmodium falciparum gametocytes: Parameter estimates from fitting a model to malariatherapy data, T. Roy. Soc. Trop. Med. H., 95 (2001), 497-501. doi: 10.1016/S0035-9203(01)90016-1.

[4]

M. B. Gravenor and D. Kwiatkowski, An analysis of the temperature effects of fever on the intra-host population dynamics of Plasmodium falciparum, Parasitology, 117 (1998), 97-105. doi: 10.1017/S0031182098002893.

[5]

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 over-estimates of growth rates, Parasitology, 117 (1998), 409-410. doi: 10.1017/S0031182098003229.

[6]

M. B. Gravenor, A. R. McLean and D. Kwiatkowski, The regulation of malaria parasitaemia: Parameter estimates for a population model, Parasitology, 110 (1995), 115-122. doi: 10.1017/S0031182000063861.

[7]

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), 7620-7624. doi: 10.1073/pnas.95.13.7620.

[8]

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. Theor. Biol., 217 (2002), 137-148. doi: 10.1006/jtbi.2002.3030.

[9]

C. Hetzel and R. M. Anderson, The within-host cellular dynamics of bloodstage malaria: Theoretical and experimental studies, Parasitology, 113 (1996), 25-38. doi: 10.1017/S0031182000066245.

[10]

M. Hou and P. C. Müller, Design of observers for linear systems with unknown inputs, IEEE Trans. Automat. Control, 37 (1992), 871-875. doi: 10.1109/9.256351.

[11]

M. Hou and P. C. Müller, Disturbance decoupled observer design: A unified viewpoint, IEEE Trans. Automat. Control, 39 (1994), 1338-1341. doi: 10.1109/9.293209.

[12]

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), 260-278. doi: 10.1137/050643271.

[13]

D. Kwiatkowski and M. Nowak, Periodic and chaotic host-parasite interactions in human malaria, Proc. Natl. Acad. Sci. USA., 88 (1991), 5111-5113. doi: 10.1073/pnas.88.12.5111.

[14]

D. P. Mason and F. E. McKenzie, Blood-stage dynamics and clinical implications of mixed plasmodium vivax-plasmodium falciparum infections, Am. J. Trop. Med. Hyg., 61 (1999), 367-374.

[15]

D. P. Mason, F. E. McKenzie and W. H. Bossert, The blood-stage dynamics of mixed plasmodium malariae-plasmodium falciparum infections, J. Theor. Biol., 198 (1999), 549-566. doi: 10.1006/jtbi.1999.0932.

[16]

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), 9161-9166. doi: 10.1073/pnas.0308256101.

[17]

L. Molineaux and K. Dietz, Review of intra-host models of malaria, Parassitologia, 41 (2000), 221-231.

[18]

L. Molineaux, M. Trauble, W. E. Collins, G. M. Jeffery and K. Dietz, Malaria therapy reinoculation data suggest individual variation of an innate immune response and independent acquisition of antiparasitic and antitoxic immunities, T. Roy. Soc. Trop. Med. H., 96 (2002), 205-209. doi: 10.1016/S0035-9203(02)90308-1.

[19]

L. B. Ochola, K. Marsh, Q. Gal, G. Pluschke and T. Smith, Estimating sequestered parasite load in severe malaria patients using both host and parasite markers, Parasitology, 131 (2005), 449-458. doi: 10.1017/S0031182005008085.

[20]

I. M. Rouzine and F. E. McKenzie, Link between immune response and parasite synchronization in malaria, Proc. Natl. Acad. Sci. USA., 100 (2003), 3473-3478. doi: 10.1073/pnas.262796299.

[21]

A. Saul, Transmission dynamics of plasmodium falciparum, Parasitol. Today., 12 (1996), 74-79. doi: 10.1016/0169-4758(96)80659-4.

show all references

References:
[1]

R. M. Anderson, R. M. May and S. Gupta, Non-linear phenomena in host-parasite interactions, Parasitology, 99 (1989), S59-S79. doi: 10.1017/S0031182000083426.

[2]

W. E. Collins and G. M. Jeffery, A retrospective examination of sporozoite- and trophozoite-induced infections with plasmodium falciparum in patients previously infected with heterologous species of plasmodium: Effect on development of parasitologic and clinical immunity, Am. J. Trop. Med. Hyg., 61 (1999), 36-43. doi: 10.4269/tropmed.1999.61-036.

[3]

M. Eichner, H. H. Diebner, L. Molineaux, W. E. Collins, G. M. Jeffery and K. Dietz, Genesis, sequestration and survival of plasmodium falciparum gametocytes: Parameter estimates from fitting a model to malariatherapy data, T. Roy. Soc. Trop. Med. H., 95 (2001), 497-501. doi: 10.1016/S0035-9203(01)90016-1.

[4]

M. B. Gravenor and D. Kwiatkowski, An analysis of the temperature effects of fever on the intra-host population dynamics of Plasmodium falciparum, Parasitology, 117 (1998), 97-105. doi: 10.1017/S0031182098002893.

[5]

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 over-estimates of growth rates, Parasitology, 117 (1998), 409-410. doi: 10.1017/S0031182098003229.

[6]

M. B. Gravenor, A. R. McLean and D. Kwiatkowski, The regulation of malaria parasitaemia: Parameter estimates for a population model, Parasitology, 110 (1995), 115-122. doi: 10.1017/S0031182000063861.

[7]

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), 7620-7624. doi: 10.1073/pnas.95.13.7620.

[8]

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. Theor. Biol., 217 (2002), 137-148. doi: 10.1006/jtbi.2002.3030.

[9]

C. Hetzel and R. M. Anderson, The within-host cellular dynamics of bloodstage malaria: Theoretical and experimental studies, Parasitology, 113 (1996), 25-38. doi: 10.1017/S0031182000066245.

[10]

M. Hou and P. C. Müller, Design of observers for linear systems with unknown inputs, IEEE Trans. Automat. Control, 37 (1992), 871-875. doi: 10.1109/9.256351.

[11]

M. Hou and P. C. Müller, Disturbance decoupled observer design: A unified viewpoint, IEEE Trans. Automat. Control, 39 (1994), 1338-1341. doi: 10.1109/9.293209.

[12]

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), 260-278. doi: 10.1137/050643271.

[13]

D. Kwiatkowski and M. Nowak, Periodic and chaotic host-parasite interactions in human malaria, Proc. Natl. Acad. Sci. USA., 88 (1991), 5111-5113. doi: 10.1073/pnas.88.12.5111.

[14]

D. P. Mason and F. E. McKenzie, Blood-stage dynamics and clinical implications of mixed plasmodium vivax-plasmodium falciparum infections, Am. J. Trop. Med. Hyg., 61 (1999), 367-374.

[15]

D. P. Mason, F. E. McKenzie and W. H. Bossert, The blood-stage dynamics of mixed plasmodium malariae-plasmodium falciparum infections, J. Theor. Biol., 198 (1999), 549-566. doi: 10.1006/jtbi.1999.0932.

[16]

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), 9161-9166. doi: 10.1073/pnas.0308256101.

[17]

L. Molineaux and K. Dietz, Review of intra-host models of malaria, Parassitologia, 41 (2000), 221-231.

[18]

L. Molineaux, M. Trauble, W. E. Collins, G. M. Jeffery and K. Dietz, Malaria therapy reinoculation data suggest individual variation of an innate immune response and independent acquisition of antiparasitic and antitoxic immunities, T. Roy. Soc. Trop. Med. H., 96 (2002), 205-209. doi: 10.1016/S0035-9203(02)90308-1.

[19]

L. B. Ochola, K. Marsh, Q. Gal, G. Pluschke and T. Smith, Estimating sequestered parasite load in severe malaria patients using both host and parasite markers, Parasitology, 131 (2005), 449-458. doi: 10.1017/S0031182005008085.

[20]

I. M. Rouzine and F. E. McKenzie, Link between immune response and parasite synchronization in malaria, Proc. Natl. Acad. Sci. USA., 100 (2003), 3473-3478. doi: 10.1073/pnas.262796299.

[21]

A. Saul, Transmission dynamics of plasmodium falciparum, Parasitol. Today., 12 (1996), 74-79. doi: 10.1016/0169-4758(96)80659-4.

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