A Refined Difference Equation for Malaria Parasite development

Suggested by Periodic and Chaotic Host-Parasite Interactions in Human Malaria, Dominic Kwiatkowski, Martin Nowak, Proceedings of the National Academy of Sciences of the United States of America, Vol. 88, No. 12 (Jun. 15, 1991), pp. 5111-5113.

The Model:

y_t+1 = d₁x_tf(x_t) x_t+1 = rd₂y_tg(x_t)

where

x_t -- number of young parasites (0-24 hours post-invasion) on day t
y_t -- number of old parasites (24-48 hours post-invasion) on day t
r -- the number of newly infected erythrocytes that arise from a single rupturing schizont
d₁ -- the probability that a parasite survives the first half of its life cycle in the absence of fever
d₂ -- the probability that a parasite survives the second half of its life cycle in the absence of fever
f(x_t) -- the fraction of young parasites that survive the damaging effect of the fever (decreasing function)
g(x_t) -- the fraction of old parasites that survive the damaging effect of the fever (decreasing function)

Experimental evidence indicates that

f(x_t)>g(x_t) (fever is more harmful to older parasites). The result?

"Initially, the parasite population replicates asynchronously, with a multiplication rate R = rd₁d₂ every two days. When parasite density increases beyond a certain level, significant fever occurs. This exerts a negative feedback force, terminating exponential growth and causing parasite density to oscillate."

Here's a lisp file that illustrates the effect, and the web version (below). See the paper for the definitions of these parameters. The starting parameters are those of Figure 2.

Inputs:

days: r: p: s: d₁: d₂: x₀: y₀:
Run (you may need to refresh the plots -- sometimes plots don't change between runs, because the images are cached by your browser....)
References
- Periodic and Chaotic Host-Parasite Interactions in Human Malaria, Dominic Kwiatkowski, Martin Nowak, Proceedings of the National Academy of Sciences of the United States of America, Vol. 88, No. 12 (Jun. 15, 1991), pp. 5111-5113. (pdf) (source)
- An Introduction to Structured Population Dynamics, J. M. Cushing, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 1998.
- Pub med: searches include:
  - malaria fever model
  - malaria plasmodium life cycle
  - malaria fever model mathematical
  Results:
  - Epidemiologic Models in Studies of Vector-Borne Diseases George Macdonald, M.D. Epidemiologic models in studies of vetor-borne diseases: The R. E. Dyer Lecture George Macdonald Public Health Rep. 1961 September; 76(9): 753764.
  - "Mice infected with Plasmodium berghei, P. chabaudi, or P. yoelii became lethargic and ceased to display normal antimosquito behavior. Periods of reduced defensiveness corresponded with maximum mosquito engorgement and with periods of maximum gametocyte infectivity to mosquitoes. Increased feeding success of mosquitoes during periods of peak gametocyte infectivity may be important to the natural maintenance of these malaria parasites." Day JF, Edman JD. Malaria renders mice susceptible to mosquito feeding when gametocytes are most infective. J Parasitol 1983;69:163170. [PubMed].
  - Cited by Darwinian medicine: Applications of evolutionary biology for veterinarians Edmund K. LeGrand and Corrie C. Brown Can Vet J. 2002 July; 43(7): 556559. "The illness associated with malaria reduces the host's ability to fend off mosquitoes, thereby increasing the chance of transmission (21) here, it is in the genetic interest of the Plasmodium sp. not to harm the mosquito vector."
  - Erythrocyte sequestration and anemia in severe falciparum malaria. Analysis of acute changes in venous hematocrit using a simple mathematical model. T M Davis, S Krishna, S Looareesuwan, W Supanaranond, S Pukrittayakamee, K Attatamsoonthorn, and N J White J Clin Invest. 1990 September; 86(3): 793800.
  - Link between immune response and parasite synchronization in malaria Igor M. Rouzine and F. Ellis McKenzie Proc Natl Acad Sci U S A. 2003 March 18; 100(6): 34733478. Published online 2003 March 5.
  - 1: Parasitology. 1998 Aug;117 ( Pt 2):97-105.
    
    An analysis of the temperature effects of fever on the intra-host population dynamics of Plasmodium falciparum.
    Gravenor MB, Kwiatkowski D.
    Department of Paediatrics, University of Oxford, John Radcliffe Hospital. michael.gravenor@paediatrics.ox.ac.uk
    Observations that growth of Plasmodium falciparum in vitro is inhibited by high temperatures have led to hypotheses that malaria fever may influence the parasite population dynamics, regulating parasite density and synchronizing parasite growth. In order to investigate the fever hypotheses, we have developed an age-structured coupled Markov chain model that describes the parasite erythrocyte cycle and its interaction with the host fever response. We estimated the model parameters using data collected from laboratory parasite cultures that were exposed to febrile or normal temperature. Using the experimental parameter values, quantitative predictions were made of the effect of fever in determining the parasite population dynamics. It was concluded from the model behaviour that, during the primary infection of a non-immune host, a typical episode of fever can effect density-dependent regulation of the parasite population, maintaining cycles of parasitaemia and promoting synchronous parasite growth.
  - 1: Parasitology. 2000 Sep;121 ( Pt 3):227-35.Links
    Mathematical modelling of the within-host dynamics of Plasmodium falciparum.
    Hoshen MB, Heinrich R, Stein WD, Ginsburg H.
    Department of Biological Chemistry, Institute of Life Sciences, The Hebrew University of Jerusalem, Israel. hoshen@vms.huji.ac.il
    The development of malaria due to Plasmodium falciparum is a complex, multi-stage process. It is usually characterized by an exponential growth in the number of parasite-infected erythrocytes, followed by marked oscillations in this number with a period of 48 h, which are eventually dampened. This course of events has been the subject of various mathematical models. In this paper we propose a new mathematical model for the in-host asexual erythrocytic development of P. falciparum malaria. Synchronicity of the infection is shown to be an inherent feature of infection, irrespective of the duration of merozoite release from the liver. It will, therefore, cause periodic symptoms, as known in malaria patients. We also simulate the effects of an induced host immune response and show how the level of immunity affects the development of disease. The simulations fit well with the clinical observations. We show how infection can become asynchronous and discuss the effect of desynchronization on the circulating and total parasitaemia and demonstrate that synchronized broods will show parasitaemia fluctuations.
    PMID: 11085243 [PubMed - indexed for MEDLINE]

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A Refined Difference Equation for Malaria Parasite development

An analysis of the temperature effects of fever on the intra-host population dynamics of Plasmodium falciparum.

Mathematical modelling of the within-host dynamics of Plasmodium falciparum.