A Refined Difference Equation for Malaria Parasite development
Suggested by Periodic and Chaotic Host-Parasite Interactios 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:
yt+1 = d1xtf(xt)
xt+1 = rd2ytg(xt)
where
- xt -- number of young parasites (0-24 hours post-invasion) on day t
- yt -- 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(xt) -- the fraction of young parasites that survive the damaging effect of the fever (decreasing function)
- g(xt) -- the fraction of old parasites that survive the damaging effect of the fever (decreasing function)
Experimental evidence indicates that
f(xt)>g(xt)
(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.