Discrete versus continuous-time models of malaria infections

Basic problem

Malaria infections have traditionally been modelled in continuous time. However the biological process is inherently discrete. Malaria parasites reproduce at a fixed age and infections consist of non-overlapping generations. The question is whether modelling infections in continuous rather than discrete time affects the outcome of models and particularly conclusions that we might draw from them. You will see that when we are interested in gametocyte production from infections, the dynamics of the two types of model can be different and lead to different predictions about how malaria parasites should optimise their transmission.

General approach

You will implement both a differential equation model and a discrete model of malaria dynamics in R. You will compare the dynamics of the two models with summary measures and look at the outcome for transmission.

What can be learnt?

Concepts:
Continuous-time versus discrete time models
Trade-offs and optimality modelling
Compartmental models

Methods:
Numerical simulation of ordinary differential equations
Simulation of difference equations

Starting point

Download the detailed description of this module, which contains an introduction to malaria biology and equations for a simple continuous-time and a discrete model. Use the R script to run the continuous model and plot the infection dynamics. Implement the discrete time model.

Interesting questions that you can investigate

• How should parameters be set to obtain similar behaviour in the discrete and continuous-time model?
• Is it possible to achieve identical behaviour in the two models?
• How do parasite growth and gametocyte production change with gametocyte investment in the two models?
• What level of gametocyte investment maximises transmission? Does this differ between the two models?

Advanced questions:

• What is the effect of implementing compartments corresponding to different development stages in the continuous model?
• How do the results of the models compare when a dynamic immune function is incorporated?
• What happens if gametocyte investment is allowed to vary over the course of infection?

Glossary

• Asexuals: in this context we use this term to refer to infected blood cells that are going to produce merozoites. Also known as schizonts.
• Gametocytes: transmission stage of malaria parasite. Some infected red blood cells develop into gamtocytes.
• Merozoites: the stage of the parasite that is released when infected cells are ruptured, and that is able to infect further cells.
• Trade-off: when an increase in a trait causes an increase in one component of fitness but a decrease in another. The trait should then evolve to an intermediate optimum at which the fitness cost and benefit are balanced.
• Compartmental model: a model in which a population is divided between a number of compartments that may be linked by migration.

Literature & Weblinks

• Caillard, V. et al. (1992) Plasmodium vinckei petteri: Identification of the stages sensitive to Arteether. Exp. Parasitology, 75:449-456.

• Koella, J.C. and Antia, R. (1995) Optimal pattern of replication and transmission for parasites with two stages in their life cycle. Theor. Popul. Biol. 47:277-291.

• McKenzie, F.E. and Bossert, W.H. (1998) The optimal production of gametocytes by Plasmodium falciparum. J. theor. Biol. 193:419-428, 1998.

• Saul, A. (1998) Models for the in-host dynamics of malaria revisited: errors in some basic models lead to large overestimates of growth rates. Parasitology, 117:405-407.

• Gravenor, M.B. and Lloyd, A.L. (1998) Reply to: Models for the in-host dynamics of malaria revisited: errors in some basic models lead to large overestimates of growth rates. Parasitology, 117:409-410.

• More on malaria in Wikipedia.