Mathematical modelling of Malaria

Today I attended a talk at UQ by Lauren Childs from Virginia Tech on efforts to understand and combat malaria, work she’d done with Caroline Buckee, Olivia Prosper and Francisco Cai. As someone working on dengue and about to move to London to work at the LSHTM, epidemiological modelling of tropical diseases has got my attention.

Malaria has been studied for a long time, at least as far back as its use as means of treating syphilis (Wagner-Jauregg 1931). For those in the audience not familiar with mathematical modelling of diseases, we got taken way back to Ross (1911) and Macdonald and others (1957), who established early models of malaria. A lot of the subsequent models are based on this early work, with susceptible-infectious-recovered (or SIR) models being the most popular approach not just for malaria but for many other diseases (Reiner et al. 2013).

Despite decades of control measures, malaria remains a significant public health problem. And despite decades of mathematical and statistical work modelling the spread of mosquito-borne diseases, the modelling of individuals, as opposed to a well-mixed continuous concentration of populations, is quite rare.

Like dengue, malaria cycles between its two hosts. A mosquito bites a human, whereupon the parasites from the bite are transferred to the liver, spending days to years in the blood replicating both sexually and asexually. Sexually reproduced parasites are then able to spread to female mosquitoes via feeding, and the cycle continues. Malaria’s not good. It can stay in your body for years as a low-level asymptomatic infection. One approach taken in this work is to look at an SIS model for humans (who can recover and become susceptible to malaria again) and an SI model for mosquitoes (who die before recovery can occur).

The talk focussed on three main topics

  1. Dissecting determinants of malaria chronicity
  2. Within-vector generation of diversity
  3. Disrupting malaria reproduction

Dissecting determinants of malaria chronicity

Malaria infections tend to either be acute, characterised by a burst of parasitemia within the body that last tens of days, or a chronic, low-density infection that can last hundreds of days. Creating high efficacy vaccines for malaria is a real challenge, and tracking the progression of an outbreak through the population can be difficult as transmission is often not directly observed (we don’t have sensors on mosquitoes). On top of that, the immune system is still not completely understood.

In L. M. Childs and Buckee (2015), the aim was to understand which host and pathogen properties were associated with chronic infections. Discrete delay-difference equations were developed to model the pathogen dynamics and the immune response. This results in a large system with a number of parameters needing to be estimated, meaning that much data is required for model calibration (or data fusion). Certainly a lot of work has been done on our end with dengue to ensure that the modelling gives realistic simulations. Whether this is by way of a Bayesian modelling framework that solves differential equation models within it or by using Latin Hypercube Sampling of parameters and calculating the distance to the observed data, marrying differential equations to data is hard.

For a virus to be successful in propagating, it needs to be virulent enough to spread but not so lethal that it kills the host and prevent transmission. Malaria has a number of tricks up its sleeve regarding the immune system, such as creating proteins on the surface of red blood cells and endothelium. As such the virus can avoid detection and trigger only certain immune responses that aid in replication but not in elimination (e.g. via the spleen). This persistence through variation is the key to its longevity in the body.

For this modelling, the parasite parameters end up being less important than the host immune response. In order to control the initial infection, you need a strong innate response at early time. The moderate responses from the variant-specific and cross-reactive responses control the long-term infection, the length of the peaks and which variant is peaking. A general adaptive immunity leads to successive decreases in the relative heights of the peaks as the infection progresses. In the absence of a strong immune system, chronic malaria infections can occur, which isn’t much fun.

Simulating within-vector generation

Stochastic modelling of the adaptation of genetic sequences was used in Lauren M. Childs and Prosper (2017) to understand how the genetic diversity of the parasite pool changes during sexual replication. The modelling occurred in two stages:

  1. Modelling the population dynamics of the different genotypes
  2. Simulating the replication and recombination processes that generate novel sequences

This extends previous work (Teboh-Ewungkem and Yuster 2010; Teboh-Ewungkem, Yuster, and Newman 2010) looking at within-vector dynamics, and considers a transition matrix for a continuous time Markov chain that explains the progression of the gamete through to the oocyst stage and its eventual bursting. The parasites then travel from the midgut to salivary gland, with the number of parasites being Poisson and the successful transition being Binomial. In this way, stochasticity is introduce in a way which reflects the individual nature of the parasites rather than the assumption of a smooth, deterministic function which can take on any real, non-negative value.

Parents of the same genotype reproducing leads to children of the same genotype. New genotypes are produced when parents of different genotypes reproduce, and the offspring are assumed to have a fitness which is the average of the parents’. As the initial diversity of the initial parasite population changes, differences in outcome are seen. A bottleneck occurs due to only 10 or so parasites initially being transmitted in a bite, but the more diverse this initial population is, the higher the resulting diversity due to recombination.

Perhaps it’s feasible to collapse this diversity by introducing a drug resistance to one of the genotypes and delivering the drug to knock out the other genotypes. It may then be possible to control that one genotype of malaria in other ways. The talk didn’t make any mention of defective interfering particles (DIs), a topic I’ve been working on at QIMR, but there has been work done on modelling the impact of DIs generally (Kirkwood and Bangham 1994) as well as using DIs to stop the replication and transmission of malaria (Urakami et al. 2017). This may well be pie in the sky from someone who’s not a biologist, but it seems that controlling the virus dynamics as well as their transmission from host to vector can be made to work together. A two-step intervention like this isn’t necessarily a simple thing, of course.

Disrupting mosquito reproduction and parasite development

Nilsson et al. (2015) focuses on replacing insecticide intervention (which prevents biting of humans) with the use of a particular hormone which is transmitted during mating. The idea behind this is that by tricking female mosquitoes into thinking that mating and egg-laying has occurred there is no need to try again in the relatively short life span of the mosquito. Not only are egg-laying and mating decreased, but it appears that susceptibility to malaria is decreased and mortality increased.

This was the most fascinating part of the talk, as it proposes a really interesting alternative to insecticide use. My only issue with the proposal is that it requires large coverage of the human population to become effective, even with low malaria prevalence. This is how herd immunity for vaccination works but vaccination requires infrequent application to humans for long-term protection. The use of this hormone is a mosquito control approach and with mosquitoes having such a short lifespan (tens of days) it seems like it’d be tricky to keep the control in place without constant application. What was really neat was that after doing the modelling, it appeared that the use of this hormone lead to an increase in mosquito populations but on further investigation the increased population were uninfected young mosquitoes and there were relatively fewer old infected mosquitoes spreading the disease.

Tarun and I asked about spatial and temporal homogeneity in the modelling, as the prevalence of a tropica disease and its vector population can be influenced by seasonal factors like temperature and humidity (White et al. 2011; Cairns et al. 2015) and that the movement of humans infected with an arbovirus is often on a greater scale than that of vectors (Stoddard et al. 2009; Acevedo et al. 2015). The models here do assume a well-mixed population, so there’s certainly scope for investigating things like cellular automata models or individual-based models with time-varying forces of infection driving the epidemiology.

Summary

While I’m moving away from arboviruses towards pneumococcus infections, the talk was very illuminating and it was very interesting to get another view on mosquito-borne diseases, particularly from a mathematical modelling approach. I had a brief chat with Prof Childs in the UQ Maths tea room after the talk, after a long chat with Matthew Holden about different approaches to interacting with biologists and science students when it comes to explaining mathematics.

References

Acevedo, Miguel A., Olivia Prosper, Kenneth Lopiano, Nick Ruktanonchai, T. Trevor Caughlin, Maia Martcheva, Craig W. Osenberg, and David L. Smith. 2015. “Spatial Heterogeneity, Host Movement and Mosquito-Borne Disease Transmission.” Edited by James M McCaw. PLOS ONE 10 (6). Public Library of Science (PLoS):e0127552. https://doi.org/10.1371/journal.pone.0127552.

Cairns, Matthew E, Patrick G T Walker, Lucy C Okell, Jamie T Griffin, Tini Garske, Kwaku Poku Asante, Seth Owusu-Agyei, et al. 2015. “Seasonality in Malaria Transmission: Implications for Case-Management with Long-Acting Artemisinin Combination Therapy in Sub-Saharan Africa.” Malaria Journal 14 (1). Springer Nature. https://doi.org/10.1186/s12936-015-0839-4.

Childs, L. M., and C. O. Buckee. 2015. “Dissecting the Determinants of Malaria Chronicity: Why Within-Host Models Struggle to Reproduce Infection Dynamics.” Journal of the Royal Society Interface 12 (104). The Royal Society:20141379–9. https://doi.org/10.1098/rsif.2014.1379.

Childs, Lauren M., and Olivia F. Prosper. 2017. “Simulating Within-Vector Generation of the Malaria Parasite Diversity.” Edited by Gordon Langsley. PLOS ONE 12 (5). Public Library of Science (PLoS):e0177941. https://doi.org/10.1371/journal.pone.0177941.

Kirkwood, T. B., and C. R. Bangham. 1994. “Cycles, Chaos, and Evolution in Virus Cultures: A Model of Defective Interfering Particles.” Proceedings of the National Academy of Sciences 91 (18). Proceedings of the National Academy of Sciences:8685–9. https://doi.org/10.1073/pnas.91.18.8685.

Macdonald, George, and others. 1957. “The Epidemiology and Control of Malaria.” The Epidemiology and Control of Malaria. Oxford University Press.

Nilsson, Sandra K., Lauren M. Childs, Caroline Buckee, and Matthias Marti. 2015. “Targeting Human Transmission Biology for Malaria Elimination.” Edited by Chetan E. Chitnis. PLOS Pathogens 11 (6). Public Library of Science (PLoS):e1004871. https://doi.org/10.1371/journal.ppat.1004871.

Reiner, R. C., T. A. Perkins, C. M. Barker, T. Niu, L. F. Chaves, A. M. Ellis, D. B. George, et al. 2013. “A Systematic Review of Mathematical Models of Mosquito-Borne Pathogen Transmission: 1970-2010.” Journal of the Royal Society Interface 10 (81). The Royal Society:20120921–1. https://doi.org/10.1098/rsif.2012.0921.

Ross, Ronald. 1911. The Prevention of Malaria. John Murray; London.

Stoddard, Steven T., Amy C. Morrison, Gonzalo M. Vazquez-Prokopec, Valerie Paz Soldan, Tadeusz J. Kochel, Uriel Kitron, John P. Elder, and Thomas W. Scott. 2009. “The Role of Human Movement in the Transmission of Vector-Borne Pathogens.” Edited by Pattamaporn Kittayapong. PLoS Neglected Tropical Diseases 3 (7). Public Library of Science (PLoS):e481. https://doi.org/10.1371/journal.pntd.0000481.

Teboh-Ewungkem, Miranda I, and Thomas Yuster. 2010. “A Within-Vector Mathematical Model of Plasmodium Falciparum and Implications of Incomplete Fertilization on Optimal Gametocyte Sex Ratio.” Journal of Theoretical Biology 264 (2). Elsevier BV:273–86. https://doi.org/10.1016/j.jtbi.2009.12.017.

Teboh-Ewungkem, Miranda I, Thomas Yuster, and Nathaniel H Newman. 2010. “A Mathematical Model of the Within-Vector Dynamics of the Plasmodium Falciparum Protozoan Parasite.” Infectious Disease Modelling Research Progress. Nova Science Publishers, New York.

Urakami, Akane, Atsuko Sakurai, Momoko Ishikawa, Moh Lan Yap, Yevel Flores-Garcia, Yasunari Haseda, Taiki Aoshi, et al. 2017. “Development of a Novel Virus-Like Particle Vaccine Platform That Mimics the Immature Form of Alphavirus.” Edited by Herman F. Staats. Clinical and Vaccine Immunology 24 (7). American Society for Microbiology:e00090–17. https://doi.org/10.1128/cvi.00090-17.

Wagner-Jauregg, Julius. 1931. “Verhütung Und Behandlung Der Progressiven Paralyse Durch Impfmalaria [Prevention and Treatment of Progressive Paralysis by Malaria Inoculation].” Handbuch Der Experimentellen Therapie Ergänzungsband, Munich.

White, Michael T, Jamie T Griffin, Thomas S Churcher, Neil M Ferguson, María-Gloria Basáñez, and Azra C Ghani. 2011. “Modelling the Impact of Vector Control Interventions on Anopheles Gambiae Population Dynamics.” Parasites & Vectors 4 (1). Springer Nature:153. https://doi.org/10.1186/1756-3305-4-153.

comments powered by Disqus