A mathematical model for studying the impact of climate variability on malaria epidemics in South Africa

Abstract

Malaria is most prevalent in tropical climates, where there are sufficient rainfall for mosquitoes to breed and conducive temperatures for both the mosquito and protozoa to live. A slight change in temperature can drastically affect the lifespan and patterns of mosquitoes, and moreover, the protozoan itself can only survive in a certain temperature range. With higher temperatures, mosquitoes can mature faster, and thus have more time to spread the disease. The malaria parasite also matures more quickly at warmer temperatures. However, if temperatures become too high, neither mosquitoes nor the malaria pathogen can survive. In addition, stagnant water is also a major contributor to the spread of malaria, since most mosquito species breed in small pools of water. The correct amount and distribution of rainfall increases the possible breeding sites for mosquito larvae, which eventually results in more vectors to spread the disease. With little rainfall, there are few places for the mosquitoes to breed. For these reasons, and in order to control mosquito population, it is important to examine the weather parameters such as temperature and rainfall which are imperative in determining the disease epidemics. Accurate seasonal climate forecasts of these variables, together with malaria models should be able to drive an early warning system in endemic regions. These models can also be used to evaluate the possible change in regions under climate change scenarios, and the spread of malaria to new regions. In this study, we develop and analyse a mosquito model to study the population dynamics of mosquitoes. Ignoring the impact of climate, the model is further developed by introducing human compartments into the model. We perform both analytical and numerical analyses on the two models and verify that both models are epidemiological and mathematical well-posed. Using the next generation matrix method, the basic reproduction number of each system is calculated. Results from both analyses confirm that mosquito- and disease-free equilibria are locally asymptotically stable whenever R0 < 1 and unstable whenever R0 > 1. We further establish the global stability of the mosquito-free equilibrium using a Lyapunov function. In order to examine the effectiveness of control measures, we calculate the sensitivity coefficients of the reproductive number of the mosquito-human malaria model and highlight the importance of mosquito biting rate on malaria transmission. In addition, we introduce climate dependent parameters of Anopheles gambiae and climate data of Limpopo province into the malaria model to study malaria transmission over the province. Climate variables and puddle dynamics are further incorporated into the mosquito model to study the dynamics of Anopheles arabiensis. The climatedependent functions are derived from the laboratory experiments in the study of Maharaj [114], and we further verify the sensitivity of the model to parameters through sensitivity analysis. Running the climate data of Dondotha village in Kwazulu-Natal province over the mosquito model, it is used to simulate the impact of climate variables on the population dynamics of Anopheles arabiensis over the village. Furthermore, we incorporate human compartments into the climate-based mosquito model to explore the impact of climate variability on malaria incidence over KwaZulu-Natal province over the period 1970-2005. The outputs of the climate-based mosquito-human malaria model are further analysed with Principal Component Analysis (PCA), Wavelet Power Spectrum (WPS) and Wavelet Cross-coherence Analysis (WCA) to investigate the relationship between the climate variables and malaria transmission over the province. The results from the mosquito model fairly accurately quantify the seasonality of the population of Anopheles arabiensis over the study region and also demonstrate the influence of climatic factors on the vector population dynamics. The model simulates the population dynamics of both immature and adult Anopheles arabiensis and increases our understanding on the importance of mosquito biology in malaria models. Also, the simulated larval density produces a curve which is similar to observed data obtained from another study. In addition, the mosquito-malaria models produce reasonable fits with the observed data over Limpopo and KwaZulu Natal provinces. In particular, they capture all the spikes in malaria prevalence. Our results further highlight the importance of climate factors on malaria transmission and show the seasonality of malaria epidemics over the provinces. The results of the PCA on the model outputs suggest that there are two major process in the model simulation. One of the processes indicate high loadings on the population of Susceptible, Exposed and Infected humans, while the other is more correlated with Susceptible and Recovered humans. However, both processes reveal the inverse correlation between Susceptible-Infected and Susceptible-Recovered humans respectively. Through spectrum analysis, we notice a strong annual cycle of malaria incidence over the province and ascertain a dominant periodicity of one year. Consequently, our findings indicate that an average of 0 to 120-day lag is generally noted over the study period, but the 120-day lag is more associated with temperature than rainfall. The findings of this study would be useful in an early warning system or forecasting of malaria transmission over the study areas.