Forcing versus feedback: epidemic malaria and monsoon rains in northwest India

Abstract

Malaria epidemics in regions with seasonal windows of transmission can vary greatly in size from year to year. A central question has been whether these interannual cycles are driven by climate, are instead generated by the intrinsic dynamics of the disease, or result from the resonance of these two mechanisms. This corresponds to the more general inverse problem of identifying the respective roles of external forcings vs. internal feedbacks from time series for nonlinear and noisy systems. We propose here a quantitative approach to formally compare rival hypotheses on climate vs. disease dynamics, or external forcings vs. internal feedbacks, that combines dynamical models with recently developed, computational inference methods. The interannual patterns of epidemic malaria are investigated here for desert regions of northwest India, with extensive epidemiological records for Plasmodium falciparum malaria for the past two decades. We formulate a dynamical model of malaria transmission that explicitly incorporates rainfall, and we rely on recent advances on parameter estimation for nonlinear and stochastic dynamical systems based on sequential Monte Carlo methods. Results show a significant effect of rainfall in the inter-annual variability of epidemic malaria that involves a threshold in the disease response. The model exhibits high prediction skill for yearly cases in the malaria transmission season following the monsoonal rains. Consideration of a more complex model with clinical immunity demonstrates the robustness of the findings and suggests a role of infected individuals that lack clinical symptoms as a reservoir for transmission. Our results indicate that the nonlinear dynamics of the disease itself play a role at the seasonal, but not the interannual, time scales. They illustrate the feasibility of forecasting malaria epidemics in desert and semi-arid regions of India based on climate variability. This approach should be applicable to malaria in other locations, to other infectious diseases, and to other nonlinear systems under forcing.

Figure 1. Malaria cases and rainfall.

(A) Monthly malaria reported cases (red) and monthly rainfall from local stations (black) for Kutch. (B) Correlation between accumulated rainfall in different time windows preceding the month of reported cases. A maximum is observed when rainfall is accumulated over 5 to 6 months. (C) Monthly reported cases as a function of accumulated rainfall in the previous five months. A threshold, nonlinear response is apparent with no effect of rainfall below a value of around 200mm and an increase in both the mean and the variance of cases above it.

Figure 2. Flow diagram for two compartment models of malaria transmission.

(A) shows the VSEIRS model and (B) shows the model. Human classes in (A) are (Susceptible), (Exposed), (Infected), and (Recovered). Mosquito classes are κ (latent force of infection) and λ (current force of infection). The possibility of transition between classes and is denoted by a solid arrow, with the corresponding rate written as . The average time of mosquitoes in the latent state is denoted by . The dotted arrows represent interactions between the human and mosquito stages of the parasite. The model in (B) adds clinical immunity , by differentiating between clinical infections that contribute to the measured cases, and less severe infections in a new class that are not clinical but remain infectious to mosquitoes at a lower level than . Clinical infections can fully recover becoming susceptible again, or remain parasitemic and transition to . Recovery from mild infections results in individuals who are fully protected from clinical disease, in class , whose further exposure to infected mosquitoes, can result again in mild infections. In time, clinical immunity can also be lost, with transitions from to , and therefore the return to full susceptibility. Only a fraction of individuals in contribute to the force of infection; the susceptibility to infection is reduced by a factor in class relative to .

Figure 3. Reported monthly malaria cases and simulations for Kutch.

Black lines show the median of ten thousand simulations; the shadowed regions correspond to the range between the 10% and 90% percentiles of the simulations. Red lines show the reported cases. (A) VSEIRS model with rainfall; (B) VSEIRS model without rainfall. Note that these curves do not represent the fit of the model one time-step ahead but the numerical simulation from estimated initial conditions at the end of 1986 for the complete twenty years' period, using observed rainfall values.

Figure 4. Hindcast predictions for the time course of epidemics for Kutch.

The malaria data are shown in red. Superimposed on these observations, we show the predicted mean cases from one to four months ahead obtained by simulating the VSEIRS model from (1) the end of August (blue dots) and (2) the end of December (green dots). Shadowed regions in respective colors correspond to the standard deviation from a set of 5000 predicted values for each given time. Notice that this procedure requires the estimation not only of the observed state (i. e. reported cases) but also of all non-observed states at each time (i.e. S,E,I,R,,). Simulations of the model require the accumulated rainfall in the previous five months: to obtain this quantity, the observed rainfall is used only until the initial time (end of August or December) and the rest of the months are completed by replacing the ‘missing’ rainfall value (given that we are predicting one to four months ahead) by its monthly average. (A) VSEIRS model with rainfall; (B) VSEIRS model without rainfall.

Figure 5. Density plot of correlation between the accumulated rainfall from May to August and the accumulated cases from September to December in 10,000 simulations from a set of 2,000 solutions for the model with rainfall (black) and without rainfall (gray).

The vertical line is the observed correlation of 0.778. For the model with rainfall, 38.63% of the simulations have a correlation with rainfall above the observed value (black circle).