Table 1.
Best-fit model parameter constants and distributions.
Table 2.
Comparison of asexual model outputs to malaria therapy data.
Table 3.
Comparison of gametocyte model outputs to malaria therapy data.
Figure 1.
Illustration of asexual, sexual, and infectivity outputs.
Our mechanistic P. falciparum infection model was used to simulate three individuals' host-parasite dynamics. (A) Schematic representation of the P. falciparum life cycle. The parasite is transmitted to humans though the bite of an infected mosquito. Motile forms (sporozoites) travel to the liver where they proliferate as liver stage parasites that and are then released into the blood. Parasites then adopt ∼48 hr cycles of red blood cell (RBC) invasion, asexual blood stage replication, and egress. Some intra-erythrocytic parasites differentiate into sexual forms (gametocytes) for uptake by further mosquito bites. Asexual parasites avoid immune capture by antigenic variation, primarily PfEMP1 cycling. (B) Individual log10 asexual parasitemias presented as a function of the number of days post emergence of parasites from the liver into the bloodstream. The inset depicts the first 50 days of infection; triangles above indicate the first day of fever. The black line is the level of detectability by microscopy (10 parasitized red blood cells (PRBC)/µL). (C) Daily gametocytemias of the same three individuals. (D) Estimated probability of human-to-mosquito transmission. Areas under the infectivity curves are equivalent to the number of fully infectious days. Although the model predicts the persistence of long-lived low-level and sub-detectable infections (as observed in malaria therapy), this panel illustrates how the bulk of infectivity usually occurs early in the course of infection.
Figure 2.
Illustration of model var dynamics.
The members of the PfEMP1 family of P. falciparum erythrocyte membrane proteins are encoded by var genes, present at ∼60 copies per genome and each expressing a different PfEMP1 type. (A) Total asexual parasitemia as a function of time post emergence from the liver was modeled and the proportion filled by each PfEMP1 variant is shown in a different color. The number of colors and their respective levels at a given time indicates the diversity of isotypes present. Results are shown from a single model output. (B) The number of isotypes circulating in the blood over time. Isotypes are ‘expressed’ only if the density of that isotype is greater than or equal to 0.02 parasitized red blood cells (PRBC) per µL (the assumed threshold for PCR detection). (C) The cumulative number of isotypes that have been expressed over time (modeled from a single infection). (D) Levels of total asexual parasitemia over time for the illustrated run, in log10 PRBC per µL.
Figure 3.
Relationships between gametocyte density and probability of human-to-mosquito infectivity.
The scatterplot data (blue circles) were collated by Carter and Graves from multiple studies [63], [64]. The blue line is a logistic regression through the Carter and Graves data. The Bousema data indicate the relationship between infectivity and density from skin feeding studies with predominantly African volunteers in endemic settings [56]. The red line indicates the infectivity of malaria therapy volunteers (‘Median Infectivity, Stage V’) [48]; this parameterization is assumed to be the default. The red dotted lines illustrate the ‘High’ (maximum) and ‘Low’ (minimum) infectivity curves used in the model. The light blue solid and dotted lines indicate the relationships between gametocyte density and infectivity, assuming only Stage VB gametocytes are infectious (see Methods). All infectivity relationships included in the model are truncated at 1 (i.e. 100% probability of infection).
Figure 4.
Comparison of model and malaria therapy cumulative distributions for two indices.
(A) These line curves show the cumulative distributions of the durations of infection for the malaria therapy data, as well as those of our mechanistic model and the compartmental models of Lawpoolsri et al. [29] and Okell et al. [12]. The distribution from the malaria therapy data comes from fitting a Gompertz probability distribution to the durations of infection from 54 patients, as reported by Sama et al. [33]. The cumulative distribution function of the best-fit Gompertz distribution is plotted in red. The mechanistic model cumulative distribution was generated by calculating the durations of infection for 1,000 runs and plotting their empirical cumulative distribution function. The distributions from Lawpoolsri et al. and Okell et al. were generated by running those compartmental models according to their mathematical assumptions. The malaria therapy and mechanistic model distributions show relatively tight fits throughout the distribution. The durations of infections for the malaria therapy data and our mechanistic model are defined as the last observable day by smear minus the first observable day; the durations for the compartmental models are defined as the durations of time in infectious compartments. (B) We reviewed a total of 262 malaria therapy charts and recorded the maximum observed gametocytemia from each patient (data were recorded as log10 values to the nearest tenth) [24], [65]. We then recorded the maximum gametocytemias from 1,000 runs of our model. Because the malaria therapy data only include individuals who recorded at least four positive gametocyte observations, we censored out model runs in which gametocyte levels never exceeded 10 per µL (N = 988) [24]. Illustrated are the empirical cumulative distributions for the two data sets after log-transformation, i.e., the proportion of data that are less than or equal to a given level of log10 gametocytemia.
Figure 5.
Mechanistic model predicted human infectivity over time and within a population.
We calculated daily human infectivity to mosquitoes, as a function of time post emergence, for 1,000 simulated individuals. (A) Asexual parasitemias from 1,000 model runs. The wide diversity of host-parasite dynamics was fitted to malaria therapy data. (B) The mean daily infectivity of 1,000 simulated individuals for the first 300 days post emergence is shown as the red curve, and the area between the 25th and 75th daily infectivity percentiles is shown in blue. (C) Net infectivity for each of 1,000 individuals. The distribution of net human infectivity is represented as a violin plot, which extends to the maximum infectivity. The red cross illustrates the arithmetic mean infectiousness, and the green box shows median infectiousness.
Figure 6.
Comparison of mean infectivity over time, D(t).
The mean human infectivity to mosquitoes was calculated as a function of time for three models: our mechanistic model as well as the stochastic representations of the models of Lawpoolsri et al. [29] and Okell et al. [12]. For each model, the mean daily infectivity of 1,000 untreated individuals was simulated. (A) Mean infectivity for the first 800 days for the three models. (B) Mean infectivity for the first 200 days only. (C) Infectivity curves for the three models, scaled so that mean infectivity is equal to that of the mechanistic model.
Figure 7.
Comparison of distributions of net human infectivity, Di.
The distributions of net human infectivity were calculated for three models: our mechanistic model as well as the compartmental models of Lawpoolsri et al. [29] and Okell et al. [12]. (A) The infectivity for each of 1,000 individuals was integrated over time for each model. The distributions of net infectivity among individuals are represented as violin plots (vertical histograms); the plots extend to the maximum infectivity. (B) Scaled distributions of net infectivity. The distributions in panel (A) were rescaled by multiplying by a scaling factor such that all three distributions had the same mean as that of the mechanistic model. The red crosses illustrate the arithmetic mean infectivity, while the green boxes show the median infectivity.