Mathematical modeling of malaria infection with innate and adaptive immunity in individuals and agent-based communities

Abstract

Background: Agent-based modeling of Plasmodium falciparum infection offers an attractive alternative to the conventional Ross-Macdonald methodology, as it allows simulation of heterogeneous communities subjected to realistic transmission (inoculation patterns).

Methodology/principal findings: We developed a new, agent based model that accounts for the essential in-host processes: parasite replication and its regulation by innate and adaptive immunity. The model also incorporates a simplified version of antigenic variation by Plasmodium falciparum. We calibrated the model using data from malaria-therapy (MT) studies, and developed a novel calibration procedure that accounts for a deterministic and a pseudo-random component in the observed parasite density patterns. Using the parasite density patterns of 122 MT patients, we generated a large number of calibrated parameters. The resulting data set served as a basis for constructing and simulating heterogeneous agent-based (AB) communities of MT-like hosts. We conducted several numerical experiments subjecting AB communities to realistic inoculation patterns reported from previous field studies, and compared the model output to the observed malaria prevalence in the field. There was overall consistency, supporting the potential of this agent-based methodology to represent transmission in realistic communities.

Conclusions/significance: Our approach represents a novel, convenient and versatile method to model Plasmodium falciparum infection.

Figure 1. Schematic representation of the model.

The population of uninfected red blood cells (x) provides the source for the infected population (y). Level I immune effector (a) is stimulated by y. Level II immune effector (b) is stimulated by y interacting with a+b. M represents the the number of merozoites, S represents an external source of inoculation.

Figure 2. Deterministic pattern versus AV pattern.

Panel A: Typical deterministic parasite density pattern (solid blue line) as predicted by the model. Also shown are innate immune effector a (blue filled curve) and adaptive immune effector b (purple filled curve). Panel B: Corresponding stochastically predicted mean parasite density (solid green line) and minimum/maximum envelope (purple fill) for the same deterministic solution (solid blue line) as shown in Panel A.

Figure 3. Typical MT-host with ‘odd-even’ envelope (purple shaded area) and its mean-curve (thick black line).

Figure 4. Typical deterministic histories starting from an immunologically naïve state with an initial inoculum.

Blue solid lines are parasitemia, the blue filled curve is immune effector a, the purple filled curve immune effector b, and the blue filled curve at the top are depleted resource cells. Deterministic histories can have single (Panel A), double (Panel B) and multiple (Panel C) wave patterns. However multiple waves patterns very rarely terminate and look very different from the MT data.

Figure 5. Two typical single wave datasets.

The solid gray lines are parasitemia. The curve with light gray fill is the model prediction.

Figure 6. Long term, multiple wave datasets calibrated with the present model.

Panels A and B depict cases where the calibration resulted in a reasonable fit. The datasets are suitable because they exhibit an initial wave of parasitemia which contains the global maximum of the entire history. Panels C and D depict cases which are less suitable because they are missing a pronounced first wave of parasitemia. The blue solid lines are the original MT data, the dashed blue lines are the best fits to the first wave of parasitemia (1st calibration step), the purple shaded areas are the AV envelopes (2nd calibration step) with its mean curve (solid green line).

Figure 7. Comparison of characteristic statistics between MT data and model prediction.

Panel A: Day of the first maximum; Panel B: Day of the last maximum; Panel C: Parasite density at first maximum; Panel D: Parasite density at last maximum; None of these characteristic features where significantly different between model and MT data.

Figure 8. AB communities subjected to stationary EIR and comparison to field data.

The data points (black) are taken from a review conducted by Beier et al. (1999) . The black line is the curve fit also given in that reference. The colored lines are model predictions based on different numbers of antigenically distinct variant clusters (m).

Figure 9. Comparison of model prediction to field observations from areas of seasonal malaria transmission.

Panel A: EIR as reported by Vercruysse (solid purple line), and reproduced as input for the model (solid blue line). Panel B: Malaria prevalence as reported by Vercruysse (solid blue line) and model prediction as monthly average (solid purple line) and envelope of monthly minima and maxima (olive fill) using as input the EIR pattern from Panel A. Panel C: EIR as reported by Gazin (solid purple line), and reproduced as input for the model (solid blue line). Panel D: Malaria prevalence as reported by Gazin (solid blue line) and model prediction as monthly average (solid purple line) and envelope of monthly minima and maxima (olive fill) using as input the EIR pattern from Panel C.