_{0}

^{1}

^{2}

^{3}

^{3}

^{*}

^{1}

^{4}

^{*}

The authors have declared that no competing interests exist.

Conceived and designed the experiments: GLJ DLS DAF. Performed the experiments: GLJ. Analyzed the data: GLJ DLS DAF. Wrote the paper: GLJ DLS DAF.

Human infection by malarial parasites of the genus _{0}_{0}

We report a new mathematical model of the progression, within a human host, of a malaria infection caused by the parasite

Approximately 2.5 billion people live in areas whose local epidemiology permits transmission of

The development of mathematical models of malaria is contingent on a detailed understanding of the parasite lifecycle. This begins in humans when motile parasite forms, termed sporozoites, enter the body through the bite of an

Many models of malaria have been developed to describe this cycle of transmission from the mosquito to the human host and back. These models can be broadly classified into two categories: compartmental and mechanistic. A compartmental model is any type of transmission model that simulates populations of individuals transitioning into different compartments at constant rates, with each compartment representing a different state of disease/non-disease. For example, an “SIR” model is a compartmental model in which individuals are grouped into three populations, namely susceptible (S), infectious (I), and recovered (R). Individuals transition between compartments at a constant rate depending on several factors that include the virulence of the disease and the immune responses of hosts. More sophisticated models include additional compartments that each represent a different disease state. For example, the infective compartment may be separated into multiple compartments (I_{1}, I_{2}, I_{3}, etc.) each with different levels of infectiousness, or other compartments may be added, for example infected but not infectious hosts (known as the E compartment)

In contrast, mechanistic malaria models incorporate the within-host mechanisms that determine human infectiousness over time. In such models, individual hosts are the primary units of analysis

Each of these two frameworks has a useful role to play. Compartmental models benefit from simplicity, identifiably, and clarity, while mechanistic ones allow for simulations of control measures that are highly non-linear. Regardless of the model type, one of the most important mathematical quantities for theories of disease control aimed at elimination is _{0}_{0}_{0}_{0}_{0}

Vectorial capacity can be estimated using a variety of techniques

Here, we report a stochastic, mechanistic, within-host model that simulates the progression of

Using this model, we have examined the levels of human-to-mosquito infectivity over time and isolated the host-related determinants of the basic reproduction number, _{0}_{0}_{0}

The model used to calculate asexual parasitemia is a within-host model that simulates the course of an infection one replication cycle after merozoites have emerged from the liver. Asexual parasitemias are modeled as a system of discrete (two-day time interval) difference equations previously elaborated by Molineaux et al.

Our model was fitted to data from malaria therapy patients, in which individuals with tertiary syphilis and with no acquired immunity to malaria were inoculated with single strains of

_{j}_{i}

Our within-host model incorporates three types of immune responses. An innate response

Each of these three responses has a characteristic effect on the progression of parasitemia. The innate response controls the initial densities of asexual parasitemia and is dominant during the initial period of infection. The second response, the variant-specific acquired response, controls the characteristic peaks and dips in parasitemia and interacts with the

_{C}

_{m}

_{v}

_{v}

_{m}

^{4.79} parasites per µL (this was the median among malaria therapy patients from

In our model, PfEMP1 variant densities are explicitly modeled. The parasite population is partitioned into 50 different subpopulations, each representing one antigenic isotype (_{j}_{i}_{i}(_{i}(

First, we assume that the PfEMP1 status of parasites is reset during the mosquito stage such that infections start with a single PfEMP1 variant

It is also possible that more than one variant, even most or all variants, are expressed at the onset of infection

In our model, we also assume that parasites expressing different PfEMP1 variants proliferate at different rates. We assign each isotype a growth rate _{i}

_{i}_{i}

Because we are interested in utilizing this model to simulate drug treatment in low-transmission areas, treatment-seeking behavior is an important consideration. In the absence of diagnostic testing, fever may serve as an indicator of infection for both patient and clinician

Specifically, we simulate the time course of an individual infection from inoculation to clearance and record the maximum parasitemia achieved (denoted _{10}(0.0002), 0) =

Our gametocytemia model equations were first articulated by Diebner et al. and Eichner et al. (the two models differ slightly; we adhere to the formulation by Eichner et al.)

As in the asexual model described above, the original gametocyte modeling work

_{S}

γ: Asexual to sexual conversion probability, peak specific; stochastic following log-normal distribution with location parameter of −6 and a scale parameter of 4 in natural log space

_{G}

_{0}

In the original Ross-Macdonald model, the infectivity of humans to mosquitoes was parameterized by a constant,

Our model of infectivity is not mechanistic in the same way that our asexual and gametocyte models are. We use the sigmoidal curve reported in

A final note regarding infectivity is that of Jeffery and Eyles in their original 1955 study of mosquito feedings on malaria therapy patients

The functional relationship between gametocyte density and infectivity used here is taken from

In the initial formulation upon which we built our asexuals model ^{4.79} asexual parasites per µL) observed among 35 malaria therapy patients ^{5.66} asexual parasites per µL) from

In the original formulation of our gametocyte model, five parameters were fitted to individual patient case histories _{S}_{G}_{0}

The first component that we fitted to data was the asexuals model. For our simulation target data, we used the distribution of durations of infection from malaria therapy

With the target data defined, we then fitted the model to these data. For a measure of the goodness-of-fit, we used the relative errors between model outputs and the min, median, and max of the 9 indices, as well as the distances between the modeled and observed durations of infection (as measured from the cumulative distribution functions). We used the log-normal distribution for maximum parasitemias with a mean of 10^{4.79} asexual parasites per µL to set _{C}_{i}

To fit the gametocyte model parameters we first needed to define our target data. We used _{S}

We had three gametocyte model parameters to fit for which we had insufficient prior information: _{G}_{0}_{0} being fixed at their population means (as reported in _{G}

Model | Parameter | Reference value | Revised value/distribution |
Description |

Asexual | _{c} |
0.2 |
0.164 | Affects levels of innate immune response to total parasitemia |

Asexual | σ | 0.02 |
0.15 | Decay rate of acquired immune response to PfEMP1 variant |

Asexual | _{c}_{c} |
Fitted to case history |
truncated ln^{4.79}), |
Asexual parasite density at the first peak of parasitemia |

Asexual | _{m}_{m} |
Fitted to case history |
Gompertz( |
First day with observed asexual parasitemia minus last observed day |

Asexual | _{i} |
truncated |
truncated |
Growth rates of different PfEMP1 variants |

Gametocyte | _{s} |
Fitted to case history |
round(truncated |
Sequestration time for gametocyte maturation |

Gametocyte | Fitted to case history |
truncated ln |
Asexual to sexual conversion probability, peak specific | |

Gametocyte | _{G} |
Fitted to case history |
Rate at which age affects gametocyte mortality | |

Gametocyte | Fitted to case history |
0.0013 | Effects of previous asexual parasitemias on gametocyte death rates | |

Gametocyte | _{0} |
Fitted to case history |
0.03 | Initial age-related component of total gametocyte mortality rate |

See ref

See ref

The best-fit parameters for the asexual and gametocyte components of our mechanistic model are shown for those parameters that have been modified from their original values

We first consider the goodness-of-fit of the asexual component of the model.

Index | Index description | Minimum observed from malaria therapy patients |
Median observed from malaria therapy patients | Maximum observed from malaria therapy patients | Minimum model values |
Median model values | Maximum model values | Relative errors, difference of minima | Relative errors, difference of medians | Relative errors, difference of maxima |

2-1 | Initial slope | 0.19 | 0.49 | 0.87 | 0.24 | 0.52 | 0.75 | −0.26 | −0.07 | 0.14 |

2-2 | Log density at first local maximum | 3.37 | 4.79 | 5.66 | 3.69 | 4.78 | 5.67 | −0.09 | 0.00 | 0.00 |

2-3 | Number of local maxima | 2 | 10 | 17 | 2 | 9 | 17 | −0.21 | 0.06 | −0.01 |

2-4 | Slope of local maxima | −0.074 | −0.013 | −0.001 | −0.091 | −0.015 | −0.007 | −0.22 | −0.12 | −9.51 |

2-5 | Geometric mean of the intervals between consecutive local maxima | 14.4 | 20.0 | 77.8 | 1.5 | 14.6 | 28.4 | 0.90 | 0.27 | 0.64 |

2-6 | Standard deviation of the logs of the consecutive local maxima | 0.03 | 0.20 | 0.47 | 0.04 | 0.31 | 0.56 | −0.31 | −0.56 | −0.20 |

2-7 | Proportion of positive observations in the first half of the interval between first and last positive day | 0.4 | 0.88 | 1.0 | 0.57 | 0.97 | 1.0 | −0.43 | −0.10 | 0.00 |

2-8 | Proportion of positive observations in the second half of the interval between first and last positive day | 0.08 | 0.46 | 0.94 | 0.12 | 0.58 | 1.00 | −0.48 | −0.27 | −0.06 |

2-9 | Last positive day | 37 | 215 | 405 | 38 | 193 | 404 | −0.02 | 0.10 | 0.00 |

See ref

Indices from best-fit model outputs. To calculate these indices from model outputs, we used a bootstrapping procedure. The mechanistic malaria model was run 1,000 times and 50 samples of 50 runs each were selected. The number of runs (50) per sample was chosen to match the sampling procedure in

For the gametocyte model, we were able to set the delay of appearance parameter _{S}_{G}_{0}_{0}_{G}_{G}

_{S} |
||||||

Malaria therapy |
Simulated |
Malaria therapy | Simulated | Malaria therapy | Simulated | |

Minimum | 4.0 | 4.0 | 2.7 E-04 | 6.5 E-05 | 1.3 | 3.2 |

Geometric Mean | 7.4 | 6.9 | 0.0064 | 0.0066 | 6.4 | 5.6 |

Maximum | 12.0 | 10.8 | 0.135 | 0.111 | 22.2 | 13.9 |

See ref _{S}

Gametocyte properties were calculated from the mechanistic malaria model outputs using best-fit gametocytemic parameters. Model values are from 50 samples of 113 runs each, from a total of 1,000 runs. The end time for all runs was 800 days.

Of note, we did not explore the entire parameter space for the asexual and gametocyte models, given computational limitations. Our final parameters were chosen as best-fits when the model outputs were qualitatively judged to be acceptably close according to the goodness-of-fit described above. We thus cannot provide precise point estimates with confidence intervals for our parameters. Nevertheless, sensitivity analyses were performed for certain parameters, as described below.

As a further check of our model outputs, we also compared our model outputs to other data not explicitly used in the model fitting. The modeled arithmetic mean duration of time between first fever and first gametocytemia detectable by smear among gametocyte-positive individuals was ∼12.9 days. This compares closely to the measured value from malaria therapy patients (10–11)

We have developed a mechanistic model of the progression of malaria within a human host, parameterized such that the model reproduces the median and extremes of the dynamics of infection observed in malaria therapy. For the asexuals model, we fitted five model parameters to the minimum, medians, and maxima of nine different malariometric indices derived from malaria therapy data. For the gametocyte model, we fitted five model parameters to the minima, geometric means, and maxima of three different indices derived from the gametocytemias of malaria therapy patients.

_{10} PRBC per µL of blood. The black line illustrates the lower limit of detectability by microscopy (10 PRBC/µL)

Our mechanistic _{10} 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.

As illustrated in

The members of the PfEMP1 family of _{10} PRBC per µL.

Another important determinant of human infectivity is the assumed relationship between gametocyte density and parasite infectivity to mosquitoes (also referred to as human-to-mosquito infectivity). A variety of functions relating gametocytes to infectivity have been described and proposed in the literature

The scatterplot data (blue circles) were collated by Carter and Graves from multiple studies

(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. _{10} values to the nearest tenth) _{10} gametocytemia.

For our model of gametocyte densities, we visually examined a total of 262 malaria therapy charts provided by Diebner et al. _{10} values to the nearest tenth). We then compared these data to the maximum gametocytemias from 1,000 runs of our model using the best-fit parameters. Because the Diebner et al. study only includes individuals who recorded at least four gametocyte-positive observations

_{10} gametocytemia. The malaria therapy values are slightly higher on average initially, with a median of 3.10 for malaria therapy versus 2.95 from the model (grey horizontal line). Our model had a broader tail than the malaria therapy data, with more elevated gametocytemias than observed in the therapy data. The mean from the malaria therapy data was 3.01, whereas the mean from the model was 2.98. However, in our model, we estimated gametocytemias every day (i.e., we captured every maximum possible), as opposed to the sparser sampling of the malaria therapy data. Further, some of the individuals included in the patient charts from

Once we were able to generate malarial infections _{0}_{0}_{0}^{−1} days. The basic reproduction number of malaria is then described by the classic formula:

In reality, neither _{0}_{0}_{i}(t)_{i}(t)

If we first consider the mean of _{i}(t)^{th} and 75^{th} percentiles of daily infectivity for these simulated individuals, as well as the mean infectivity over time (in red). The mean infectivity

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 25^{th} and 75^{th} 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.

If we integrate _{i}(t)_{i}_{i}

If we integrate either the mean human infectivity over time _{i}_{0}

Once we had computed _{i}

We first compared the ^{−1} with a constant daily human-to-mosquito infectiousness (^{−1}) and each compartment had a different proportional infectivity (1.90, 3.08, 1.53, 0.28) of the average daily infectivity

We can derive

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.

The differences in

Integrating over time, we find that the

To calculate the net human infectiousness _{0}_{1} is the clearance rate of infectivity, _{1} and

Given our calculations of

In the previous section we calculated the mean responses of individuals over time for the models of Lawpoolsri et al. _{i}_{i}_{i}_{i}_{i}_{i}

The distributions of net human infectivity were calculated for three models: our mechanistic model as well as the compartmental models of Lawpoolsri et al.

We ran a variety of sensitivity analyses by varying the model parameters and observing the changes in model output. For the asexuals model, we adjusted

For the gametocyte model, we examined the effects of varying the _{G}_{G}_{G}_{G}_{G}_{G}_{G}_{G}

Regarding the relationship between gametocyte density and human-to-mosquito infectivity, our default model outputs assumed the relationship from Stepniewska et al. as fitted from malaria therapy

To develop the six other possible relationships between gametocyte densities and infectivity, we utilized additional information regarding the biology of

The mean net infectivity values for seven of the parameterizations are 70.0, 41.1, 23.6, 64.1, 33.3, 16.2, and 36.3 net infectious days for the Stage VB, High; Stage VB, Median; Stage VB, Low; Stage V, High; Stage V, Median; Stage V, Low; and Carter & Graves parameterizations, respectively (without the Jeffery-Eyles corrections and with

Also of note, our model calculated

Here we describe the development of a novel, stochastic, within-host model of the progression of malaria in patients with no acquired malarial immunity. This model utilizes the difference equations originally developed by Molineaux and Dietz to simulate the progression of asexual and sexual parasitemias

Once our mechanistic model was formulated, we revisited the analytic Ross-Macdonald model to examine how human infectiousness enters into the formula for the basic reproduction number _{0}_{i}_{0}

Our study included a review of the mathematical literature to determine whether we could impute these quantities from other modeling work to provide a baseline for comparison. We examined the models of Lawpoolsri et al. _{i}_{0}_{0}

In addition to our calculation of the invariant _{0}

In addition to the usefulness of these results for mapping and control efforts, the modeling platform and analytic framework described herein will help clarify the different assumptions among malaria models. Further, because we calculate asexual and sexual parasite densities daily, and because the model reproduces the entire variability of host-parasite dynamics observed in malaria therapy, our modeling framework provides a powerful new tool for exploring the effects of antimalarial treatments on transmission. As malaria decreases worldwide, our model results will become more relevant to more regions of the world, thus helping to improve targeting of control efforts.

Standalone model designed for Windows operating systems.

(ZIP)

Standalone model designed for Macintosh operating systems.

(ZIP)

Source code for malaria model and graphical user interface components.

(ZIP)

Graphical user interface of standalone model software.

(PDF)

User guide for use of the model software.

(PDF)

Computing resources were provided in part by the ‘Hotfoot’ High Performance Computing Cluster at Columbia University.