Endemicity response timelines for Plasmodium falciparum elimination

Abstract

Background: The scaling up of malaria control and renewed calls for malaria eradication have raised interest in defining timelines for changes in malaria endemicity.

Methods: The epidemiological theory for the decline in the Plasmodium falciparum parasite rate (PfPR, the prevalence of infection) following intervention was critically reviewed and where necessary extended to consider superinfection, heterogeneous biting, and aging infections. Timelines for malaria control and elimination under different levels of intervention were then established using a wide range of candidate mathematical models. Analysis focused on the timelines from baseline to 1% and from 1% through the final stages of elimination.

Results: The Ross-Macdonald model, which ignores superinfection, was used for planning during the Global Malaria Eradication Programme (GMEP). In models that consider superinfection, PfPR takes two to three years longer to reach 1% starting from a hyperendemic baseline, consistent with one of the few large-scale malaria control trials conducted in an African population with hyperendemic malaria. The time to elimination depends fundamentally upon the extent to which malaria transmission is interrupted and the size of the human population modelled. When the PfPR drops below 1%, almost all models predict similar and proportional declines in PfPR in consecutive years from 1% through to elimination and that the waiting time to reduce PfPR from 10% to 1% and from 1% to 0.1% are approximately equal, but the decay rate can increase over time if infections senesce.

Conclusion: The theory described herein provides simple "rules of thumb" and likely time horizons for the impact of interventions for control and elimination. Starting from a hyperendemic baseline, the GMEP planning timelines, which were based on the Ross-Macdonald model with completely interrupted transmission, were inappropriate for setting endemicity timelines and they represent the most optimistic scenario for places with lower endemicity. Basic timelines from PfPR of 1% through elimination depend on population size and low-level transmission. These models provide a theoretical basis that can be further tailored to specific control and elimination scenarios.

Figure 1

A diagram of human infection dynamics in the Ross-Macdonald model, which was used for planning during the GMEP. Changes in the fraction of infected humans, denoted X, were described by two simple rules. The incidence rate of new infections or "force of infection", denoted h, describes the per capita rate that uninfected humans would become infected. Clearance of existing infections, the per capita rate that infected humans lose infections, was denoted r. Human infection dynamics are described by an equation: = h(1 - X) - rX, where the superdot represents the derivative with respect to time.

Figure 2

A diagram of the queuing models, which extend the Ross-Macdonald model by tracking changes in the MOI. The fraction of the whole population with MOI of m, denoted x_m, changes when new infections occur or when existing infections clear, and in any short interval of time, one individual's MOI increments or decrements by one [18]. The rate that new infections arise may depend on MOI, denoted h_m. The rate of loss may depend on MOI, denoted ρ_m, and ρ₁ = r denotes the rate a simple infection is lost. Changes in the fraction of the population that is uninfected are described by an equation: = -h₀x₀ + rx₁. Changes in the fraction of people who are already infected with a given MOI are described by a set of equations: = -h_mx_m + h_m-1x_m-1 - ρ_mx_m + ρ_m+1x_m+1. These equations describe a family of queuing models: each queuing model makes different assumptions about infection and clearance. In "infinite strain" models, h_m = h, and in "finite strain" models where M denotes the maximum number of types, h_m = h(1-m/M). Models considered parasite types that cleared independently, or with competition or facilitation. For independent clearance, individual types were unaffected by concurrent infection with other parasite types, so ρ_m = rm. Competition and facilitation were modelled by letting ρ_m = rm^σ, where σ >1 described competition and σ <1 implies facilitation. Compared with independent clearance, per-strain clearance rates are faster with competition (i.e. ρ_m> rm), and slower with facilitation (i.e. ρ_m<rm). Clearance rates per person increase with MOI in all the models; if no new infections occurred, the expected waiting time to lose all the existing infections would be the sum of times to progressively decrement MOI: 1/r+1/ρ₂+1/ρ₃+...+1/ρ_m.

Figure 3

a) The decline in PfPR for six different models with completely interrupted transmission: the Ross-Macdonald model (black), infinite strains, homogeneous biting, and independent clearance (red), two strains, homogeneous biting and independent clearance (blue), infinite strains, homogeneous biting and competition (purple), four strains with homogeneous biting and facilitation (orange), and senescing infections (green). b)The daily decay rates for several models have been plotted as a function of declining PfPR, which is plotted on the vertical axis. The decay rate is the log of the PR ratio on consecutive days. Diamonds along these trajectories are plotted one year apart. The models have the same colors as in the panel above: the Ross-Macdonald model (black), infinite strains, homogeneous biting and independent clearance (red), and two strains, homogeneous biting and independent clearance (blue). The trajectories have also been plotted for completely interrupted transmission as above and for two larger values of R_C (0.5, and 0.75), which reach different asymptotic decay rates. c)The same graph has been re-plotted on a logarithmic scale for PfPR to show that the decay rates remain constant at r(R_C-1) after PfPR is below approximately 1%. d)The relationship between baseline PfPR and the predicted MOI for the model with heterogeneous biting at various levels: α = 2, dotted; α = 3, dashed; α = 4.2, solid; α = 6, dash-dot; and homogeneous biting, red. The more aggregated the biting, the higher the average MOI for the same PfPR. e)The PfPR over time for the same models as panel dand completely interrupted transmission starting from a baseline PfPR of 70%, and compared with the Ross-Macdonald model (black). f)The decline in PfPR for a model with infinite strains, homogeneous biting, and independent clearance, but with different values of R_C (solid, R_C = 0; dashed, R_C = 0.5; and dotted, R_C = 0.75).

Figure 4

a) Using a stochastic model, PfPR was simulated over time starting from 1% and following through to elimination (defined as when no one remains infected) in a population of 100,000 people using the Ross-Macdonald model for R_C = 0 (red), R_C = 0.5 (blue), and R_C = 0.75 (green). The solid line shows the median PfPR over time from the ensemble, the dashed lines show the 5^th and 95^th quantiles, and the solid grey lines (sometimes covered by the median PfPR) show the same results for the deterministic models. b)The same plots for ensembles of stochastic simulations using Gamma distributed infectious periods (n = 4), for which infections effectively "senesce". The grey line (under the solid blue line) shows the Ross-Macdonald model for completely interrupted transmission. c, d)For the same ensemble of 500 simulations corresponding to the panels above, these violin plots show a kernel density plot for the distribution of extinction times, in years. The white dots show the median time to extinction, the thick black lines show the inter-quartile range. In the left-hand panel, the yellow dots show the predicted values from Equation 6, which are fairly close to the average waiting time to reach 1/H from the ensemble of simulations.

Figure 5

The parasite rate in two parts of Taveta [14]: the forest (top) and the village (bottom). The data track the PfPR in 2–10 year olds starting from 1954 and extending through 1959 – the grey shows the binomial confidence intervals by the exact test. Spraying started late in 1955 with effects that extended through 1959 (shown in pink). The blue line shows the Ross-Macdonald model with R_C = 0 that was used for planning during the GMEP, the purple line shows the Ross-Macdonald model with R_C = 0.7, the orange and red line shows the infinite queuing model with heterogeneous biting (α = 3) and R_C = 0 (orange) and with R_C = 0.7 (red).