A malaria transmission model was developed to closely track the observed transmission intensity over an eight-year period of seven hamlets in the Tanaosri subdistrict, Suanphung district in Ratchaburi Province, a mountainous area along the Thai-Myanmar border. The study site covered an area of around 50 km² with approximately 3,500 inhabitants in about 500 households. Free diagnosis and treatment of malaria has been provided for people in the area since 1997 by the Rajanagarindra Tropical Disease International Center (RTIC), operated by Mahidol University, Thailand, the only malaria clinic in this area. Individuals commonly receive treatment for febrile malaria within two days of an acute attack. After deployment of the early detection and treatment programme, the peak incidence of clinical malaria has gradually decreased from approximately four cases per 100 persons in 1999 to about two cases per 100 persons from 2003 through 2005. A single dose of mefloquine had been used as the first-line drug for treatment of uncomplicated falciparum malaria until the year 2005; the standard regimen was then changed to a two-day artesunate-mefloquine combination therapy, as recommended by the Thai government, because of the increased mefloquine resistance in the area 17. A single dose of 30 mg primaquine is given to all P. falciparum positive patients on the last day of the treatment course. In Thailand, the haemolysis after primaquine administration among glucose-6-phosphate dehydrogenase (G6PD) deficiency patients is relatively mild 1819. A test for G6PD deficiency is not required before a single dose primaquine administration.

Malaria transmission in this area is markedly seasonal with a peak of transmission during the rainy season (April to July), mainly due to the fluctuation of the mosquito population. However, malaria transmission does continue at a low level during the dry season. The P. falciparum parasite rate (PfPR) in the area is very low, according to regular active surveillance surveys that have sampled more than half of the study population since 2003 (Table 1). Parasites were not detected in three surveys in 2005. Although some infections with low parasite densities might have gone undetected, the low PfPR and yearlong continuous transmission suggest that asexual blood-stage infection is rare and thus gametocyte carriage in the absence of asexual parasites must play an important role in maintaining mosquito infection and the persistence of malaria transmission in the area.

A deterministic model of the infection dynamics of human and mosquito populations in the study area was developed. In the models, the human population was divided into five compartments: state variables tracked the proportion that were susceptible (S_h), liver-stage only (L_h), asymptomatic infection with gametocyte carriage (A_h), clinical episode (C_h), and gametocyte carriage only (G_h). In this low malaria transmission setting, malaria immunity and super-infection are rare, so an immune population was not included in the model. In the model for the mosquito population, seasonal mosquito population dynamics was considered, so the model tracked the population density of susceptible (M), infected (Y), and infectious (Z) mosquitoes (Figure 1). The human and mosquito infection dynamics were connected by human blood feeding of uninfected mosquitoes on infectious humans, and of infectious mosquitoes on uninfected humans. The dynamics of malaria infections in the two host populations are, thus, described by the following differential equations.

Equations for the human population:

Equations for the mosquito population:

Susceptible humans become infected when exposed to infectious mosquitoes at the rate baZ, where a is the human feeding rate and b is the probability of transmission 2021. A large proportion of infected patients (P) become symptomatic at the rate ν, an inverse of the latent period 22, these symptomatic individuals are subsequently detected and treated by passive surveillance at the clinic. Because of the delayed development of P. falciparum gametocytes, the rate that patients become infectious (ξ) after an acute attack depends on the duration of the gametocyte maturation process 78. Individuals normally remain infectious for a period of time even if the infection is treated and cured. Infections with gametocytes lose their infectivity to mosquitoes at the rate σ, which depends in part on the different treatment regimens 1323.

Asymptomatic asexual blood-stage infections are occasionally observed in low-endemic malaria areas 24. Therefore, a small proportion of infected individuals (1-P) was assumed to become asymptomatically infected. In addition, asymptomatic cases imported from neighboring areas occur at the rate μ. These asymptomatic patients are likely to remain untreated and infectious for a long period, until the gametocytes are naturally eliminated at rate r 7. The imported rate of asymptomatic cases was assumed to be equal to the exported rate of susceptible individuals, so that the population size remains constant.

For the mosquito population, adult mosquitoes emerge from larval habitat, which is modeled with a sinusoidal function for seasonal forcing, λ(1+sin(2πt/365). Mosquito infection occurs at rate ac, where c is the probability of transmission when they feed on infectious humans 21. These infected mosquitoes subsequently become infectious at the rate q. A constant death rate, g, was applied to all mosquito classes.

All parameter estimates were obtained from published literature and unpublished data from the study area; except for the average mosquito birth rate, λ, which was estimated by fitting the model to the observed malaria occurrence data, and the waiting time to clear gametocytes under different treatment regimens, which was computed using another model (see below). Details and value estimates of parameters in the models are described in Table 2.

The duration of infectivity under different drug regimens was computed using a within-host model of gametocytogenesis (Figure 2). The dynamics of the model are described by the following differential equations.

In P. falciparum, the erythrocytic stage takes approximately two days, at the end of which each asexual parasite, A_g, produces approximately 16 merozoites (θ) 2526. In each cycle of erythrocytic-schizogony, all merozoites were assumed to have an equal chance of undergoing gametocytogenesis; however, only a proportion, p = 0.02, of merozoites commit to gametocytogenesis 27. The mortality rate (α) is applied to the asexual parasite population due to the schizonticidal treatment. In the model, patients were assumed to receive treatment immediately after asexual parasite density reached 10⁴/μL of blood, the level that normally causes clinical symptoms among people in low-endemic areas 282930. After artesunate-mefloquine combination treatment, the density of asexual parasites is reduced by a factor of about 1,000 per 2-day schizogony cycle (α), i.e., the asexual parasites density is dropped from 10⁴/μL to 10/μL at the first cycle after treatment 313233. The surviving merozoites at each cycle then either convert to gametocytes or multiply into newly merozoites that continue to the next schizogony cycle. Therefore, the initial density of early stage gametocytes is a product of the conversion proportion and the net number of merozoites produced with each cycle (pθ). The early stage of immature gametocytes (I_g), which are indistinguishable from asexual parasites, remain in circulation for about one day (δ^-1), then sequester on blood vessels while continuing the maturation process (sequestered immature gametocytes; S_g). It takes about 8 days (β^-1) for gametocytes to mature and release to the peripheral blood (visible mature gametocytes; G_g). The longevity of a mature gametocyte in the blood stream is approximately 3.5–4 days (ρ^-1) 8.

The mortality rate of mature gametocytes due to primaquine treatment (τ) depends on the day of primaquine administration relative to the day of initial schizonticidal treatment (asexual parasites reach 10⁴/μL of blood). Plasma concentration of primaquine was estimated to decrease at elimination rate 0.5³ per day (an 8-hour elimination half-life). Since the plasma concentration of primaquine in vivo is difficult to determine, primaquine was assumed to remain effective in killing 90% of mature gametocytes when the plasma concentration was above 10^-5 of the maximum concentration, for a net gametocyte reduction ratio of 100 per 2-day cycle, i.e., gametocyte density, which is a function of the mature gametocytes that arise from newly-emerging merozoites at each two-day cycle, eight days earlier, are reduced by 100-fold. The untreated gametocytes remaining in the blood over time were then used to determine the duration of infectiousness. Multiple realizations with different timeframes of primaquine administration were performed to determine the different durations of individual infectiousness after primaquine treatment

The duration of infectiousness is related to mature gametocyte densities and mature gametocyte longevity. The duration of infectiousness was computed based on the density of gametocytes in an individual host, using a log sigmoid relationship between gametocyte density (G_g) and infectivity to mosquitoes 34.

With a 3.6 day half-life, gametocyte densities decline by 90% every eight days. If gametocyte densities were very high initially, then there would be virtually no drop-off in infectivity until the densities reach the S-portion of the logistic curve (Figures 3A and 3B). The time that a person remains infective (probability of infectivity more than zero) differs by about 10 days for every 10-fold difference in the initial gametocyte density (Figure 3C). Thus, for every 10-fold reduction in initial gametocyte densities achieved by curing asexual blood-stage infections alone there will be about 10-day reduction in the duration of human infectivity to mosquito. Therefore, a switch from mefloquine to artemisinin-mefloquine combination therapy (ACT) would reduce infectivity by about 10 days for every 10-fold reduction in mature gametocyte densities. Similarly, a two-day delay in appearing at the clinic could result in a 10-fold increase in gametocyte densities and a 10-day increase in infectiousness. Further reductions can only be achieved by a follow-up treatment with primaquine to reduce the densities of mature gametocytes (Figure 3D).

The impact of optimally timing primaquine treatment was assessed by comparing the initial basic reproductive number for malaria derived from the population when no intervention was applied (R₀) with the basic reproductive numbers for malaria at different scenarios of the treatment intervention (R_c). R₀ is an estimate of expected number of hosts infected by a single infectious person during his or her entire infectious period 35. The magnitude of R₀ provides insight into the transmission intensity of the disease and is often used to justify the effect of intervention programmes 3637. The classic formula for R₀ of Ross and Macdonald is shown in equation 5 35.

Where V denotes vectorial capacity, detailed definitions of other parameters are shown in Table 2.

The basic reproductive number for the control programme was calculated by modifying the Ross and Macdonald formula to consider the proportion treated with ACT (P), or with the follow-up with primaquine (Q):

Where c and r indicate the individual infectivity and duration of gametocyte carriage in natural infections, respectively. Parameters c_P and 1/σ_P indicate the individual infectivity and duration of gametocyte carriage, respectively, when a schizonticidal drug regimen is applied to a population; the reductions are due to the clearance of asexual parasites. The parameters c_Q and 1/σ_Q indicate the individual infectivity and duration of gametocyte carriage, respectively, when a primaquine follow-up regimen is applied to the population. A product of c_P and 1/σ_P, or c_Q and 1/σ_Q is the cumulative duration of infectiousness for the two different treatment regimens, which is defined as a function of gametocyte density in equation 4.

The impact of the follow-up primaquine regimen was represented by the magnitude of a ratio between R₀ and R_c (R₀/R_c). The vectorial capacity (V) and the transmission probability from mosquitoes to human (b) were assumed to be the same regardless of the intervention, so they cancel out in the ratio. Therefore, R₀/R_C can be calculated by the following equation.

The larger the magnitude of R₀/R_C, the greater the reduction in potential transmission. In addition, the relationship between the percent coverage with the primary treatment (P) and with the follow-up treatment (Q) and the ratio of R₀ and R_c was examined.

Finally, the optimal timing of primaquine administration was computed by finding the timing that produced the shortest infectious period in the gametocyte model. By assuming 100% coverage of follow-up primaquine treatment among symptomatic patients, the new infectious period was replaced in the initial transmission model to examine the effect of optimal timing of primaquine administration at the population-level. Also the possibility of malaria elimination in the area was determined when different elimination strategies were implemented.

The transmission model provided a good estimate of the dynamics of malaria transmission in the study area. Over the eight-year period, the estimated incidence and observed incidence were comparable in all years except 2000, when the model significantly underestimated malaria incidence (Figure 4A). Malaria transmission in the area was strongly seasonal. Although the model assumed that 99% of the infected population received standard malaria treatment, malaria transmission only decreased gradually in the first three years before settling into a lower, stable orbit, with the annual peak incidence about 1.8 per 100 persons. The switch from mefloquine to ACT in 2005 resulted in a significant reduction in the incidence in year 2006, to less than one per 100 persons. However, estimates from the model suggest that malaria transmission will continue even after the deployment of ACT.

The changes in transmission dynamics are illustrated in Figure 4B. As is typical of low malaria transmission settings, the proportion of asymptomatic gametocyte carriers was low compared with the proportion of symptomatic individuals, which varied seasonally. While the proportion of people with gametocytaemia almost reached zero during the dry season, the model predicts that about three percent of the population remained gametocytaemic when the environment was suitable for mosquito vectors. This small proportion of gametocytaemic people could play an important role in maintaining transmission of the parasite in the area.

Different malaria dynamics were observed when primaquine treatment was given at different times. Changes in the density of each parasite developmental stage over time with different primaquine treatment regimens are shown in Figure 5A. The simulated primaquine plasma concentration remained above the killing concentration for up to three days after administration. The duration and density of individual gametocytaemia differed substantially according to the timing of primaquine administration. When primaquine was given on the same day as other anti-malarial drugs, there were no mature gametocytes in the blood to be killed by primaquine, and infectiousness of the patient was not changed in comparison with no primaquine treatment. The duration of individual infectivity to mosquitoes could be as long as 14 days. In contrast, duration of infectiousness would be greatly reduced to two days if the primaquine treatment was delayed until the majority of mature gametocytes were circulating in the bloodstream (day 7 or later). On the other hand, delaying primaquine administration until day 10 or later results in an increased duration of infectiousness as mature gametocytes circulate for several days (Figure 5B).

The optimal timing of primaquine treatment showed a substantial effect in reducing malaria transmission in the area. However, the added value of the follow-up primaquine treatment was strongly correlated with the proportion of patients treated with anti-malarial drugs (P). The follow-up primaquine treatment showed a greater effect when a large fraction of clinical malaria was detected and treated. In an area with an effective detection and treatment control programme (P = 99%), the basic reproductive rate (R_c) could be reduced up to 35 times by implementing the optimally timed primaquine treatment programme, compared with the initial basic reproductive rate (R₀). More importantly, the proportional reduction in transmission that was achieved through an 8-day follow-up with primaquine in 99% of the patients was approximately equal to the proportional reductions achieved from the initial treatment. In addition, the follow-up primaquine treatment programme appeared to be effective at reducing transmission only when the coverage of the follow-up primaquine treatment was above 90% (Figure 6).

The transmission model was re-simulated by replacing the duration of infectivity (σ^-1) from 10 days to 2 days, according to the gametocyte model. The model indicated that if all individuals received primaquine at day 8 after initial acute attack, the malaria incidence in the area would be reduced substantially (Figure 7). However, changing the primaquine administration regimen alone may not be enough to eliminate P. falciparum malaria in the area; the incidence of malaria still persisted during the wet season, largely because of imported malaria. By including an intervention focused on the effective detection and treatment of imported asymptomatic infections, the incidence of malaria in the area could reach zero within three years after the combination programme was introduced (Figure 7).