A model for malaria treatment evaluation in the presence of multiple species
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A model for malaria treatment evaluation in the
presence of multiple species
J.N. Walkera , R.I. Hicksona,b,c , E. Changa , P. Ngord,e , S. Sovannarothd , J.A.
Simpsonf , D.J. Pricef,g , J.M. McCawa,f , R.N. Pricee,h,i , J.A. Flegga , A.
arXiv:2205.01838v1 [q-bio.PE] 4 May 2022
Devineh,f
a
School of Mathematics and Statistics, University of Melbourne, Australia
b
Australian Institute of Tropical Health and Medicine, and College of Public Health,
Medical & Veterinary Sciences, James Cook University, Australia
c
Health and Biosecurity, CSIRO, Australia
d
Cambodian National Center for Parasitology, Entomology and Malaria Control,
Cambodia
e
Mahidol-Oxford Tropical Medicine Research Unit, Faculty of Tropical Medicine, Mahidol
University, Thailand
f
Centre for Epidemiology and Biostatistics, Melbourne School of Population and Global
Health, University of Melbourne, Australia
g
Department of Infectious Diseases, University of Melbourne, at the Peter Doherty
Institute for Infection and Immunity, Australia
h
Division of Global and Tropical Health, Menzies School of Health Research and Charles
Darwin University, Australia
i
Centre for Tropical Medicine and Global Health, Nuffield Department of Clinical
Medicine, University of Oxford, UK
Abstract
Plasmodium (P.) falciparum and P. vivax are the two most common causes
of malaria. While the majority of deaths and severe morbidity are due to P.
falciparum, P. vivax poses a greater challenge to eliminating malaria outside
of Africa due to its ability to form latent liver stage parasites (hypnozoites),
which can cause relapsing episodes within an individual patient. In areas
where P. falciparum and P. vivax are co-endemic, individuals can carry par-
asites of both species simultaneously. These mixed infections complicate
dynamics in several ways; treatment of mixed infections will simultaneously
affect both species, P. falciparum can mask the detection of P. vivax , and
it has been hypothesised that clearing P. falciparum may trigger a relapse of
dormant P. vivax. When mixed infections are treated for only blood-stage
parasites, patients are at risk of relapse infections due to P. vivax hypno-
Preprint submitted to arXiv May 5, 2022zoites.
We present a stochastic mathematical model that captures interactions
between P. falciparum and P. vivax, and incorporates both standard schizon-
tocidal treatment (which targets blood-stage parasites) and radical treatment
(which additionally targets liver-stage parasites). We apply this model to as-
sess the implications of different treatment coverage of radical cure for mixed
and P. vivax infections and a so-called “unified radical cure” treatment strat-
egy for P. falciparum, P. vivax and mixed infections. We find that a uni-
fied radical cure strategy, with glucose-6-phosphate dehydrogenase (G6PD)
screening, leads to a substantially lower incidence of malaria cases and deaths
overall. We perform a one-way sensitivity analysis to highlight important
model parameters.
Keywords: Malaria, Unified treatment, Plasmodium falciparum,
Plasmodium vivax
∗
Corresponding author: james.walker2@unimelb.edu.au
1. Introduction
Almost half of the world’s population is at risk of malaria, with ongoing
transmission reported in 79 countries [59]. In 2020 there were an estimated
241 million cases and 627,000 malaria deaths, with funding for control and
elimination estimated at US$3.3 billion [59]. Over the last decade substan-
tial gains have been made in reducing the burden of disease. In 2014 the
leaders of 18 malaria endemic countries in the Asia Pacific committed to
eliminating the disease in the region by 2030 [35]. In this region the two
parasite species that cause the greatest burden of malaria are Plasmod-
ium falciparum (P. falciparum) and Plasmodium vivax (P. vivax ). Most
research and intervention efforts have been focussed on P. falciparum, the
most pathogenic parasite species. However, outside of Africa P. vivax is
now the predominant cause of malaria, almost invariably co-existing with
P. falciparum. While malaria control measures impact both species, these
are often suboptimal for P. vivax due to its ability to form dormant liver par-
asites (hypnozoites) that can reactivate, causing future infections (relapses).
P. vivax also forms sexual stages early in infection and is able to transmit to
the mosquito vector before the patient seeks treatment. Its generally lower
parasite density makes it more difficult to detect.Primaquine is the only widely-used drug available that clears hypnozoites.
The combination of primaquine and schoizontocidal drugs, such as chloro-
quine (CQ) or artemisinin-based combination therapies (ACT), is known as
radical cure. Primaquine can cause drug induced haemolysis particularly
in individuals with G6PD deficiency, an inherited enzymopathy present in
up to 30% of malaria endemic populations. For this reason the WHO cur-
rently recommends screening for G6PD deficiency prior to administration of
primaquine to reduce the risk of severe primaquine-induced haemolysis [58].
The effectiveness of primaquine is limited by healthcare providers reluctance
to prescribe it, and patient adherence to complete a course of treatment. New
point-of-care tools for diagnosing G6PD deficiency have recently come onto
the market but have yet to be introduced widely into clinical practice. The
challenges in safely and consistently treating P. vivax with radical cure has
resulted in its relative rise as a proportion of malaria cases [9]. One modelling
study indicated that over 80% of P. vivax cases in the Greater Mekong Sub-
region (GMS) arise from relapses [2], highlighting the importance of radical
cure to reduce the burden of disease [21].
Successful malaria elimination campaigns in co-endemic settings will re-
quire widespread use of safe and effective radical cure to patients present-
ing with P. vivax as well as the hidden reservoirs of infection. Failure
to consider P. vivax malaria as a target for elimination may compromise
P. falciparum elimination campaigns because communities that continue to
experience cases of malaria, even if due to another type of parasite, may show
a reduced willingness to participate in future interventions designed to pre-
vent re-introduction of P. falciparum. In an effort to accelerate P. falciparum
malaria elimination in the GMS, the use of mass drug administration (MDA)
or mass screening and treatment is now being investigated [28]. These inter-
ventions do not include radical cure, but all stages (blood and liver) of all
parasite species will need to be eradicated to eliminate malaria.
Cambodia aims to eliminate all species of malaria by 2025. In 2019, mixed
infections of both P. falciparum and P. vivax accounted for 16.6% of malaria
infections [11]. Mixed infections can change treatment outcomes in several
ways: P. falciparum malaria can mask a P. vivax malaria co-infection [1, 5]
and an episode of P. falciparum malaria is associated with a greater risk of
P. vivax infection in the subsequent weeks after treatment [15, 26, 30]. It has
been hypothesized that the fever and haemolysis caused by acute falciparum
malaria may trigger reactivation of P. vivax hypnozoites and subsequent re-
lapse. Whereas current radical cure policy is reserved for patients presenting
3with P. vivax malaria, a unified treatment policy, in which patients presenting
with either P. vivax or P. falciparum are prescribed radical cure has poten-
tial to reduce recurrent episodes of malaria and target hidden resevoirs of
infection [40].
While a range of mathematical models for malaria have been proposed,
implemented, analysed and used to support policy decisions over the last 100
years—as reviewed recently [34, 51]—few models have included the parasite
dynamics of both P. falciparum and P. vivax [3, 41, 42, 48]. To our knowl-
edge, only one of these modelling investigations explored interactions between
species [48]. Silal and co-authors developed a deterministic metapopulation
model of P. falciparum and P. vivax,and incorporated key interactions be-
tween P. falciparum and P. vivax, including “treatment entanglement” (any
treatment affecting the other parasite species), “triggering” (P. vivax hypno-
zoite activation following an episode of P. falciparum), and “masking” (where
non-P. falciparum rapid diagnostic test (RDT) results are either missed or
falsely attributed to be P. falciparum). The remaining models [3, 41, 42]
effectively consider the dynamics of the two species to be completely inde-
pendent.
We present the first stochastic agent-based model for the transmission of
both P. falciparum and P. vivax , which addresses the dynamics of mixed
infections, parasite interactions and antimalarial treatments. Our model con-
siders humans as discrete agents which transition between compartments ac-
cording to a continuous-time Markov chain (CTMC) model. The CTMC is
coupled with a set of ordinary differential equationss (ODEs) that govern
the mosquito population, where the transmission rate both from mosquitoes
to humans and humans to mosquitoes are held constant over small time-
steps. The stochastic feature is important as infectious disease models are
known to be highly stochastic as they approach elimination, and when a
population is divided into many compartments relatively small numbers are
expected in some. This model has 6 compartments for P. falciparum and
7 for P. vivax , representing a model with lower complexity than the other
multi-species model with interactions, which has 14 and 17 compartments,
respectively [48]. The reduction in model complexity partially comes from
removing age-stratification from the model. One of the main effects of age is
in the acquisition of immunity to prevent developing clinical malaria, which
is captured in our model through lower probabilities of clinical malaria upon
reinfection or relapse (i.e., in the following, the probability of clinical malaria
upon reinfection or relapse is 0.5, compared to 0.95 for a naive P. falciparum
4infection). Even with the reduction in model complexity, we note that the
model requires many input parameters, not all of which are well defined in
literature. Hence, we perform a univariate sensitivity analysis to understand
the impact of each parameter with respect to the model outputs, malaria
cases and deaths.
As an example, we consider scenarios with Cambodia-like P. falciparum
and P. vivax prevalence and parameters, since both P. falciparum and P.
vivax are present and the Anopheles (An.) populations are able to trans-
mit both. This model is applied to assess the effect of standard blood-stage
treatment, differing coverage of radical cure prescription, and a unified treat-
ment policy in which radical cure is prescribed to patients presenting with
P. vivax , P. falciparum and mixed infections. For each of these treatment
scenarios we also consider a MDA intervention, where a proportion of the
population are prescribed standard blood-stage treatment, which allows for
asymptomatic infections to be treated.
2. Methods
2.1. Transmission model
To capture the transmission dynamics of both P. falciparum and P. vivax,
we use a stochastic agent-based approach for the human population coupled
with a deterministic system of ODEs for the mosquito population. Each
human agent has their status with respect to both P. falciparum and P. vi-
vax tracked over time, which allows mixed infections to be captured. The
agent-based model is implemented by holding rates constant over discrete
time-steps for computational efficiency and for ease of coupling to the ODEs
that govern the mosquito population.
Each individual’s state is bivariate to specify their state with respect to
P. falciparum and one for P. vivax . For each Plasmodium species, humans
are regarded as being susceptible (S), infectious with clinical symptoms (I),
infectious but asymptomatic (A), recovered with no hypnozoites (R), re-
covered with hypnozoites (L for latent: not applicable for P. falciparum),
undergoing standard blood-stage treatment with no radical cure (T ), or un-
dergoing treatment with radical cure (G). Radical cure is defined as low-
dose primaquine (3.5mg/kg total) administered over 14 days. A simplified
schematic of the human transitions are depicted in Figure 1. Given the large
number of connections between states required to describe the transmission
5and treatment dynamics, the model schematic uses a single line between
connectors where multiple exist and does not depict interactions between
the species. The full list of possible transition rates and stoichiometries is
provided in the Supplementary Table 2.
The dynamics of a human individual infected with type x malaria are
briefly described here for x = f (P. falciparum) and v (P. vivax ). Indi-
viduals susceptible (S) to type x malaria are infected at rate λx . Upon
infection they develop clinical symptoms (I) with probability pc,x or are oth-
erwise asymptomatic (A). Individuals are symptomatic for a mean duration
of 1/sigmax , at which point they either become asymptomatic (A) or die
without treatment with probability pI,x . The individuals with clinical symp-
toms may be treated at rate cx τx , where cx is the probability that they are
able to access healthcare and τx is the rate at which medical attention is
sought if it is readily available. Individuals with asymptomatic malaria will
clear all blood-stage parasites at rate αx and, for P. vivax , will be left with
hypnozoites with probability ph,v . When an individual with vivax malaria
seeks treatment they are prescribed radical cure (G) with probability pN,x ,
otherwise they receive standard treatment (T ).
Any infectious individual may additionally be treated at rate ηx (t) via an
intervention program (such as MDA); the form of ηx (t) will be discussed in
Section 2.3. When an individual is treated this way they are prescribed rad-
ical cure with probability pM,x , otherwise they receive standard treatment.
An individual undergoes treatment for an average of 1/ψ days (14-days for
primaquine) at which point they may: die with probability pG,x , remain with
asymptomatic blood-stage malaria with probability pT f P , if x = v they are
left with latent hypnozoites (L) with probability pP,v , otherwise they recover
(R). Similarly, an individual ends standard treatment after an average of
1/ρx days (3-days for ACT) at which point they may: die with probability
pT , remain with asymptomatic blood-stage malaria with probability pT f A , if
x = v they are left with latent hypnozoites with probability pA,v , otherwise
they recover. Latent stage P. vivax infected individuals experience a relapse
at rate νv or they are reinfected at rate λv rv (where rx represents a possible
reduction in susceptibility due to anti-parasite immunity) [20]. Upon relapse
or reinfection from L the individual gets clinical malaria with probability
pL,v (where pL,v < pc,v ). Recovered individuals (R) are reinfected with rate
λx rx at which point they become a clinical case with probability pR,x (where
pR,v < pc,v ). In addition to the possibility of relapse or reinfection, recovered
6and latent individuals lose immunity and hypnozoites at rates ωx and κv ,
respectively.
For individuals with mixed infections, there are several transitions in the
model where the individual’s state with respect to one species of malaria is
not independent of the individual’s state with respect to the other; we refer
to these dependencies as species “interactions”. When an individual with a
mixed infection is treated, they move from states in {I, A, L} to a state in
{T, G} for both species (depending on the treatment); this is referred to as
“treatment entanglement”. Similarly, when a patient stops treatment with
respect to one malaria species, they are moved to one of the post-treatment
states with respect to the other. Death with respect to one species will cause
a transition to death with respect to the other. The model allows treatment
efficacies to vary for mixed infections, however, in this work we have assumed
antimalarial efficacy against each species to be equivalent to the efficacy
against mono-infections. We model P. vivax relapses triggered by the recov-
ery of P. falciparum (“triggering”) by setting the relapse rate for a person
that is recovered (R) from P. falciparum but has latent stage (L) P. vivax to
ν̂f v = zf νv , where zf > 1. The model also allows blood-stage P. vivax to be
masked by blood-stage P. falciparum (“masking”) by treating a mixed in-
fection as though it were a P. falciparum infection only with probability hv .
Explicitly, for an individual with P. falciparum and P. vivax both in states I
or A, the probability of receiving radical cure is pN,f v = hv pN,f +(1 − hv )pN,v .
2.2. Transmission intensity and vector species
The dynamics of the mosquito population are governed by a system of
ODEs (presented in the Supplementary Section ). The mosquitoes follow
standard SEI dynamics with the addition of a seasonally varying death rate
and the ability for mosquitoes to carry and spread mixed infection in a single
bite (known as simultaneous inoculation). Asymptomatic individuals tend
to have a lower peripheral parasitaemia and therefore were assumed to be
less infectious to mosquitoes than symptomatic individuals with a relative
infectiousness of 0.1.
We consider model parameters that would be representative of a Cambodia-
like context, where both P. falciparum and P. vivax circulate, and the
mosquito species present (An. dirus, An. minimus, An. maculatus, and An.
7Figure 1: Simplified schematic of the human transmission model
for a single parasite species. The model compartments are S (sus-
ceptible), I (symptomatic infectious), A (asymptomatic infectious),
T (undergoing standard treatment), G (undergoing radical cure), R
(recovered and partially-immune) and L (latent stage hypnozoites for
P. vivax only). Solid lines represent rates, the dashed lines probabil-
ities, and the circles designate where a rate is split by probabilities.
The probability parameters are not explicitly shown in the figure as,
in many cases, the probability of each outcome depends on the cur-
rent state (for example, the probability of symptoms upon infection
is lower for recovered individuals than susceptible individuals).
8barbirostris) are able to transmit both parasite species [49] allowing for a
simplification of the mosquito dynamics.
2.3. Treatment Scenarios
We simulate three treatment scenarios: current practice, accelerated rad-
ical cure, and unified radical cure. Note that we assume in each scenario
that when a person tests positive for malaria the species is always identified
correctly for mono-infections, since specialised RDTs have been shown to
have high sensitivity and specificity, particularly for P. falciparum (see, for
example, [1]).
1. Current practice: Under this scenario, P. falciparum and most P. vivax
were treated with standard blood-stage treatment with only 16% of
P. vivax prescribed radical cure. The low coverage of radical cure for
was chosen to match the rates reported from a study in Cambodia
where radical cure was prescribed conservatively to 16% of detected
P. vivax cases [27]. That is, the probability that an individual receives
radical cure when being treated for malaria x, is
0,
for x = f
pN,x = 0.16, for x = v
(1 − hv )0.16, for x = f v,
where hv is the probability that P. vivax is masked by P. falciparum
when the individual is tested.
2. Accelerated radical cure: Under this scenario, any eligible G6PD
normal person diagnosed with P. vivax is prescribed radical cure with a
low-dose 14 day course of primaquine (total dose 3.5 mg/kg) alongside
a 3 day course of blood-stage treatment. Anyone who is >6 months
old and is not pregnant or lactating is considered eligible for radical
cure. We assume that the G6PD RDTs have a sensitivity of 94% and
a specificity of 91% [29], 6% of the population have G6PD enzyme
activity7). The probability of receiving radical cure under this scenario is
0,
for x = f
pN,x = 0.82, for x = v
(1 − hv )0.82, for x = f v.
3. Unified radical cure: Under this scenario, radical cure is prescribed
to any eligible person that is detected with malaria parasites, with eligi-
bility as defined and calculated in the accelerated radical cure scenario.
The probability of receiving radical cure in this scenario is
0.82, for x = f
pN,x = 0.82, for x = v
0.82, for x = f v.
The three treatment scenarios and the probability of receiving radical cure is
summarised in Table 1 with an assumed probability of masking of hv = 0.5.
Table 1: Treatments and radical cure coverage by species for each
scenario, with an assumed probability of masking of 0.5. Here, radical
cure coverage is defined as the probability of receiving radical cure
given a detected infection. Treatments ACT, CQ, PQ1 and PQ14
denote a 3-day course of artemisinin-based combination therapy, a 3-
day course of chloroquine, a 1-day course of primaquine and a 14-day
course of primaquine, respectively.
Current Practice Accelerated RC Unified RC
Treatments
P. falciparum ACT + PQ1 ACT + PQ1 ACT + PQ14
P. vivax CQ + PQ14 CQ + PQ14 ACT + PQ14
Mixed ACT + PQ14 ACT + PQ14 ACT + PQ14
Radical Cure Coverage (given detected infection)
P. falciparum 0 0 0.82
P. vivax 0.16 0.82 0.82
Mixed 0.08 0.41 0.82
For each of the three treatment scenarios we also consider the impact
of a mass-drug-administration (MDA) intervention where a proportion of
10the population are given a standard blood-stage treatment irrespective of
infective status, thus allowing a proportion of asymptomatic blood-stage in-
fections to be treated. We assume that a proportion, p, of the population
are prescribed a standard blood-stage treatment over a fixed period of time,
∆t = t2 − t1 , so that the treatment rate of an individual with species x due
to MDA is: (
−ln(1−p)
∆t
, t ∈ (t1 , t2 ),
ηx (t) =
0, otherwise.
We assume that people will not be screened for G6PD status nor prescribed
radical cure during MDA, based on concerns about haemolytic risks out-
weighing the benefits in patients who do not have malaria [38]. That is, the
probability that an individual receives radical cure under MDA, given that
they are treated for malaria type x is pM,x = 0 for all x.
2.4. Implementation
We present the impact of different treatment and intervention strategies
on the number of malaria cases and deaths from 2021 to 2030, which is the
regional target for malaria elimination.
For each treatment and intervention scenario we run 50 model simula-
tions and record the model compartments over time. Given the relatively
short time frame, we ignore background human demographic dynamics in
our model to reduce computational complexity.
For the current practice scenarios we assume that radical cure is pre-
scribed conservatively and does not increase the risk of haemolysis (as a
best case scenario). For the accelerated radical cure and unified radical cure
scenarios we set the probability of death to be slightly increased for pa-
tients administered primaquine, by considering the probability of being a
patient that is not pregnant, lactating or under 6 months old (0.96), having
G6PD deficiency (0.06), having a false negative G6PD test (0.06) [29], hav-
ing haemolysis given G6PD status (0.109) [39], being unable to be treated
appropriately for haemolysis (0.1) and dying from the haemolysis without
treatment (0.1) [18]. Similarly, we increase the probability of radical cure
failure to clear hypnozoites, to account for the chance that a course of pri-
maquine cannot be completed due to haemolysis (these details are given in
11Supplementary Section 7).
For the MDA scenario we assume that half of the population receive stan-
dard blood-stage treatment over a 30 day period. We let the MDA roll out
occur twice yearly, before and after the yearly peak.
Initial conditions are set to be similar to eastern Cambodia, because the
prevalence of symptomatic and asymptomatic infections in the region is well
understood [45] and levels of immunity in eastern Cambodia were studied in a
2005 sero-survey [16]. We note that some parameter values were based on ex-
pert elicitation, a limited evidence base, and some parameter estimates vary
greatly between studies. As such, the scenarios presented here are indicative
of population dynamics of multi-species infections over time that accommo-
dates interaction between P. falciparum and P. vivax infections and expected
trends in impact for different interventions. Accordingly, the scenarios here
should not be interpreted as forecasts of malaria cases and deaths in Cam-
bodia and other similar malaria-endemic regions over the next 10 years. All
parameters and initial conditions are given in Supplementary Tables 3 and 4.
We performed a sensitivity analysis on the model, where we modified
each model parameter separately and recorded the relative change in model
outputs. To implement the sensitivity analysis, we considered the baseline
value of each parameter (given in Tables 2 and 3) and ran simulations with
the parameter scaled down to 80% and up to 120% while all other parameters
remained fixed. If scaling a probability parameter up to 120% compared to
baseline led to a probability being greater than 1 the value was instead held
at 1. Similarly, the relative susceptibility of partially-immune individuals
compared to susceptible individuals was not scaled up, so as not to exceed 1.
For each set of parameter values, 50 repeats of the simulation were run and
various outputs were recorded, including: the total number of P. falciparum
infections, P. vivax infections, symptomatic P. falciparum infections, symp-
tomatic P. vivax infections, P. vivax relapses, deaths, standard treatments
administered and radical cure treatments administered.
123. Results
3.1. Scenario modelling
Figure 2 gives the total number of infectious individuals in the popula-
tion over time (the median, minimum and maximum of the 50 simulations)
and figures presenting all model compartments through time are presented
in Supplementary Figures 6 and 7. The different treatment strategies had
little effect on the prevalence of P. falciparum, but were associated with a
slight decrease in the unified treatment scenario. For P. vivax we find that
increasing the coverage of radical cure has a large impact on the prevalence
of P. vivax , which appears to be approaching elimination. The MDA in-
tervention greatly reduced the prevalence of P. falciparum but had less of
an effect on P. vivax . No scenarios led to elimination of malaria over the
ten year period, although elimination may have been achieved over a longer
time-frame.
Figure 3 shows boxplots of the cumulative number of infections and deaths
by species. The unified treatment strategy with G6PD testing of all individ-
uals resulted in fewer infections and fewer deaths overall, despite a slight
increase in the risk of haemolysis from radical cure.
3.2. Sensitivity analysis
In Figure 4, the results of the sensitivity analysis are presented in terms
of the ten most influential parameters on cumulative symptomatic infections.
Sensitivity analyses with respect to all parameters and other outcomes are
given in Supplementary Figure 8. These figures present the mean, minimum
and maximum relative outcome compared to baseline over the 50 simulations,
for each parameter set and orders them based on their relative sensitivity (in
terms of absolute difference between the 80% and 120% scenarios).
P. falciparum and P. vivax symptomatic infections were most sensitive
to many of the vector related parameters, including: the bite rate (b), the
death rate of mosquitoes (δ0 ), the probability of transmission given an infec-
tious bite (from human to mosquitoes and vice versa, M,x and H,x ) and the
rate at which exposed mosquitoes become infectious (γx ). These parameters
are well known to be sensitive for mosquito-spread infectious diseases.
13Current Practice Accelerated RC Unified RC
P. falciparum P. vivax
20,000
without MDA
10,000
Prevalence
0
20,000
MDA
10,000
0
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
Year
Figure 2: Clinical infections over 10 years for P. falciparum (left
panels) and P. vivax (right panels) with Clinical treatment only (top
row), MDA (bottom row).
14Current Practice Accelerated RC Unified RC
P. falciparum P. vivax
600,000
Cumulative Infections
400,000
200,000
4,000
Cumulative Deaths
3,000
2,000
1,000
0
without MDA MDA without MDA MDA
Treatment scenario
Figure 3: Total number of clinical infections and deaths over a 10
year period.
15Aside from the important vector related parameters, we identified several
important human related parameters, including: the relative infectiousness of
asymptomatic carriers (ζA,x ), the relative susceptibility of partially-immune
individuals (r), the rate at which asymptomatic infections are cleared (αx ),
the rate of treatment seeking (τx ) and accessibility of treatment (c). The
parameters ζA,x and αx determine the expected number of secondary infec-
tions generated by asymptomatic individuals. The parameter r is related
to anti-parasite immunity, it represents a possible lower rate of infection in
recovered individuals. The parameters τx and c determine the rate at which
symptomatic cases get treated and the probability that they are treated at all.
In addition to the parameters that were influential on symptomatic in-
fections of both species, P. falciparum symptomatic infections were sensitive
to the probability that partially-immune individuals become symptomatic
upon reinfection (pR,x ) and P. vivax symptomatic infections were sensitive
to the probability that hypnozoites are cleared when blood-stage parasites
are cleared without treatment (ph ). This emphasises the role of relapses in
generating continuing P. vivax malaria burden. Other model outcomes were
broadly sensitive to the same model parameters without any additions or
omissions.
4. Discussion
Capturing dynamics of multiple malaria species concurrently is policy-
relevant, but has drawn little attention to-date. We developed a model of
sufficient complexity to capture these dynamics, and showed how it can be
used to inform health policy. Our model incorporates the dynamics of both
P. falciparum and P. vivax in a way that captures masking, treatment en-
tanglement and triggering. This is the second multi-species model which
meaningfully captures dependencies between P. falciparum and P. vivax [48]
and is the only stochastic, agent-based model to do so. The stochasticity
makes it particularly well suited to model P. falciparum and P. vivax in low
transmission settings, small populations, or as malaria is approaching elimi-
nation. The model also has fewer compartments, and fewer parameters, than
the only other model with similar features [48].
Our scenario analysis explored the effect of different coverage rates of
radical cure treatment, assuming that individuals are tested for G6PD prior
16Parameter value relative to baseline: 0.8 1.2
b
delta0
epsilonM pf
epsilonH pf
Parameter
zetaA pf
r pf
alpha pf
gamma pf
tau pf
c
0 1 2 3 4
Parameter
Cumulative value relative
symptomatic to baseline:
falciparum infections relative
0.8 to 1.2
baseline
b
delta0
epsilonM pv
epsilonH pv
Parameter
alpha pv
zetaA pv
ph
gamma pv
r pv
tau pv
0 1 2 3 4
Cumulative symptomatic vivax infections relative to baseline
Figure 4: Sensitivities of P. falciparum (top) and P. vivax (bottom)
cumulative symptomatic infections with respect to varying model pa-
rameters. These are presented in terms of the mean relative outcome,
compared to baseline, when each parameter is scaled by 0.8 and 1.2.
Blue and red dots represent the mean outcomes given a parameter
scaling of 0.8 and 1.2, respectively. Error bars represent the mini-
mum and maximum relative outcome, compared to baseline. Each
minimum, mean and maximum calculated
17 from 50 simulations.to treatment. The scenario analysis showed that a unified radical cure can
reduce the prevalence of malaria cases and deaths overall, even when ac-
counting for the risk of death due to haemolysis. The radical cure is effective
because it both blocks transmission and kills dormant hypnozoites. A uni-
fied radical cure strategy avoids issues associated with masking when admin-
istering targeted treatment, allows for a consistent protocol and messaging
around malaria treatment, and does not require the species of malaria to be
determined prior to treatment. Modelling a MDA allowed us to assess the
additional impact achieved by treating asymptomatic infections as a means
to reduce malaria burden. We found that MDA is an effective way to re-
duce prevalence, but it will not necessarily lead to elimination if coverage
is too low. This is in line with a report from WHO, based on a systematic
review of 270 literature reports, which states that for MDA to be effective,
at least 80% of the population should be treated [57], noting that this may
be naturally limited by compliance. We considered only MDA with stan-
dard blood-stage cure here, due to safety concerns about treating individuals
with G6PD deficiency with radical cure. As a result, we found that MDA
decreased P. falciparum prevalence but had little effect on P. vivax preva-
lence. Targeted interventions may allow radical cure to be administered en
masse (such as focal screen and treat, or mass screen and treat) [57]. Our
modelling framework easily allows these other kinds of interventions to be
incorporated through the time-varying treatment function, ηx (t).
The sensitivity analysis shows that, although the model has many param-
eters, the outputs are largely sensitive to relatively few parameters. The most
sensitive parameters for both species were those related to vector-dynamics,
the bite rates, transmission probabilities, mosquito death rate and the in-
fectious period of mosquitoes, which are well known to be influential for
mosquito-spread diseases [12]. Additionally, the relative infectiousness of
asymptomatic individuals (ζA,x ), the rate at which asymptomatic infections
are cleared (αx ), the relative susceptibility of partially-immune individuals
(r), the rate of treatment seeking (τ ) and accessibility of treatment (c) were
all found to be influential. Note that τ and c determine the proportion of
symptomatic infections that go untreated and become asymptomatic. Fur-
ther, ζA,x and αx determine the expected number of secondary infections
generated by asymptomatic individuals. These sensitivities highlight how
important asymptomatic individuals can be in driving malaria burden and
the need for interventions that target asymptomatic infections, such as MDA
18(explored in this work) or better diagnostics that can detect infections with
low level parasitaemia. The number of P. vivax infections was sensitive to
the probability of asymptomatic carriers naturally clearing hypnozoites, re-
inforcing the notion that relapses contribute significantly to malaria burden,
as has been shown empirically in an analysis of 68,361 patients [19].
We performed a one-dimensional sensitivity analysis, therefore, the re-
sults should only be interpreted as the output sensitivity with respect to
each parameter in isolation, and not interpreted as a full quantification of
model output uncertainty. A full probabilistic sensitivity analysis is appro-
priate for assessing output uncertainty, particularly if using the model to
inform public health policy.
The simulations in our scenario analyses show behaviour comparable to
Cambodia with parameters consistent with literature and expert elicitation.
Many of the model parameters are location-specific such as the bite rate,
relative infectiousness of asymptomatic carriers, probability of death from
radical cure and the initial model state. In the future, we aim to provide
a statistical framework for fitting this model, so that it may be applied in
contexts where parameters may differ. The complexity of the multi-species
model poses a challenge to jointly fitting all model parameters because of
the high dimension of the parameter space and the run time, which was on
the scale of minutes. Optimised approximate Bayesian inference methods
such as Bayesian Optimization for Likelihood-Free Inference (BOFLI) and
Likelihood-Free Inference by Ratio Estimation (LFIRE) may provide solu-
tions to both of these challenges [25, 52]. If the run time of the stochastic
model becomes prohibitive for inference, as may be the case when applied to
larger populations with high prevalence (where capturing small fluctuations
in low numbers is less important), a deterministic or hybrid model equivalent
could be applied instead.
This modelling framework provides the basis for future malaria modelling
studies to evaluate the impact of interventions in malaria endemic regions
where both P. falciparum and P. vivax are prevalent. In particular, param-
eters in the model can be adjusted to consider other treatments, such as
single-dose tafenoquine, high-dose 7-day primaquine, and triple ACTs. The
multi-species malaria model was developed in a way that enables economic
analyses through the separation of different treatments and outcomes for in-
19dividuals given their treatment. In the future, costs and quality of life metrics
can be evaluated alongside the impact on cases and deaths. For example,
this model could be used to identify under which circumstances a unified
treatment for malaria would be cost-effective. Lastly, the modelling frame-
work could be expanded to include other species of malaria, such as zoonotic
P. knowlesi.
4.1. Role of the funding source
ACREME funded the salary of RIH, and contributed to the costs of data
cleaning and organisation by PN and the CNM.
Acknowledgements
This work is supported in part by the Australian Centre for Research Ex-
cellence in Malaria Elimination (ACREME), funded by the NHMRC (1134989).
J.A. Simpson is funded by an Australian National Health and Medical Re-
search Council of Australia (NHMRC) Investigator Grant (1196068). J.M.
McCaw’s research is supported by the ARC (DP170103076, DP210101920)
and ACREME. J.A. Flegg’s research is supported by the ARC (DP200100747,
FT210100034). A. Devine’s research is supported by DFAT.
20A model for malaria treatment evaluation in the presence
of multiple species: Supplementary information
This Supplementary document provides further details on the mathematical
model and model parameters.
5. Human population dynamics
The model considers the human population in a stochastic, agent-based,
framework which is coupled to a system of ordinary differential equations
(ODEs) that describe the mosquito population. The human dynamics are
already described in the main document, so here we supplement that descrip-
tion with the full table of stoichiometries for the human model dynamics in
Table 2 with model parameters defined in Table 4.
The implementation of treatment entanglement in this paper assumes an-
timalarial treatment efficacy is the same against individual species in mixed
infections as it is for mono-infections, but it does introduce new transi-
tions; for example, an individual with mixed malaria may recover from both
P. falciparum and P. vivax simultaneously. These modified rates and tran-
sitions are described in Table 3.
6. Mosquito population dynamics
The adult female mosquito population is based on standard SEI-type dy-
namics with births, deaths, seasonality and mosquitoes that can carry mixed
malaria.
Let V , Wx and Yx denote the number of mosquitoes that are susceptible
to all species, exposed to species x and susceptible to other species, and,
infectious with species x and susceptible to other species, respectively. Let
ZYf ,Wv and ZWf ,Yv denote the number of mosquitoes that are infectious with
P. falciparum and exposed to P. vivax and vice versa. Let the force of in-
fection from species x on the mosquito population be denoted by λM,x ; the
equations for λM,x are given by Equations (13)-(14). The mosquito popula-
tion is modelled by the following system of ODEs:
21Table 2: Table of transitions and stoichiometry for the human agent
based model, for species x. For P. falciparum there is no latent
compartment. The total population is N = Sx + Ix + Ax + Rx +
Tx +Lx +Gx , and λH,x = bM,x (Yx +ZYx ,Wx̄ +Yf v )/N . Rates which
are affected by triggering and masking interactions in the model are
highlighted in red and blue respectively (the interactions are defined
in Table 3). Parameters are defined in Table 4
From→to Rate Description
Sx → Ix pc,x λH,x Clinical infection of naive individual
Sx → Ax (1 − pc,x )λH,x Asymptomatic infection of naive in-
dividual
Ix →death pIx ,x σx Death due to malaria
Ix → Ax (1 − pIx ,x )σx Loss of clinical symptoms
Ix → Tx pN,x cx τx + pM,x ηx (t) Standard treatment
Ix → Gx (1 − pN,x )cx τx + (1 − pM,x )ηx (t) Treatment including radical cure
Ax → Lx ph,x αx Recovered with hypnozoites
Ax → Rx (1 − ph,x )αx Recovered with no hypnozoites
Ax → Tx pM,x ηx (t) Standard treatment via MDA or
FSAT
Ax → Gx (1 − pM,x )ηx (t) Radical cure treatment via MDA or
FSAT
Rx → Ix pRx ,x rx λx Clinical infection of semi-immune
Rx → Ax (1 − pRx ,x )rx λx Asymptomatic infection of semi-
immune
Rx → Sx ωx Waning immunity
Lx → Ix pRx ,x rx λx Clinical infection of hypnozoite car-
rier
Lx → Ax (1 − pRx ,x )rx λx Asymptomatic infection of hypno-
zoite carrier
Lx → Ix pLx ,x νx Relapse to clinical infection
Lx → Ax (1 − pLx ,x )νx Relapse to asymptomatic infection
Lx → Sx κx Hypnozoite “death”
Tx →death pTx ,x ρx Standard treatment outcome is
death
Tx → Ax pT f A (1 − pTx ,x )ρx Treatment completed but fails to
fully clear blood-stage parasites
Tx → Rx (1 − pAx ,x )(1 − pT f A )(1 − pTx ,x )ρx Treatment succesfully completed
Tx → Lx pAx ,x (1 − pT f A )(1 − pTx ,x )ρx Treatment completed but hypno-
zoites remain
Gx →death pGx ,x ψx Radical cure treatment outcome is
death
Gx → Ax (1 − pGx ,x )pT f P ψx Treatment with radical cure com-
pleted but blood-stage parasites re-
main
Gx → Rx (1 − pP,x )(1 − pT f P )(1 − pGx22
,x )ψx Treatment with radical cure com-
pleted and successful
Gx → Lx pP,x (1 − pT f P )(1 − pGx ,x )ψx Treatment with radical cure com-
pleted but hypnozoites remaindV
= δ0 M − (λM,f + λM,v + λM,f v )V − δ(t)V , (1)
dt
dWf
= λM,f V − (γf + δ(t))Wf , (2)
dt
dWv
= λM,v V − (γv + δ(t))Wv , (3)
dt
dWf v
= λM,f v [V + Wf + Wv ] + λM,f Wv + λM,v Wf − (γf + γv + δ(t))Wf v ,
dt
(4)
dYf
= γf Wf − δ(t)Yf , (5)
dt
dYv
= γv Wv − δ(t)Yv , (6)
dt
dZYf ,Wv
= γf Wf v + (λM,v + λM,f v )Yf − (γv + δ(t))ZYf ,Wv , (7)
dt
dZWf ,Yv
= γv Wf v + (λM,f + λM,f v )Yv − (γf + δ(t))ZWf ,Yv , (8)
dt
dYf v
= γf ZWf ,Yv + γv ZYf ,Wv − δ(t)Yf v , (9)
dt
where (10)
2π(t − φ)
δ(t) = δ0 1 − ξ cos + π/2 , (11)
365
M = V + Wf + Yf + Wv + Yv + Wf v + ZYf ,Wv + ZWf ,Yv + Yf v , (12)
and parameters are described in Table 4. The mosquito dynamics are de-
picted in Figure 5.
Here we derive the force of infection equations for the mosquito popu-
lation. The force of infection equations are presented in terms of a force of
infection for P. falciparum, for P. vivax and for mixed infections. The mono-
infection terms need to consider successful infection from single species in-
fectious individuals and a partially-successful infection from individuals with
mixed infection, whereas, the mixed force of infection accounts for successful
infection by both species.
First, let Ax be the set of infectious states in the single species model for
species x, that is, Ax = {Ix , Ax , Tx and Gx }, and let Acx be the compliment.
23Figure 5: Schematic of the mosquito portion of the transmission
model, where V are the susceptible mosquitoes, W are those ex-
posed but not yet infectious, Y are those infectious, Z are for
those in combinations of W and Y , and the subscripts are defined
as f=P. falciparum and v=P. vivax. This is also colour coded as
blue=susceptible, red=infectious, orange=latent. The system is fully
described by Equations (1)-(14). Despite appearances, this is a simple
susceptible-exposed-infectious mosquito model, just with all possible
combinations of those for the two parasite species.
24Let x̄ denote the other species of malaria and, for any sets B and C, let
(Bx , Cx̄ ) denote a set of states in the multispecies model. We define f (a)
as a function which takes a state, a = (ax , ax̄ ), and returns the number of
individuals in that state. Lastly, let the probability of transmission of species
x from a human, in state a, be H,ax = ζax H,x .
The force of infection resulting in mosquitoes being infected by P. falci-
parum, P. vivax and mixed malaria are:
b X X
λM,x = H,ax f (a) + H,ax (1 − H,ax̄ )f (a) , (13)
N c
a∈(Ax ,Ax̄ ) a∈(Ax ,Ax̄ )
for x = f or v, and
b X
λM,f v = H,af H,av f (a) . (14)
N
a∈(Af ,Av )
7. Radical cure coverage and outcomes
This section outlines calculations relating to coverage of radical cure in
the accelerated radical cure and unified radical cure scenarios, based on eli-
gibility, G6PD status and RDT accuracy.
Let pinel , pg6pd , psense and pspec be the probabilities that individuals are in-
eligible for treatment (Falciparum human compartments over time Falciparum human compartments over time
600 600
400 400
T
T
200 200
0 0
1000 1000
750 750
G
G
500 500
250 250
0 0
1250
1000 1000
750 750
I
I
500
500
250
250
0
25000 25000
20000 20000
Count
Count
Standard 15000 Standard
15000
A
A
10000 Pv Radical 10000 Pv Radical
5000
5000 Unified Unified
0
0.050 0.050
0.025 0.025
0.000 0.000
L
L
−0.025 −0.025
−0.050 −0.050
20000
40000
15000 30000
S
S
10000 20000
5000 10000
80000
80000
70000
R
R
76000
60000
72000
50000
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
Year Year
Figure 6: The state of humans in the multispecies model over time
for P. falciparum under the regular treatment scenario (left) and the
MDA scenario (right).
The probability a patient that is prescribed radical cure dies is given by
prcdeath = prchaem puntreated pdeath .
For the parameters considered here pnorc , prchaem and prcdeath are 0.18, 0.0005
and 0.000005, respectively.
8. Full model time-series
This section gives figures of the total number of individuals in each com-
partment over time for P. falciparum and P. vivax for the scenarios presented
in the main text (see Figures 6 and 7). Note that the total infections pre-
sented in Figure 2 is the sum of the I and the A compartments given in
Figures 6 and 7.
26Vivax human compartments over time Vivax human compartments over time
600 600
400 400
T
T
200 200
0 0
600 600
400 400
G
G
200 200
0 0
800 800
600 600
400 400
I
I
200 200
0 0
7500 7500
Count
Count
Standard Standard
5000 5000
A
A
Pv Radical 2500 Pv Radical
2500
0 Unified 0 Unified
4000 4000
3000 3000
2000
L
L
2000
1000 1000
0 0
70000
70000
60000
60000
S
S
50000
50000
40000 40000
50000 50000
45000
40000 40000
R
R
35000
30000
30000
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10
Year Year
Figure 7: The state of humans in the multispecies model over time
for P. vivax under the regular treatment scenario (left) and the MDA
scenario (right).
27Table 3: Table delineating how the interaction parameters affect
transmission. Note: RBC competition affects concurrent infections,
Cross immunity affects sequential infections. All interaction parame-
ters are dimensionless.
Symbol Explanation Modifies Transitions affected Value(s) Reference(s) Notes
Treatment Entanglement
- Simultaneous treat- New flows. Simultaneous [43, 48, 55] Whenever an individ-
ment for mixed treatment: ual with a mixed in-
infections. (If , Iv ) → (Tf , Tv ), fection would enter a
(If , Av ) → (Tf , Tv ), state with treatment,
(Af , Iv ) → (Tf , Tv ), they will instead be
(Af , Av ) → treated with respect
(Tf , Tv ), to both species.
28
(If , Lv ) → (Tf , Tv ),
(Af , Lv ) →
(Tf , Tv ).
hiddenlatexjutsu.
The flows are
changed similarly
for radical cure.
Continued on next pageTable 3 – Continued from previous page
Symbol Explanation Modifies Transitions affected Value(s) Reference(s) Notes
- Simultaneous end of New flows. (Tf , Tv ) → - The efficacy of treat-
treatment for mixed (Af , Rv ), ments for each strain
infections. (Tf , Tv ) → are assumed equal to
(Rf , Av ), those of monoinfec-
(Tf , Tv ) → tions. That is, the
(Af , Av ), probability of each
(Tf , Tv ) → treatment outcome is
(Af , Lv ), equal to the product
(Tf , Tv ) → of the two transi-
(Rf , Rv ), tion probabilities for
(Tf , Tv ) → monoinfections.
29
(Rf , Lv ).
hiddenlatexjutsu.
The flows are
changed similarly
for radical cure.
Continued on next pageTable 3 – Continued from previous page
Symbol Explanation Modifies Transitions affected Value(s) Reference(s) Notes
- Infection during treat- New flows. (Tf , Sv ) → (Tf , Tv ), - If, while an individual
ment. (Sf , Tv ) → (Tf , Tv ), with monoinfection is
(Tf , Rv ) → undergoing treatment,
(Tf , Tv ), they are infected by
(Rf , Tv ) → the other species of
(Tf , Tv ). malaria, they will be
hiddenlatexjutsu. treated for both.
The flows are
changed similarly
for radical cure.
Masking
30
Continued on next pageTable 3 – Continued from previous page
Symbol Explanation Modifies Transitions affected Value(s) Reference(s) Notes
hv Probability that The prob- (If , Iv ) → (Tf , Tv ), 0.5 (0.2, [1, 5, 48] This is the proba-
masking occurs ability of (If , Av ) → (Tf , Tv ), 0.8) bility that a mixed
receiving (Af , Iv ) → (Tf , Tv ), infection is treated as
standard (Af , Av ) → though it were a P.
treatment, (Tf , Tv ). falciparum infection,
given treated, hiddenlatexjutsu. either through only
pN,f v = P. falciparum being
hv pN,f + (1 − detected, or health
hv )pN,v , workers not adher-
and ing to radical cure
pM,f v = guidelines. The tran-
31
hv pM,f + (1 − sition probabilities
hv )pM,v . for radical cure,
given treated, are
also modified to stay
complimentary to
the probability of
standard cure, given
treated.
Triggering
Continued on next pageTable 3 – Continued from previous page
Symbol Explanation Modifies Transitions affected Value(s) Reference(s) Notes
zf Increase in P. vi- The rate of (Rf , Lv ) → 3.5 (2.0, [15, 26, 30, Increased rate of P.
vax relapse rate due ν̂v = zf νv (Rf , Iv ), 6.0) 56] vivax relapse follow-
to triggering. (Rf , Lv ) → ing P. falciparum in-
(Rf , Av ). fection.
hiddenlatexjutsu.
32Table 4: Table of parameters.
Symbol Description P. falciparum P. vivax Units Source Location Notes
Initial Conditions (Humans)
N (human population) Population size 100,000 100,000 people Assumed.
I0 Clinical (propor- 0.01 0.005 per capita Mondul Kiri We initialise wit
tion) mixed terms set t
zero for mosquitoe
and humans (thes
terms are not pre
sented in this table)
Cross sectional surve
[45].
A0 Asymptomatic 0.25 0.05 per capita Mondul Kiri Cross sectional surve
33
(proportion) [45].
R0 Immunity (pro- 0.7 0.4 per capita East Cambodia Serosurvey [16].
portion)
L0 Liver-stage (pro- - 0.03 per capita Assumed
portion)
T0 Undergoing 0.01 0.005 per capita Roughly calibrated s
ACT Treatment early dynamics alig
(proportion) with Mondul Kir
data.
G0 Undergoing rad- 0 0 per capita Assumed.
ical cure (pro-
portion)
Continued on next pagTable 4 – Continued from previous page
Symbol Description P. falciparum P. vivax Units Source Location Notes
Initial Conditions (Mosquitoes)
M/N ratio of 1/3 1/3 unitless Assumed.
mosquitoes
to humans
W Exposed (pro- 0.1 0.1 per capita Mondul Kiri Cross sectional surve
portion) [45].
Y Infectious (pro- 0.1 0.1 per capita Mondul Kiri Cross sectional surve
portion) [45].
Species-independent parameters
ξ Amplitude of 0.05 0.05 unitless Asia-Pacific re- Calibrated. Range
seasonality gion from [48].
34
b Number of 0.38 (0.1, 0.5) 0.38 (0.1, 0.5) per day Ranges from Calibrated. Informe
mosquito bites Senegal by [44] for P. vivax
per human per [48] for all.
day
φ Day of peak 300.0 (1.0, 300.0 (1.0, day Cambodia Calibrated so that in
transmission 365.25) 365.25) cidence peaks in Octo
from mosquitos ber.
δ Inverse of 0.0714 (0.028, 0.0714 (0.028, per day Mount Average life ex
average life 0.125) 0.125) Cameroon pectancy of 14 day
expectancy of region, Indone- [36, 48, 53].
mosquitoes sia
Continued on next pagTable 4 – Continued from previous page
Symbol Description P. falciparum P. vivax Units Source Location Notes
µ Inverse of aver- 4.053072e-05 4.053072e-05 per day Cambodia Average life ex
age human life (3.933693e- (3.933693e- pectancy is 67.5 year
expectancy 05, 05, [17].
4.680086e-05) 4.680086e-05)
Simplifying assumption of species-independent parameters
pT f A Probability 0.03 (0.0, 1.0) 0.03 (0.0, 1.0) unitless Gambia and Assuming 3 da
standard treat- Kenya course of ACT [37].
ment fails to
clear gameto-
cytes
pT f P Probability 0.03 (0.001, 0.03 (0.001, unitless Gambia and
Assuming 14 day pr
35
radical cure 0.1) 0.1) Kenya maquine with thre
fails to clear days of ACT has th
gametocytes. same efficacy agains
blood-stage malaria a
the standard 3 da
treatment of ACT.
α Inverse of 0.0167 (0.05, 0.0167 (0.05, per day Northern Ghana Average asymp
average asymp- 0.15) 0.15) tomatic infectiou
tomatic infec- period of 130 day
tious period [48].
Continued on next pagTable 4 – Continued from previous page
Symbol Description P. falciparum P. vivax Units Source Location Notes
ω Inverse of aver- 0.00038 (0.0, 0.00038 (0.0, per day Tanzania and Calculated from
age duration of 0.005) 0.005) The Gambia year half-life from
natural immu- [22].
nity
r Relative sus- 1.0 (0.0, 1.0) 1.0 (0.0, 1.0) unitless Assume no ant
ceptibility of parasite immunit
those with some with respect t
immunity to this susceptibility to in
species com- fection. Anti-parasit
pared to those immunity is cap
without tured via a reductio
36
in infectiousnes
in asymptomati
carriers.
ζA Relative in- 0.1 (0.05, 0.8) 0.1 (0.05, 0.8) unitless Assumed.
fectiousness of
asymptomatic
cases compared
to clinical
Continued on next pagTable 4 – Continued from previous page
Symbol Description P. falciparum P. vivax Units Source Location Notes
ζG Relative in- 0.0 (0.0, 0.1) 0.0 (0.0, 0.1) unitless Assumed.
fectiousness of
cases undergo-
ing radical cure
(primaquine-
based treat-
ment) compared
to clinical
ζI Relative in- 1.0 (1.0, 1.0) 1.0 (1.0, 1.0) unitless Assumed.
fectiousness of
clinical cases
37
compared to
clinical with a
P. falciparum-
only
ζT Relative in- 0.0 (0.0, 0.33) 0.0 (0.0, 0.33) unitless Assumed.
fectiousness of
cases undergo-
ing standard
treatment com-
pared to clinical
cases with no
treatment
Continued on next pagTable 4 – Continued from previous page
Symbol Description P. falciparum P. vivax Units Source Location Notes
c treatment cover- 0.3 (0.0, 1.0) 0.3 (0.0, 1.0) unitless Assumed.
age level
Species-dependent parameters
γ Inverse of 0.1 (0.028, 0.0833 (0.028, per day Mixture, South Average latent perio
duration of 0.2) 0.33) and South-East of 10 days for P. fa
latent period Asia ciparum and 12 fo
in mosquitoes P. vivax [23, 48] fo
(AKA the all, [10] for P. vivax
extrinsic incuba- all ranges from [12].
tion period)
pc Proportion of 0.95 (0.8, 1.0) 0.8 (0.8, 1.0) unitless USA, sub- [14, 24, 48] for all, [4
38
non-immune Saharan Africa, for P. vivax.
expected to Columbia
develop clinical
malaria
pR Proportion im- 0.5 (0.0, 0.77) 0.2 (0.0, 0.66) unitless Assumed. Informe
mune expected by Columbian exper
to develop clini- iment [4] which ha
cal malaria upon 0.66 for P. vivax i
reinfection a small population o
young healthy vo
unteers, Cambodia
data [27, 31] an
ranges from [48].
Continued on next pagTable 4 – Continued from previous page
Symbol Description P. falciparum P. vivax Units Source Location Notes
ρ Inverse of av- 0.33 (0.125, 0.33 (0.125, per day Gambia and Assume a 3 day cours
erage duration 0.33) 0.33) Kenya of ACT. [37, 48].
for regular
treatment
M Transmission 0.5 (0.0, 0.8) 0.3 (0.0, 0.8) unitless Informed by exper
probability: elicitation. Partiall
mosquito to informed by [47, 48
human (per bite 50] for all, [8, 44] fo
from an infec- P. vivax.
tious mosquito)
H Transmission 0.1 (0.0, 0.5) 0.1 (0.0, 0.5) unitless Informed by exper
39
probability: hu- elicitation. [33, 48] fo
man to mosquito P. falciparum, [44, 46
(per bite on an for P. vivax, range
infectious hu- from [12].
man)
κv Inverse of aver- - 0.0025 (0.002, per day South East Asia On average, hypno
age time until 0.003) zoites die out after 40
hypnozoites die days [48, 54]. Th
naturally 1/500 day limit i
from [44].
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