Appendix
This appendix includes details about the model formulation and the parameters that were used in
simulation. It is organized as follows (sections are hyperlinked):
I.
Epidemiological Model
II.
Vector Ecology Model
III.
Insecticide Resistance Model
IV.
Model limitations and extensions
V.
Considering Population Growth in the Epidemiological Model
1
Epidemiological Model
With regard to malaria infection by two strains, drug-resistant (r) or nonresistant (n), individuals
are either:
i. Susceptible (𝑆)
ii. Incubating (𝐻𝑛 and 𝐻𝑟 )
iii. Infectious and symptomatic (𝐼𝑛 and 𝐼𝑟 )
iv. Recovered (𝑅)
v. Infectious but asymptomatic (𝑄𝑛 and 𝑄𝑟 )
The compartmental flow diagram for the disease, accounting for age-structure, is depicted in
figure S1.
Formally, the equations characterizing malaria epidemiology among the human population
are as follows: Individuals die at age-specific rate 𝑑(𝑎), regardless of disease status; as in
previous malariological models of this sort [1-3], we make the simplifying assumption that there
is no disease-induced mortality. Susceptibles are infected with nonresistant and resistant strains
of malaria at age-specific rates 𝜆𝑛 (𝑎) and 𝜆𝑟 (𝑎) respectively. Recovered, untreated individuals
return to susceptible status at rate 𝜂, which is the rate of immunity loss. Fractions 𝜔𝑆𝑃 and 𝜔𝐴𝐶𝑇
of sick individuals are treated with SP and ACT respectively; these individuals recover at rates
𝑟𝑆𝑃 and 𝑟𝐴𝐶𝑇 respectively, assuming they are not infected with a resistant strain in the case of SP.
The differential equation for the number of individuals of age 𝑎 who are susceptible at time 𝑡 is
thus:
𝜕𝑆 𝜕𝑆
+
= 𝜂𝑅 + (𝜔𝑆𝑃 𝑟𝑆𝑃 + 𝜔𝐴𝐶𝑇 𝑟𝐴𝐶𝑇 )𝐼𝑛 + 𝜔𝐴𝐶𝑇 𝑟𝐴𝐶𝑇 𝐼𝑟 − [𝜆𝑛 (𝑎) + 𝜆𝑟 (𝑎) + 𝑑(𝑎)]𝑆
𝜕𝑡 𝜕𝑎
The average incubation time for either strain of malaria in the model is 𝜏𝐻 , and incubating
individuals become symptomatic at a constant rate. It is assumed that individuals who have
2
naturally recovered from the illness may be reinfected and become symptomatic at rates
𝜆𝑖 (𝑎)𝜃𝐼 𝑅, where 𝜃𝐼 ∈ [0,1], using the subscript 𝑖 = 𝑛 , 𝑟 to denote, nonresistant and resistant
strains respectively. The differential equations for the number of incubating and symptomatic
individuals are therefore:
𝜕𝐻𝑖 𝜕𝐻𝑖
1
+
= 𝜆𝑖 (𝑎)[𝑆 + 𝜃𝐼 𝑅] − [ + 𝑑(𝑎)] 𝐻𝑖
𝜕𝑡
𝜕𝑎
𝜏𝐻
𝜕𝐼𝑖 𝜕𝐼𝑖 𝐻𝑖
+
=
− {(1 − 𝜔𝐴𝐶𝑇 − 𝜔𝑆𝑃 𝑟𝑆𝑃 𝟏[𝑖 = 𝑛])𝑟𝑖 + 𝜔𝑆𝑃 𝑟𝑆𝑃 𝟏[𝑖 = 𝑛] + 𝜔𝐴𝐶𝑇 𝑟𝐴𝐶𝑇 + 𝑑(𝑎)} 𝐼𝑖
𝜕𝑡 𝜕𝑎 𝜏𝐻
where 𝟏[𝑖 = 𝑛] is the indicator function for the nonresistant strain. Individuals who have
recovered “naturally” from malaria (i.e. without the aid of SP or ACT), and are thus in the 𝑅
compartment, become reinfected with malaria at the same rates 𝜆𝑛 (𝑎) and 𝜆𝑟 (𝑎) as susceptibles.
As mentioned above, a fraction 𝜃𝐼 of the individuals manifest symptoms, and the remaining
portion 1 − 𝜃𝐼 is asymptomatic, putting them in the 𝑄𝑛 and 𝑄𝑟 compartments. Asymptomatic
infections recover at rates 𝑞𝑛 and 𝑞𝑟 for nonresistant and resistant strains, respectively. The
differential equations for the naturally recovered and infected, asymptomatic individuals is thus:
𝜕𝑅 𝜕𝑅
+
= (1 − 𝜔𝑆𝑃 − 𝜔𝐴𝐶𝑇 )𝑟𝑛 𝐼𝑛 + (1 − 𝜔𝐴𝐶𝑇 )𝑟𝑟 𝐼𝑟 − [𝜆(𝑎) + 𝜆𝑟 (𝑎) + 𝜂 + 𝑑(𝑎)]𝑅 + 𝑞𝑛 𝑄𝑛 + 𝑞𝑟 𝑄𝑟
𝜕𝑡 𝜕𝑎
𝜕𝑄𝑖 𝜕𝑄𝑖
+
= (1 − 𝜃𝐼 )𝜆𝑖 (𝑎)𝑅 − [𝑞𝑖 + 𝑑(𝑎)]𝑄𝑖
𝜕𝑡
𝜕𝑎
for 𝑖 = 𝑛 , 𝑟.
In the MATLAB implementation of the model, policy controls such as the fractions 𝜔𝑆𝑃
and 𝜔𝐴𝐶𝑇 of symptomatic individuals who are treated with SP or ACT may directly depend on
age or time, or may be formulated as “feedback” controls, dependent on the state variables of the
system. In the paper, we only consider coverage levels which do not vary over time.
The boundary conditions for this subsystem of partial differential equations consists of
initial conditions on the age-profiles of the above state variables and a characterization of
3
population recruitment using an age-specific fertility rate 𝑓(𝑎). Denoting the population density
as 𝑃 ≡ 𝑆 + 𝐻𝑛 + 𝐻𝑟 + 𝐼𝑛 + 𝐼𝑟 + 𝑅 + 𝑄𝑛 + 𝑄𝑟 , recruitment into the newborn susceptible class is
∞
characterized by 𝑆(𝑡, 0) = ∫0 𝑓(𝑎)𝑃(𝑡, 𝑎)𝑑𝑎. We make the simplifying assumption that there is
no recruitment into other disease compartments, i.e. there is no vertical transmission.
Vector Ecology Model
The vector ecology component of the model determines the forces of infection 𝜆𝑖 (𝑎) in the
epidemiological model above, which link insecticide spraying impacts on the vector population
with epidemiological impacts. The aggregate mosquito population dynamic is assumed to follow
the differential equation:
𝑑𝑀
= Π − 𝜇𝑀
𝑑𝑡
where Π is density-independent recruitment per unit time and 𝜇 is the vector mortality rate [3].
Vectors’ disease status with regard to the naïve and drug resistant malaria strains is either
susceptible (𝑉), incubating (𝑋𝑛 and 𝑋𝑟 ), or infectious (𝑌𝑛 and 𝑌𝑟 ). The epidemiological
dynamics among the vector population are similar to those of the human population, except that
we do not specify age-structure for vectors. Vectors are infected by nonresistant and resistant
malaria strains at rates 𝜆𝑛𝑋 and 𝜆𝑟𝑋 respectively, at which point they begin incubating parasites. At
a constant rate, incubating mosquitoes become infectious to humans, such that 𝜏𝑆 is the average
incubation time of malaria in mosquitoes (a.k.a the sporogonic cycle).
𝑑𝑉
= Π − (𝜆𝑛𝑋 + 𝜆𝑟𝑋 + 𝜇)𝑉
𝑑𝑡
𝑑𝑋𝑖
1
= 𝜆𝑖𝑋 𝑉 − ( + 𝜇) 𝑋𝑖
𝑑𝑡
𝜏𝑆
𝑖 = 𝑛 ,𝑟
4
𝑑𝑌𝑖 𝑋𝑖
= − 𝜇𝑌𝑖
𝑑𝑡
𝜏𝑆
Note that the differential equation for 𝑉 is redundant, and can be eliminated using the identity
𝑉 ≡ 𝑀 − 𝑋𝑛 − 𝑋𝑟 − 𝑌𝑛 − 𝑌𝑟 .
In the model, the rates 𝜆𝑖𝑋 are determined by the prevalence of malaria in vectors and
humans, as well as human-vector contacts determined, for example by biting behavior. For a
given probability 𝜌ℎ𝑚 (resp. 𝜌𝑚ℎ ) of passing parasites from (to) mosquitoes to (from) a human of
age 𝑎, and also given a likelihood ratio 𝑚(𝑎) of biting a human of age 𝑎 relative to an arbitrary
base age, we model these rates as:
𝜆𝑖 (𝑡, 𝑎) = 𝑌𝑟 (𝑡)
ℎ
𝜌ℎ𝑚 (𝑎)𝑚(𝑎)
⋅ ∞
𝜏𝐺 ∫ 𝑚(𝛼) 𝑃(𝑡, 𝛼)𝑑𝛼
𝑖 = 𝑛 ,𝑟
0
∞
𝜆𝑖𝑋 (𝑡)
ℎ ∫ 𝜌𝑚ℎ (𝛼)𝑚(𝛼) [𝐼𝑖 (𝑡, 𝛼) + 𝜃𝑄𝑖 (𝑡, 𝛼)]𝑑𝛼
= ⋅ 0
∞
𝜏𝐺
∫ 𝑚(𝛼) 𝑃(𝑡, 𝛼)𝑑𝛼
0
where 𝜃 is the fraction of asymptomatic human infections which are infectious to mosquitoes.
The age dependence of the transmission probabilities 𝜌ℎ𝑚 and 𝜌𝑚ℎ is due in this model to the
effect of bednet coverage, which may vary by age. Specifically, if the baseline probabilities are
0
0
𝜌ℎ𝑚
and 𝜌𝑚ℎ
, and the protective effect of nets in terms of lowering contact is to reduce both of
these probabilities by a fraction 𝛽, then for age-specific levels of net coverage 𝐶𝑛𝑒𝑡𝑠 (𝑎), we
have:
0
𝜌ℎ𝑚 (𝑎) = [1 − 𝛽𝐶𝑛𝑒𝑡𝑠 (𝑎)]𝜌ℎ𝑚
0
𝜌𝑚ℎ (𝑎) = [1 − 𝛽𝐶𝑛𝑒𝑡𝑠 (𝑎)]𝜌𝑚ℎ
As with other controls, 𝐶𝑛𝑒𝑡𝑠 may also be time dependent, either directly or through feedback
dependence on the state variables.
5
Weather affects vector ecology by controlling vector recruitment, the gonotrophic, and
the sporgonic cycle. In this model we follow Worrall, Connor et al. [3] formulate mosquito
recruitment at time 𝑡 as
Π(𝑡) ∝ 𝑝𝑟𝑒𝑐𝑖𝑝(𝑡)
where 𝑝𝑟𝑒𝑐𝑖𝑝(∙) is a continuous function. The proportionality constant implied by this
relationship is a parameter to be fitted; see table 1 in the main text. The specific function
𝑝𝑟𝑒𝑐𝑖𝑝(∙), displayed in Figure 2 of the main text, is obtained from a Fourier regression of
precipitation on month over several decades, such that 𝑝𝑟𝑒𝑐𝑖𝑝(∙) is periodic over one year
intervals.
Weather has also been hypothesized to affect biting behavior, through effects on
mosquitoes’ reproductive frequency. In this model we again follow Worrall, Connor et al. [3]
and formulate the gonotrophic cycle as
𝜏𝐺 (𝑡) = 𝑣𝐺 +
𝑓𝐺
𝑡𝑒𝑚𝑝(𝑡) − 𝑔𝐺
where 𝑣𝐺 is the minimum time required for gonotrophy, 𝑔𝐺 is the minimum temperature required
for gonotrophy, and 𝑓𝐺 minimum degree-time required for gonotrophy. Without good data to
independently estimate these parameters, we use estimates of these parameters from Worrall,
Connor et al. [3]. Note that this function can be expressed as log-linear in the deviations
[𝜏𝐺 (𝑡) − 𝑣𝐺 ] and [𝑡𝑒𝑚𝑝(𝑡) − 𝑔𝐺 ]. If we define “excess time until gonotrophy” as the time
above the minimum, and the “excess temperature” similarly, it is apparent here that a 1%
increase in excess temperature yields a 1% decrease in the excess time required for gonotrophy.
Similarly to the gonotrophic cycle, we model the length of the sporogonic cycle as a
function of temperature as follows:
6
𝛼𝑆
𝑓𝑆
𝜏𝑆 (𝑡) = 𝑣𝑆 + [
]
𝑡𝑒𝑚𝑝(𝑡) − 𝑔𝑆
where 𝑣𝑆 , 𝑔𝑆 , and 𝑓𝑆 are directly analogous to 𝑣𝐺 , 𝑔𝐺 , and 𝑓𝐺 . The parameter 𝛼𝑆 controls the
relative impact of excess temperature increases on the excess length of the sporogonic cycle; we
allow this parameter to vary, because restricting it to unity yields a poor fit with the data. We
estimated these parameters independently of the rest of the model using data reported in
Teklehaimot, Lipsitch [4]. The resulting estimates are displayed in Table S1. As with the
gonotrophic cycle model, this function can be expressed as log-linear in deviations from the
minimum time and temperature required for sporogony. However, our estimates indicate that
magnitude of the effect is greater than unity: A 1% increase in excess temperature yields a
3.07% decrease in the excess time until sporogony.
As with precipitation, the specific function temp(∙), displayed in figure 2 of the main
text, is obtained from a Fourier regression of temperature on month over several decades, such
that temp(∙) is periodic over one year intervals.
Insecticide Resistance Model
To incorporate insecticide resistance, we provide a mathematical characterization of how
insecticides impact mosquito mortality under different levels of resistance as well as a
characterization of how resistance spreads through the vector population. Here, resistance is
characterized by diploid genetics; vectors are assumed to express one of two alleles (resistant or
susceptible) at a single locus on a pair of chromosomes. An example of such a polymorphism in
a mosquito vector is the “knockdown resistance” (kdr) gene in A. gambiae. The dynamic
variable of interest here is the fraction 𝑠 of the vector population which possesses at least one
copy of the susceptible allele.
7
Average vector mortality 𝜇 is a weighted mean of the mortality rates of individual
genotypes, and depends on 𝑠, as well as the level of insecticide exposure 𝐶 :
𝜇 = 𝜇0 + 𝜇𝑟 (1 − 𝑠)2 + 𝐷𝜇𝑟 2𝑠(1 − 𝑠) + 𝐶ℎ𝑔[(1 − 𝐷)2𝑠(1 − 𝑠) + 𝑠 2 ]
Here, 𝐷 ∈ [0,1] is the relative dominance of the resistant gene in heterozygotes. The extra
mortality term 𝜇𝑟 for resistant genotypes captures the possibility of an evolutionary “fitness cost”
associated with possessing resistance to the insecticide, discussed further in the following
paragraph. The insecticide-induced evolutionary advantage of the resistant types is captured by
assuming 𝑔 > 0. This representation of how vectors respond to insecticide exposure is relatively
standard in the literature on insecticide resistance [e.g. 5].
To model the evolution of 𝑠, a “replicator dynamic” is used to characterize selection for
the resistant or susceptible genotypes [6, 7]. In the present context, such a dynamic takes the
form:
𝑑𝑠
= 𝜖𝜏𝐺 (𝑓 − ℎ𝑔𝐶)[𝐷𝑠 + (1 − 𝐷)(1 − 𝑠)]𝑠(1 − 𝑠)
𝑑𝑡
where 𝜖 is the speed of insecticide resistance accumulation, 𝑓 is the total fitness cost of the
resistant type relative to the susceptible type. Here, this fitness cost is defined as 𝑓 ≡ 𝜇𝑟 +
1
𝜏𝐺
𝐺
ln 𝐺𝑆 , where 𝐺𝑆 and 𝐺𝑅 are respectively measures of the fecundity of sensitive and resistant
𝑅
genotypes respectively. This definition of the fitness cost is a generalization of current
definitions in the literature examining the accumulation of insecticide resistance in pest and/or
vector populations. Overall insecticide exposure 𝐶 is modeled as:
∞
𝐶(𝑡) ≡ 𝜂𝑛𝑒𝑡𝑠
∫0 𝐶𝑛𝑒𝑡𝑠 (𝑎,𝑡)𝑃(𝑎,𝑡)𝑑𝑎
∞
∫0 𝑃(𝑎,𝑡)𝑑𝑎
+ 𝜂𝐼𝑅𝑆 𝐶𝐼𝑅𝑆 (𝑡)
where 𝜂𝑛𝑒𝑡𝑠 and 𝜂𝐼𝑅𝑆 are the fractions of time vectors are exposed to ITNs or IRS respectively,
𝐶𝑛𝑒𝑡𝑠 (𝑎, 𝑡) is the coverage level of nets among individuals of age 𝑎 at time 𝑡, 𝑃(𝑎, 𝑡) is the total
8
population density of age 𝑎 at time 𝑡, 𝐶𝐼𝑅𝑆 (𝑡) is the fraction of the total population covered by
IRS at time 𝑡. So total insecticide coverage is the sum of coverage levels due to spraying and
nets. And
A note on numerical simulation of the model
As mentioned in the text, the full model was implemented using exponential functions as
opposed to linear differences, or some other differential equation simulation method. As an
example of the use of exponential functions, consider the ordinary differential equation (ODE)
for the overall mosquito population
𝑑𝑀
𝑑𝑡
= Π − 𝜇𝑀. This ODE can be simulated as:
(𝑒 −𝜇Δ − 1)
𝑀(𝑡 + Δ) − 𝑀(𝑡)
=Π+
𝑀(𝑡)
Δ
Δ
where Δ is a small increment of time. We can rearrange this to get:
𝑀(𝑡 + Δ) = ΔΠ + 𝑒 −𝜇Δ 𝑀(𝑡)
The simulation of partial differential equations in this context can be done similarly. For
example, the PDE for asymptomatic infections
𝜕𝑄
𝜕𝑡
𝜕𝑄
+ 𝜕𝑎 = (1 − 𝜃𝐼 )𝜆(𝑎)𝑅 − [𝑞 + 𝑑(𝑎)]𝑄𝑖 can be
approximated using:
𝑄(𝑡 + Δ, 𝑎 + Δ) = {1 − exp[−(1 − 𝜃𝐼 )𝜆(𝑎)Δ]}𝑅(𝑡, 𝑎) + exp{−[𝑞 + 𝑑(𝑎)]Δ} 𝑄(𝑡, 𝑎)
Thus, this method generalizes to all of the differential equations in this paper.
Model limitations and extensions
In terms of modeling extensions that may be warranted, the first task is to allow for timedependent portfolios, e.g. allowing for “pulses” of IRS or abandoning SP following the advent of
widespread resistance. Additionally, while deterministic, differential equations models of
9
malaria transmission remain the standard in the literature [e.g. 8], allowing for uncertainty and
stochasticity in transmission dynamics seems increasingly necessary from a policy standpoint:
In the standard deterministic models, disease eradication can only accomplished in the long-run
and through the perpetual employment of sufficiently intensive interventions. A stochastic
model can allow analysts to think in more concrete terms about the “probability of eradication,”
and, because true eradication can be achieved in finite time in such models, there may be a large
economic benefit (in addition to the obvious, moral benefits) to achieving eradication, since
future malaria interventions could become unnecessary and thus abandoned, saving costs. A
related issue is the theory underlying mass-action or pseudo-mass action disease models [9, 10]:
Such models approximate random transmission well in settings with large numbers of infected
individuals. Such a setting certainly describes the malaria distribution in many African
countries, including Tanzania. However, when we begin to consider the possibility of malaria
elimination or eradication, we must analyze transmission when there are relatively few infected
individuals, pointing towards the necessity of a stochastic model.
All of the limitations addressed above are not isolated to this modeling exercise: They
are limitations of all current-generation deterministic, Macdonald-Ross models of vector borne
transmission. Thus, one lesson from this modeling exercise is that generalizing malaria
transmission models to account for the above issues has direct import for cost-effectiveness
analysis of malaria control programs.
Considering Population Growth in the Epidemiological Model
The assumption of a stabilized human population clearly does not hold in many malaria endemic
countries, including Tanzania. High fertility and mortality, as well as major migrations across
national borders and between rural and urban areas, can have dramatic implications for the cost10
effective control of malaria. These implications are only just beginning to be analyzed
systematically [11]. One of the most pressing issues for making theoretical models of
transmission more applicable for on-the-ground disease control is resolving the problem of
increasing human-vector population ratios: Even in current-generation models of vector-borne
transmission, an increasing human population inevitably leads to malaria elimination in the longrun, due to the low probability of a mosquito landing on an infected human. One potential
solution to this problem is to assume a constant ratio of humans-to-mosquitoes. Even though
vector recruitment is assumed by convention to be independent of human population size (i.e.
independent of the availability of the prey), a constant human-vector ratio could be justified on
spatial terms. For example, in the Auger and Kouokam’s [12] model of vector-borne disease
transmission in a “patchy environment,” one can generate a constant human-vector ratio by
assuming that the number of patches is proportional to human population, and that recruitment in
each patch is identical. As a preliminary investigation into this issue, we translate this
assumption into the present model, assuming that vector recruitment Π is proportional to total
∞
human population ∫0 𝑃(𝑎)𝑑𝑎.
11
Table S1: Fitted parameters for temperature-dependence of sporogonic and gonortophic cycles
Symbol
𝑣𝑆
Best Fit
4.664656106
Description
Minimum days required for sporogony
Units
Days
𝑓𝑆
16.177
Degree-days required for sporogony
Degree-days
𝑔𝑆
12.456
Minimum temperature required for sporogony
Degrees celsius
𝛼𝑆
3.0757
Shape parameter on accumulation of degree days
Elasticity
12
Table S2: Baseline model parameters from the literature or assumed1
Symbol
Baseline Value
Description
Units
DEMOGRAPHY AND EPIDEMIOLOGY
𝑃0
23,860
Total human population. Arbitrary
humans
𝜂
1.38629
Rate of immunity loss from Águas, White et al. [1]
per year
𝑟𝑆𝑃
0.749
Recovery rate those successfully treated with SP [13]
per day
𝑟𝐴𝐶𝑇
1.52
Recovery rate those successfully treated with ACT [13]
per day
𝑟𝑛 /𝑟𝑟
0.2
Relative untreated recovery rate of drug-resistant cases; fitness cost [14-16].
ratio
𝑞𝑛 /𝑞𝑟
0.2
Relative untreated recovery rate of drug-resistant cases; fitness cost [14-16].
ratio
𝛽
0.8
fraction in [0,1]
𝑤𝑟𝑒𝑠𝑡𝑖𝑛𝑔
0.55
VECTOR ECOLOGY, TRANSMISSION PARAMETERS
Physical protectiveness of nets; higher = more protective. Arbitrary: see
sensitivity analysis.
Fraction of time vectors are resting or attempting to lay eggs: exposed to IRS [17].
𝑤ℎ𝑢𝑚𝑎𝑛𝑠
fraction in [0,1]
0.22
Fraction of time vectors are attempting to feed on humans: exposed to ITNs [17].
fraction in [0,1]
𝜇0
0.2712
Baseline vector mortality [3, 18]
Mosquitoes per day
𝑔
0.833
Insecticide-induced vector mortality for susceptible mosquitoes [3, 18]
Mosquitoes per day
𝑣0
0.15
Initial genetic frequency of DDT/pyrethroid resistance in vectors.
fraction in [0,1]
𝑓
0
Aggregate fitness cost of insecticide resistance [19-21]
Rate per day
𝐷
1
Genetic dominance of resistant genotypes [19-21]
fraction in [0,1]
2.272
0.464022
0.5
Speed of evolution of insecticide resistance in vectors [17]
Probability of passing parasites from humans to mosquitoes. Assumed.
Fraction of vector bloodmeals on humans (human blood index, HBI)
Dimensionless
Probability
fraction in [0,1]
1
Ability of asymptomatic malaria cases to infect mosquitoes; assumed.
dimensionless weight
1 − 0.0056 ∙ 𝑒 −0.998∙age
Age-specific relative likelihood of mosquito bites [1].
dimensionless weight
𝜀
0
𝜌𝑚ℎ
ℎ
𝜃
𝑚(age)
1
The references in this table either provided values for the parameters, or guidance in choosing reasonable values.
13
Table S3: Univariate sensitivity analysis of cost-effective portfolios (CEPs)1
PARAMETER VARIED
(IF ANY)
Baseline, for comparison
VECTOR CONTROL
TREATMENT
ITN
IRS
ACT
SP
ANNUALIZED
COSTS PER
CAPITA
5 years
100%
60%
60%
40%
$0.42
10 years
100%
0%
80%
20%
$0.50
20 years
100%
0%
100%
0%
$1.91
POLICY
HORIZON
SENSITIVITY ANALYSIS OF ECONOMIC PARAMETERS
Discount rate doubled
(𝛿 = 0.06)
Discount rate quadrupled
(𝛿 = 0.12)
Discount rate halved
(𝛿 = 0.015)
Infection cost halved
($40 per case)
Infection cost doubled
($160 per case)
5 years
100%
60%
60%
40%
$0.44
10 years
100%
0%
80%
20%
$0.56
20 years
100%
0%
100%
0%
$1.67
5 years
100%
60%
60%
40%
$0.49
10 years
100%
0%
80%
20%
$0.69
20 years
100%
0%
100%
0%
$1.32
5 years
100%
60%
60%
40%
$0.40
10 years
100%
0%
80%
20%
$0.48
20 years
100%
0%
100%
0%
$2.05
5 years
100%
60%
40%
60%
$0.34
10 years
100%
0%
60%
40%
$0.41
20 years
100%
0%
100%
0%
$1.58
5 years
100%
60%
100%
0%
$0.55
10 years
100%
0%
100%
0%
$0.67
20 years
100%
0%
100%
0%
$2.58
SENSITIVITY ANALYSIS OF INTERVENTION-RELATED BIOLOGICAL PARAMETERS
Positive fitness cost of
insecticide resistance
(𝑓/𝜏𝐺 = 0.849)
No fitness cost of drug
resistance
(𝑟𝑛 /𝑟𝑟 = 𝑞𝑛 /𝑞𝑟 = 0)
Inverted time resting versus
feeding on humans
(𝑤𝑟𝑒𝑠𝑡𝑖𝑛𝑔 = 0.22, 𝑤ℎ𝑢𝑚𝑎𝑛𝑠 = 0.55)
40% decrease in physical
protectiveness of nets
(𝛽 = 0.5)
1
5 years
80%
100%
60%
40%
$0.40
10 years
60%
100%
60%
40%
$0.27
20 years
0%
100%
60%
40%
$0.19
5 years
100%
60%
60%
40%
$0.42
10 years
100%
0%
100%
0%
$0.52
20 years
100%
0%
100%
0%
$1.94
5 years
100%
0%
60%
40%
$0.37
10 years
60%
0%
80%
20%
$2.08
20 years
100%
0%
100%
0%
$3.49
5 years
80%
60%
60%
40%
$0.51
10 years
100%
20%
80%
20%
$2.20
20 years
100%
40%
0%
20%
$4.10
Magnitudes of parameter changes based on references cited in Table S2.
14
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