A voluntary use of insecticide treated nets can stop the vector transmission of Chagas disease

One of the stated goals of the London Declaration on Neglected Tropical Diseases is the interruption of domiciliary transmissions of Chagas disease in the region of the Americas. We used a game-theoretic approach to assess the voluntary use of insecticide treated nets (ITNs) in the prevention of the spread of infection through vector bites. Our results show that individuals behave rationally and weigh the risks of insect bites against the cost of the ITNs. The optimal voluntary use of ITNs results in predicted incidence rates that closely track the real incidence rates in Latin America. This means that ITNs are effective and could be used to control the spread of the disease by relying on individual decisions rather than centralized policies. Our model shows that to completely eradicate the vector transmission through the voluntary individual use of ITNs, the cost of ITNs should be as low as possible.


Introduction
American trypanosomiasis, in humans known as Chagas disease, is one of the world's most important neglected tropical diseases [1] and is most prevalent in Latin America where it is the vectors, Rhodnius prolixus, and the host (human) populations denoted by r and h subscripts, respectively. The model involves two separate areas: the palm plantation and human settlement. The palm plantation acts as a potential reservoir of vectors. Since R. prolixus usually bites humans at night [46], the transmission is modelled only through contact between vectors and humans in the settlement. For simplicity, we do not consider any infected vectors in the plantation area, although infected R. prolixus can be found in the palm trees as well [47]. We also do not consider any human dynamics in the plantation area.
There is no recovery from Chagas disease [16] so the total human population is divided into (1) susceptible (S h ), i.e. not infected with T. cruzi, (2) exposed (E h ), i.e. infected with T. cruzi but not yet infectious, and (3) infectious (I h ). The population size is normalized so that N h = S h + E h + I h = 1. The vector population is divided into vectors living in the plantation (M r ) and the settlement (N r ). The population in the plantation is always susceptible. The vector population in the settlement is divided into susceptible (S r ), exposed (E r ) and infectious (I r ).
Humans are born as susceptible at the per capita rate α. While there is a disease induced mortality [48], we follow [44] in omitting this for the sake of simplicity of the model. To keep the human population constant, the human natural death rate is considered the same as the birth rate and the same across all classes. The vectors are born as susceptible at per capita rate β; the birth rate is considered the same in the plantation and the settlement. The vectors follow a logistic growth with carrying capacity K p in the plantation and K s in the settlement. The vectors migrate from the plantation to the settlement at rate ω. The natural death rate of vectors, μ, is considered the same across all classes. We assume μ < β.
The disease is transmitted from an infectious vector to susceptible humans at rate (1 − p)a 0 and from an infectious human to susceptible vectors at rate (1 − p)b 0 . Here, p is the frequency of ITN use while a 0 and b 0 are the transmission rates without ITN use. After the incubation period, the exposed individuals become infectious at rate δ for humans and σ for vectors.
The values and ranges of model parameters are summarized in Table 1, details are presented below.

Parameter estimation
The vector-to-human transmission rate, a 0 , is estimated as 5 PLOS NEGLECTED TROPICAL DISEASES literature, [49] estimated the probability of vector-to-human transmission (per contact) as 5.8 × 10 −4 (with 95% CI: [2.6, 11.0] × 10 −4 ). This estimate is consistent across triatomine species, robust to variations in other parameters, and corresponds to 900-4,000 contacts per case. Moreover, [59] estimates the biting rate to be between 0.2 − 0.4 and [60] and [49] further confirm this by stating that the biting rate on average is 0.3 per day. We note that [61] found feeding rate of T. infestants to be between 0.3 and 0.6 per day.
Similarly, from the data presented in [50, Table 7]. This yields an average 0:056 7 ¼ 0:008 per day and the range [0.0033, 0.016] per day. We note that [63] assume the death rate (for T. infestants) to be 0.0046 per day with a reference to [52]; however we believe that this discrepancy is likely caused by [63] considering the standard deviation 0.034 instead of the average rate 0.056 per week in [52]. Also, [1] uses the death rate 1.73 per year (i.e. 0.0047 per day) with a reference to [62]. For the purpose of our study we will use 0.008 per day as the average death rate with the range [0.0033, 0.016] per day.
The incubation period for humans after exposure to a triatomine bite is 5-14 days [13]. As noted by [12], definitive data is not available because persons who live in areas of active transmission are generally continually at risk for exposure to the vectors. We assume that the β Birth rate of vectors 0.022 [52] μ Natural death rate of vectors 0.008 [52] δ Incubation rate in humans 0.1 [13] σ  [53] who found that 3-4 days post-infection, the T. cruzi parasite population begins to colonize the triatomine hosts to reach a climax at day 7 post-infection, which is maintained during the next two weeks.
The carrying capacity of vectors in the field is 31, 900 per km 2 [55]. The small plantation size is about 5ha [69], yielding 1600 bugs, i.e. after renormalization for a 8 person household, we get K p = 200.
The migration rate, ω, was set to 0.01 per day on average with the range [0, 0.02] [56]. We note that [56] cites [70] as the source for that number, but we were not able to locate the information in [70]. Also, [44] and [2] use ω = 0.05. However, for the other values of the parameters, most notably the vector birth rate β and death rate μ such a high value of ω would result in no bugs in the plantation area. Consequently, we adapted the value 0.01 from [56]. Moreover, as seen from the sensitivity analysis, the results are not overly sensitive to values of ω.

Analysis
The model of the transmission dynamics shown in Fig 1 yields the following system of differential equations.

Equilibria of the ODE system (1)-(7)
There are four possible equilibria of the dynamics (1)- (7): (i) an unstable disease-free equilib- ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi and finally (iv) the endemic equilibrium and m � ¼ m þ m s N � r denotes the vector mortality rate in the settlement when the population levels reach the equilibrium. See the section below for detailed calculations.

Step-by-step calculations for equilibria
Let us set N r = S r + E r + I r and investigate the system that results from (4) and from the sum of eqs (5) and (6), and (7).

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The system (16) and (17) has three equilibria: (a) (0, 0) which is always unstable (if β > μ), ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi which is defined and locally stable if ω < β − μ. Now, let us proceed to solve for the equilibria of the system (1)- (7). Based on above calculations, we have to solve the following system of algebraic equations where M � r is given by (18) and N � r is given by (19). Let us set m � ¼ m þ m s N � r . By adding (20),(21), and (22), we get We have the disease-free equilibrium with S h = 1, E h = 0, I h = 0, S r = N r , E r = 0, I r = 0 whenever any of the following happens: (1) I r = 0 (in particular when N � For the rest of the section, we will assume that p < 1, I h > 0 and I r > 0. By (22), and so by (26) and (27), By (25), Using Eq (24) By (20), Finally, and thus and the remaining values are given by PLOS NEGLECTED TROPICAL DISEASES

The basic reproduction number
The basic reproduction number, i.e. the number of secondary infections caused by a single infectious individual in an otherwise disease-free population is given by where is the number of secondary infections caused by a single infectious individual if nobody uses ITN. The formula can be derived as follows. An infectious individual lives on average for a time α −1 . During that time, they expose susceptible vectors at the rate ð1 À pÞb 0 N � r . Each of the exposed vectors become infectious with probability s sþm � . Each of the infectious vectors then lives for the time μ �−1 during which it exposes susceptible individuals at the rate (1 − p)a 0 N h = (1 − p)a 0 . Each of those exposed individuals will become infectious with probability d dþa . The equilibrium E 1 0 is always unstable. The disease-free equilibrium E 2 0 is locally stable if ω > β − μ and R 0 < 1. The disease-free equilibrium E 3 0 exists if ω < β − μ and is stable if R 0 < 1. The endemic equilibrium E � is defined and locally stable if ω < β − μ and R 0 > 1.

Herd immunity
It follows from (43) that @R 0 @p < 0, i.e. R 0 is decreasing in p. Consequently, the population will reach herd immunity at the smallest value of p 2 [0, 1] for which R 0 � 1, i.e.

Model setup
In this section, we set up and solve a game-theoretic model of individual ITN use decisions. We will assume that the system (1)- (7) is in an equilibrium. Individuals can either use or not use an ITN. We assume that all individuals are rational and act in their own self-interest [21]. As usual in vaccination games, see for example [35], individuals weigh the perceived cost of ITN use against the risks of infection. The risk of infection depends on the population-wide ITN use rate. This results in a public goods game in which individuals base their ITN use decision on the decisions of others. For simplicity, we will consider only the actual monetary costs of ITN use, but we note that perceived costs could involve other factors such as possible discomfort associated with limited air circulation, and possible side effects of the insecticide [45]. The cost varies, it can be about $5 in Mexico and $9 in Columbia [58]. An ITN lasts about 2 years [57], so the annual cost is C ITN = $2.50 in Mexico and C ITN = $4.5 in Columbia. The annual cost of a T. cruzi infection, C Chagas , in Latin America is estimated as $383 with a range $207-$636 [7]. From the perspective of an individual, the expected cost of not using ITNs when the probability of ITN use in the overall population is p, denoted by C noITN (p), is given as a product of C Chagas and the probability of getting infected (the probability of moving from the S h compartment to the compartment I h ), where the herd immunity level of ITN use, p HI , is given in (45).

Dependence of C noITN on p
Here we show that C noITN (p) is decreasing in p; this is illustrated in Fig 2. By (46) r > 0 and, as seen below, @I � r @p < 0, we will get that @C noITN @p < 0. By (37), and thus, as S � r and I � h are decreasing in p, E � r should be decreasing in p, contradiction with an already established fact that @E � r @p > 0.

Nash equilibria
When the ITN use p is such that C noITN (p) = C ITN , the ITN use is at Nash equilibrium, p NE ; this means that no individual has an incentive to deviate from their current ITN usage. Because C noITN (p) is decreasing in p, p NE is in fact a convergently stable Nash equilibrium which indicates that the population will evolve toward it, see [71]. When p < p NE , it is beneficial for the individual to use the ITN (the cost of using the ITN is smaller than the expected cost of infection); when p > p NE , it is beneficial for the individual not to use the ITN (the cost of the net is larger than the expected cost of infection).

Results, sensitivity analysis and model validation
The values of p HI and p NE are close together, see Figs 2 and 3. For the parameters as in Table 1, the model predicts p HI = 0.5026 and p NE = 0.5017.
More than 50% of the households in endemic areas of Colombia use bednets (although some were not insecticide treated) [ [74]. When we assume C ITN = 4.5, to more closely match the price in Colombia, our model predicts incidence rate around 8.37 person per year per 100,000 individuals, again in close agreement with the real incidence rate of 11 person per year per 100,000 individuals [74]. This all indicates that individuals behave rationally as predicted in general by [21]. Our findings also agree with [75] whose results showed that peoples' acceptance of ITN use is related to the perception of an immediate protective effect against vectors. Our crucial result is that the incidence rate is essentially linear and increasing with the cost of the ITNs, see Fig 4A. It follows that to reduce the incidence of Chagas disease, one should reduce the cost C ITN as much as possible. Fig 4B shows how the incidence rates depend on the number of triatomines at home (K s ). When K s is small, the incidence rate is 0. Once K s increases above a certain threshold, the incidence rate increases rapidly, but then it decreases in K s . This agrees with [39]; they showed that it is best to have no dogs in the household (low K s ) but that once there are dogs in the  Table 1. Herd immunity is achieved when the expected cost reaches 0. The Nash equilibrium ITN use is achieved at the intersection with the dotted line (the cost of the ITN). It follows that p NE < p HI , but also p NE � p HI . https://doi.org/10.1371/journal.pntd.0008833.g002

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house, human infection declines with the number of dogs (i.e. with increasing K s ), allowing the dogs to sufficiently divert vectors away from the humans. In [11] they also conclude that reducing the population of triatomines and keeping domestic animals out of the households is the best way to decrease the risk of human infections. Table 2 shows the sensitivity indices of p HI , p NE , the difference p HI − p NE and the incidence i on model parameters. Since p NE and p HI are very close to each other, their sensitivity indices are almost identical. We note that, if we disregard human birth rate, α, that cannot be easily individually adjusted, the herd immunity level p HI is most sensitive to the vector birth rate β,  Table 1. For those parameters, p HI = 0.5026 and p NE = 0.5017.
https://doi.org/10.1371/journal.pntd.0008833.g003 PLOS NEGLECTED TROPICAL DISEASES settlement carrying capacity K s and the transmission rates a 0 and b 0 . In all cases, the sensitivity index is about 0.5 (or − 0.5 in the case of β), meaning that 1% increase of the parameter causes p HI to increase (decrease in the case of β) by about 0.5%. Since p NE increases (decreases) slightly more than p HI , the sensitivity index of p HI − p NE and of the incidence rate have reverse signs. The incidence rate is most sensitive to the cost of the ITNs, C ITN , and the vector birth rate, β. A decrease of the ITN cost causes the incidence rate to decrease. The dependence of β is more complex. The disease is endemic only for medium values of β 2 (0.009, 0.068); there are not enough vectors for low β (in fact no vectors for β < μ) or not enough infected vectors for high β. In the endemic state, there is a critical birth rate β 0 � 0.018 = μ + ω where reducing β below β 0 , while still having it above 0.009, may actually increase the incidence rate. See Fig 4C.

Conclusions and discussion
In this paper, we modeled Chagas disease dynamics using the compartmental model developed in [44]. We parameterized the model based on values found in literature. We applied a gametheoretical approach, developed by [23], and determined the optimal voluntary use of  Table 1.
https://doi.org/10.1371/journal.pntd.0008833.g004 Table 2. The sensitivity analysis. The sensitivity index SI y of a variable y on a parameter x was calculated as ð x y Þ � @y @x À � , see for example [72]. The numbers were rounded to the three decimal places. Parameters are as specified in Table 1. The sensitivity index − 0.5 means that a 1% increase of a parameter value x will result in the 0.5% decrease of the variable y.

PLOS NEGLECTED TROPICAL DISEASES
insecticide treated nets (ITNs) to prevent the spread of infection through vector bites. We validated our model by predicting incidence rates that closely track the real incidence rates in Latin America, Mexico and Columbia. Our results confirm that individuals behave rationally and weigh the risks of insect bites against the cost of ITNs. Our model gives two main predictions. We show that to completely eradicate the vector transmission through the voluntary use of ITNs, the cost of ITNs should be as low as possible. We also show that coupling ITN use with other means of vector control to decrease the vector presence in the households is very effective. On the other hand, in agreement with [39], if one cannot reduce the vector's presence (or the vector birth rate) in the household below a critical threshold, increasing the vector presence may lead to a slightly lower incidence rate.
The use of ITNs has many advantages: it protects against multiple diseases such as malaria, leishmaniasis, and dengue [18], and it can be easily integrated into community health work [58,76]. Compared to residual insecticide spraying, the use of ITNs does not require qualified spraying teams and it also requires considerably less insecticide [77]. Moreover, [78] showed that residual insecticide spraying was less effective than expected mainly because of moderate insecticide resistance and the limited effectiveness of selective treatment of infested sites only. The vectors can navigate past the nets; but most vectors that traversed the nets were early-stage nymphs, which are less likely to carry T. cruzi [79]. Furthermore, the spread of triatomine insects can be slowed down even if ITNs is used only on animal cages [79].
Our model can be extended in several ways. One can include disease related mortality which was omitted here for the sake of simplicity. A disease related mortality could cause a backward bifurcation and an existence of endemic equilibria even for R 0 < 1 [80,81]. One can also relax the assumption about vector migration and allow the vectors to migrate from the settlement. Finally, one can consider transmission other than between vectors and humans. Yet the findings of our simple model agree with more complex models such as [11,39,42,50] which found that the best way to decrease risk of human infection is by decreasing the number of triatomine in a given area and reducing the number of domestic animals.
Although every math model has many limitations, these models can help us to understand diseases and implications of various control measures. We hope that this model helps to serve as a tool in showing the importance of ITN use to prevent Chagas disease and to minimize the domestic transmission in Latin America as stated by the London Declaration.