A game-theoretic model of lymphatic filariasis prevention
A game-theoretic model of lymphatic filariasis prevention
Rychtář, Jan;Taylor, Dewey
2022-09-22 00:00:00
Lymphatic filariasis (LF) is a mosquito-borne parasitic neglected tropical disease. In 2000, WHO launched the Global Programme to Eliminate Lymphatic Filariasis (GPELF) as a pub- lic health problem. In 2020, new goals for 2030 were set which includes a reduction to 0 of a1111111111 a1111111111 the total population requiring Mass Drug Administrations (MDA), a primary tool of GPELF. a1111111111 We develop a mathematical model to study what can happen at the end of MDA. We use a a1111111111 game-theoretic approach to assess the voluntary use of insect repellents in the prevention a1111111111 of the spread of LF through vector bites. Our results show that when individuals use what they perceive as optimal levels of protection, the LF incidence rates will become high. This is in striking difference to other vector-borne NTDs such as Chagas or zika. We conclude that the voluntary use of the protection alone will not be enough to keep LF eliminated as a OPENACCESS public health problem and a more coordinated effort will be needed at the end of MDA. Citation: Rychtař J, Taylor D (2022) A game- theoretic model of lymphatic filariasis prevention. PLoS Negl Trop Dis 16(9): e0010765. https://doi. org/10.1371/journal.pntd.0010765 Author summary Editor: Keke C. Fairfax, University of Utah, UNITED STATES We adapt a compartmental ODE model of lymphatic filariasis (LF) transmission and Received: May 2, 2022 focus our attention on what happens after Mass Drug Administrations (MDA) is termi- nated. We add a game-theoretic component to the model and study whether LF transmis- Accepted: August 23, 2022 sion can be substantially interrupted by voluntary use of personal protection strategies Published: September 22, 2022 such as using insect repellents. We identify optimal voluntary protection levels and dem- Copyright:© 2022 Rychta ´ř, Taylor. This is an open onstrate that LF incidence rates will become too high. access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. 1 Introduction Data Availability Statement: All data are in the manuscript. Lymphatic filariasis (LF), also known as elephantiasis, is a mosquito-borne parasitic disease Funding: The author(s) received no specific caused by microscopic filarial roundworms Wuchereria bancrofti, Brugia malayi and Brugia funding for this work. timori [1]. The roundworms are transmitted to humans by mosquitoes of the genera Aedes, Anopheles, Culex and Mansonia [1]. LF is one of the leading causes of chronic disability world- Competing interests: The authors have declared that no competing interests exist. wide [2]. PLOS Neglected Tropical Diseases | https://doi.org/10.1371/journal.pntd.0010765 September 22, 2022 1 / 18 PLOS NEGLECTED TROPICAL DISEASES A game-theoretic model of lymphatic filariasis prevention Fig 1. World map of LF and MDA status in 2020. Data collected from [7] and map was made with the aid of borders.m file [8] in MATLAB. https://doi.org/10.1371/journal.pntd.0010765.g001 In 2000, WHO launched its Global Programme to Eliminate Lymphatic Filariasis (GPELF) as a public health problem [3]. The primary strategy for LF control and elimination is the WHO recommended preventive chemotherapy [4]. The entire population at risk is treated by mass drug administration (MDA) for at least five consecutive years. In 2020, 863 million peo- ple in 50 countries were living in areas that require MDA [3]; see Fig 1. At the same time, GPELF set new goals for the new NTD Road Map (2021-2030) that include reduction to 0 of the total population requiring MDA and 100% of endemic countries implement post-MDA or post-validation surveillance [3]. MDA has already ended and was successful in Dominican Republic [5] but it was not so successful in Haiti [4] and American Samoa [6]. It is therefore important to plan ahead and estimate what can happen at the end of MDA. Mathematical modeling is a standard and indispensable tool for NTDs elimination efforts [9, 10]. The main mathematical models of LF transmission and control are LYMFASIM [11], EPI- FIL [12, 13] and TRANSFIL [14]. The models and their implications for the LF control and elimination through MDA are discussed in [15, 16] or [17]. Furthermore, [18] and [19] created an SI-SI model to investigate the long-term effects of targeted medical treatment in Indonesia. [20] developed an SEI-SI model which was extended by [21] to include possible vaccination and chemoprophylaxis. [22] developed model with vaccination. [23] constructed an SEIQ-SI LF model with quarantine and treatment as control strategies. Also, [24] modeled LF-tuberculosis coinfections and [25] considered global stability and backward bifurcation of their LF transmis- sion model. The cost-effectiveness of different intervention strategies is considered in [26]. In our paper, we adapt a SEI-SI compartmental model by [27] which investigated the effect of MDA on LF transmission in the Philippines. Unlike previous LF modeling papers, we focus our attention on what happens when MDA is terminated and no longer in place. We are inter- ested to see whether the LF transmission can be substantially interrupted by voluntary use of personal protection strategies such as using insect repellents. The research is inspired by [28] PLOS Neglected Tropical Diseases | https://doi.org/10.1371/journal.pntd.0010765 September 22, 2022 2 / 18 PLOS NEGLECTED TROPICAL DISEASES A game-theoretic model of lymphatic filariasis prevention and [29] who showed that a voluntary use of DEET can help eliminate dengue or zika virus infections. We apply the game-theoretic framework developed in [30] and subsequently applied to many diseases, including COVID-19 [31]; see [32] for a recent review. The framework is useful in instances when individuals choose to protect against the mosquito bites and consequently the disease on their own rather than when there are centralized efforts directed towards disease elimination or mosquito control [33]. It has been long established that individuals act in a way that maximizes their self-interests, rather than the interests of the entire group [34]. Voluntary disease protection is prone to free-riding because it produces public goods (reduction of dis- ease prevalence) that have the following two main characteristics [35]: non-rivalry (consump- tion of a good by one person does not affect the quantities consumed by other individual) and non-exclusion of consumption (impossible to restrict the benefits to certain individuals). The “free-riders” avoid the costs associated with disease prevention while benefiting from other individuals’ actions [36]. Individuals try to balance the real or perceived costs of disease protec- tion against the costs of the disease [37]. The outcomes of different choices of a specific indi- vidual depend on the actions chosen by the rest of the population since the behavior of the rest of the population determines the prevalence of the disease and thus the risk of infection to a focal individual. A solution of this game is a concept of Nash equilibrium, a strategy from which nobody prefers to deviate. We identify such optimal voluntary protection levels and demonstrate that under such con- ditions, LF incidence rates become too high. Thus, we conclude that voluntary use alone is not a sufficient tool to keep LF eliminated as a public health concern after the end of MDA. 2 Mathematical model In this section we build a mathematical model for the voluntary use of insect repellents and other personal protection means to prevent LF. We first introduce the compartmental model of LF transmission. Then, we add the game-theoretic component that will allow us to investi- gate individuals’ optimal decisions on choosing their level of protection. Finally, we will cali- brate the model based on data from the literature. 2.1 Compartmental model We consider the situation at the hypothetical termination of the MDA treatments. We adapt an ODE compartmental model for LF transmission that was introduced in [27]. Their com- partmental model simplified by the absence of MDA but extended by the presence of exposed vectors is shown and described in Fig 2. The parameters are explained in Table 1. As derived in 3.1, the effective reproduction number is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b y y n as vh hv v ð1Þ R ¼ : b b ðb þ aÞðb þ sÞ h v h v When R < 1, then the disease-free equilibrium is locally asymptotically stable and when R > 1, then the endemic equilibrium is locally asymptotically stable [38]. Furthermore, if R > 1, then the force of infection at the endemic equilibrium is given by R 1 l ¼ by : vh vh by b þ s ð2Þ vh 2 v þ R b n s h v PLOS Neglected Tropical Diseases | https://doi.org/10.1371/journal.pntd.0010765 September 22, 2022 3 / 18 PLOS NEGLECTED TROPICAL DISEASES A game-theoretic model of lymphatic filariasis prevention Fig 2. (a) Life cycle of W. bancrofti. Image courtesy of Public Health Image Library, Centers for Disease Control and Prevention (https://phil.cdc.gov/Details.aspx?pid= 3425). (b) Scheme of the ODE compartmental model for LF transmission from [27] with no treatment (after the termination of MDA). The human population is divided into uninfected U , latent L , and infectious I ; the total population is N = U + L + I . Mosquitoes are either uninfected U , exposed E , or infected I ; the total h h h h h h h v v v population is N = U + E + I . Solid arrows represent the transition of humans and mosquitoes between different states of infection. The letters next to the arrows v v v v specify the rates of the transitions. All new members of both populations enter their respective uninfected classes at per capita rates b and b . Both humans and h v mosquitoes leave their respective population through natural death at per capita ratesδ andδ . The uninfected mosquitoes become infected at rate l ¼ by . The h v hv hv uninfected humans become latent at the rate l ¼ by , the force of infection. The latent individuals progress to infectious at rateα. The exposed vectors become vh vh infectious at rate σ. Dashed lines represent the transfer of parasites from human to mosquito and vice versa through a mosquito bite. https://doi.org/10.1371/journal.pntd.0010765.g002 2.2 Game-theoretic component At this point, we add a game-theoretic component to study individual prevention strategies and introduce the following game inspired by the framework introduced in [30]. The players of the game are uninfected individuals who repeatedly chose to protect them- selves against mosquito bites. Their strategy is given by a number c2 [0, 1] that specifies a pro- portion of the time the individual uses personal protection such as insect repellent to prevent mosquito bites. The strategy c influences the mosquito biting rate,β =β(c). For illustrative pur- poses, we assumeβ(c) =β (1 − c) whereβ is the maximal mosquito biting rate without any 0 0 protection. However, our analysis and qualitative results will stay valid for any non-negative decreasing functionβ(c) satisfyingβ (c)� 0 on [0, 1]. The protection does not come for free and we assume that to use a strategy c, the individual has to pay the cost k(c). In our examples, we assume k(c) =κc whereκ is the cost of complete and maximal protection. However, our analysis and qualitative results stay valid for any non- negative increasing function k(c) satisfying k (c)� 0 on [0, 1]. We assume that the cost k(c) is PLOS Neglected Tropical Diseases | https://doi.org/10.1371/journal.pntd.0010765 September 22, 2022 4 / 18 PLOS NEGLECTED TROPICAL DISEASES A game-theoretic model of lymphatic filariasis prevention Table 1. Model parameters. The rates are per capita per week. The parameter values are discussed in Section 2.3. The range shows the bounds we used in sensitivity and uncertainty analysis in Section 4.1. Symbol Description Value Range −4 −4 −3 b Human birth rate 6 × 10 [10 , 10 ] −4 −4 −3 δ Human natural death rate 4.2 × 10 [10 , 10 ] δ Mosquito natural death rate 0.1 [0.05, 0.15] b Mosquito birth rate δ + b −δ v v h h c Proportion of the time the individuals use protection variable in [0, 1] β Maximal mosquito bite rate 1 [0.5, 1.5] β(c) Mosquito bite rate when protecting at c β (1 − c) −4 −3 θ Probability of transmission from mosquito to human 7.5 × 10 [0, 10 ] vh θ Probability of transmission from human to mosquito 0.37 [0.2, 0.4] hv α Progression rate from L to I 0.0288 [0.02, 0.05] h h σ Progression rate from E to I 2/3 [0.1, 1] v v n Number of mosquitoes per human 3 [0, 5] κ Cost of maximal protection (relative to cost of LF) 0.1 [0, 1] k(c) Cost of protection (relative to cost of LF) when using c κc https://doi.org/10.1371/journal.pntd.0010765.t001 relative to the cost of the disease, i.e., k(c) = 1 means that the cost of the protection equals the cost of the disease. The solution of the game, called the Nash equilibrium, is the population-level value c at NE which no individual can increase their own benefits by deviating from the population strategy. The individual’s benefits, or payoffs, depend on the individual’s strategy but also on the prevalence of LF in the population, i.e., on the strategies of other players. Following [30], we assume that all individuals are provided with the same information such as prevalence of LF in the population, the cost of contracting LF, and the cost of protection. We will also assume that they all use the information in the same and rational way to assess costs and risks. 2.3 Model calibration We adopt most parameter values from [27] and references therein. All rates are expressed per −4 capita per week. We set the human birth rate as b = 6 × 10 and the human death rate asδ = h h −4 4.2 × 10 to agree with the population dynamics of the Caraga region, the Philippines. As in [39], we set the mosquito death rate asδ = 0.1. In line with [27], to keep the mosquito popula- tion to be a constant multiple of N , we set b =δ + b −δ . The number of mosquitoes per h v v h h humans was estimated as n = 3. We assume the progression rate from L to I isα = 0.0288 v h h [17]. Also, we assume the maximal mosquito bite rate isβ = 1 [39]. The probability of trans- mission from human to mosquitoes is θ = 0.37 [13]. In vectors, L1 stage larvae needs 1.5 hv weeks to mature into infectious L3 stage larvae [40], i.e., the rate of progression from E to I is v v σ = 2/3. We differ from [27] by setting the probability of transmission from mosquito to human as −4 −4 θ = 7.5 × 10 = 6.6 × 1.13 × 10 where 6.6 is the mean saturation level of L3 larvae in mos- vh −4 quitoes [41] and 1.13 × 10 is the proportion of L3 filarial parasites entering a host which −4 develop into adult worms [13]. We note that [27] used a value θ = 1.13 × 10 , but that gives vh R � 1:3. Our values of θ yields R � 3:43. Such a value is more in line with [42] which esti- vh e e mates R values for LF to be between 2.7 and 30. Finally, we assume that the cost of (complete) protection, relative to the cost of LF, is given byκ = 0.1. We arrived at this estimate as follows. In 2000, a chronic LF patient could lose up to PLOS Neglected Tropical Diseases | https://doi.org/10.1371/journal.pntd.0010765 September 22, 2022 5 / 18 PLOS NEGLECTED TROPICAL DISEASES A game-theoretic model of lymphatic filariasis prevention $50 annually due to LF [43]. We adjusted it to $100 annually for today’s value. At the same time, the cost of full protection by DEET was estimated in [29] to about $10. We investigate the dependence of our result on the parameter values in Section 4.1. 3 Analysis To solve the game, i.e., find the Nash equilibrium and the optimal voluntary protection level, we assume that all players use the same strategy, c , and only the strategy of the focal player, pop c, may vary. We assume that human and mosquito populations are large enough so that the behavior of a single individual does not significantly affect the number of infected mosquitoes. The effective reproduction number depends on c . Specifically, pop sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b ðc Þy y n as pop vh hv v ð3Þ R ðc Þ ¼ : e pop b b ðb þ aÞðb þ sÞ h v h v Assumingβ(c ) =β (1 − c ), we get pop 0 pop R ðc Þ ¼ ð1 c ÞR ð0Þ: ð4Þ e pop pop e When R ðc Þ � 1, the population will reach disease-free equilibrium. When e pop R ðc Þ > 1, i.e., when c 2 [0, c ] where e pop pop max c ¼ 1 ; ð5Þ max R ð0Þ the population will reach the endemic equilibrium. Here, c is the maximal protection level max at which R � 1 and the disease-free equilibrium is not stable. We will assume R ð0Þ > 1 and e e c 2 [0, c ] as otherwise the disease is eliminated and thus there is no need for a further pop max analysis. As common in game-theoretical models, we will assume that the population actually is in the endemic equilibrium [30]. An uninfected focal individual in U using a strategy c when everyone else uses a strategy I I v v c contracts the infection and moves to L at rate bðcÞy . Note that the ratio i ¼ pop h vh v N N h h depends on the strategy c , see Eq (47) in Section 3.1. The rate is thus given by pop l ðc; c Þ ¼ bðcÞy i ðc Þ ð6Þ vh pop vh v pop where R ðc Þ