https://orcid.org/0000-0002-9579-4522
Figures
Abstract
Visceral Leishmaniasis (VL) is a deadly, vector-borne, parasitic, neglected tropical disease, particularly prevalent on the Indian subcontinent. Sleeping under the long-lasting insecticide-treated nets (ITNs) was considered an effective VL prevention and control measures, until KalaNet, a large trial in Nepal and India, did not show enough supporting evidence. In this paper, we adapt a biologically accurate, yet relatively simple compartmental ordinary differential equations (ODE) model of VL transmission and explicitly model the use of ITNs and their role in VL prevention and elimination. We also include a game-theoretic analysis in order to determine an optimal use of ITNs from the individuals’ perspective. In agreement with the previous more detailed and complex model, we show that the ITNs coverage amongst the susceptible population has to be unrealistically high (over 96%) in order for VL to be eliminated. However, we also show that if the whole population, including symptomatic and asymptomatic VL cases adopt about 90% ITN usage, then VL can be eliminated. Our model also suggests that ITN usage should be accompanied with other interventions such as vector control.
Citation: Davis C, Javor ER, Rebarber SI, Rychtář J, Taylor D (2024) A mathematical model of visceral leishmaniasis transmission and control: Impact of ITNs on VL prevention and elimination in the Indian subcontinent. PLoS ONE 19(10): e0311314. https://doi.org/10.1371/journal.pone.0311314
Editor: Bibi Razieh Hosseini Farash, Mashhad University of Medical Sciences, IRAN, ISLAMIC REPUBLIC OF
Received: August 26, 2023; Accepted: September 17, 2024; Published: October 4, 2024
Copyright: © 2024 Davis et al. This is an open 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.
Data Availability: All relevant data are within the paper and its Supporting information files.
Funding: C.D., E.R.J., and S.I.R. were supported by the VCU REU program in mathematics funded by the National Science Foundation grant number DMS1950015 awarded to D.T. The work was also supported by the National Science Foundation grant number DMS 2327790 awarded to D.T. The funders had no role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: Jan Rychtar: I have read the journal’s policy and the authors of this manuscript have the following competing interests: I serve on the editorial board of PLOS ONE. This does not alter our adherence to PLOS ONE policies on sharing data and materials. Other authors declare no conflict of interest.
Introduction
Visceral Leishmaniasis (VL) is a deadly, vector-borne, parasitic, neglected tropical disease found primarily in the Indian Subcontinent, East Africa, and Brazil [1]. In the Indian subcontinent, which accounts for over two thirds of VL cases [2], VL is caused by parasites belonging to Leishmania donovani complex [3]. There is no known animal reservoir and parasites are transmitted from human to human by female sandfly Phlebotomous argentipes [4].
Because VL treatment is expensive and sometimes ineffective [5], VL prevention is critical. There is currently no available vaccine [6, 7] and the prevention focuses on active case detection [8], vector control [9], and bite prevention [10]. The efficacy of long-lasting insecticide-treated nets (ITNs) in the prevention of VL was evaluated by KalaNet, a cluster randomized controlled trial in Nepal and India [11]. Since the distribution of ITNs during the KalaNet trial did not reduce the risk of L. donovani infection or clinical VL [12], ITNs are not part of the standard government VL control programme in India [13].
Mathematical modeling is now a common and necessary tool used to understand epidemics and disease elimination efforts [14, 15]. In contrast to most neglected tropical diseases [16], there are many different models of VL; see for example [17–20] for recent reviews. A comprehensive model of VL for the Indian Subcontinent was developed in [21]. [13] built on [21] to quantify the role of ITN use in VL transmission. They have shown that in order to eliminate VL, the ITN usage would have to be above 96%. For comparison, at the end of KalaNet trial, the ITNs use was still only around 91% [12].
The model of [13] included twelve human and three sandfly compartments and a long list of parameters. The purpose of this paper is to use a simpler model with a shorter list of parameters to see whether this could yield to different predictions. This is in line with [22] who discusses the tension between model realism and mathematical tractability and proposed rules for simplifying the models in an approach that can be particularly beneficial for modeling diseases affecting developing countries. We also explicitly investigate how the ITNs usage amongst the infected population influences the outcomes.
Model
We adopt a model with seven human and two sandfly compartments from [17] and extend in by a game-theoretic component which incorporates the voluntary use of insecticide treated nets (ITNs) as in [13]. This significantly simplifies the model from [13] while still being in line with a biologically realistic model from [17]. The underlying disease transmission model makes all of the standard assumptions and has all of the typical limitations of compartmental models as discussed, e.g., in [23].
Disease transmission model
We will consider the human and sandfly population. The human population is divided into seven groups: susceptible (S), exposed (E), infected asymptomatic (IA), infected symptomatic (also called Kala-Azar, IK), infected dormant (ID), PKDL (IP) and fully recovered individuals (R). The total human population is N where N = S + E + IA + IK + ID + IP + R.
The sandfly population is divided into susceptible (SF) and infectious (IF). The total sandfly population is NF = SF + IF. For simplicity, we assume NF = nFN; here nF is the number of sandflies per person.
The dynamics is as follows. Human individuals are all born as susceptible at rate Λ. Susceptible individuals become exposed at a rate of
(1)
where (1 − p) is the proportion of the human population that does not use ITNs, β−1 is the duration of the sandfly feeding cycle, and iF is the probability that an infected sandfly infects a human.
After an incubation time , the exposed individuals become infectious asymptomatic (IA). The average duration of asymptomatic infection is
and the individuals can (a) develop Kala-Azar (with probability fAK), (b) dormant infection (with probability fAD), (c) PKDL (with probability fAP), or (d) recover (with probability fAR = 1 − (fAK + fAD + fAP).
The symptomatic infection, Kala-Azar, is a serious infection that results in an additional mortality μK. If we assume that the patient survived, the infection lasts for an average time . With probability fKD, the symptoms resolve and the individual moves to a dormant infection stage; with probability 1 − fKD, the individuals fully recover.
Cases with dormant infection develop PKDL after an average time (assuming they survived). After the PKDL infection which lasts
on average, the individuals recover. Recovered individuals maintain immunity for time ρ−1, after which they become susceptible.
Independent of VL, humans die at a rate μ from every compartment. As mentioned above, cases with Kala-Azar suffer from an additional mortality rate μK.
For simplicity, we assume that the sandfly population remains constant NF. The flies live for an average time and are born at rate μFNF as susceptible.
The susceptible sandflies become infected at rate
(2)
where β−1 is the duration of the feeding cycle, and, for x ∈ {A, K, D, P}, ix is the proportion of sandfly bites of humans in infected compartment Ix that transmits the VL infection, and px is the proportion of cases in compartment Ix that uses ITNs.
The dynamic is summarized in Fig 1. The notation is summarized in Tables 1 and 2.
Human population is shown in the top seven compartments, and sandflies at the bottom two. Solid arrows represent transitions into and out of each compartment, the formulas next to the arrows are the transmission rates. All humans are born as susceptible (S). They become exposed (E) at rate λ = (1 − p)βiFIF given in (1) when bitten by an infectious sandfly (IF). The green dotted line represents this causation. Exposed cases then become asymptomatic (IA). The asymptomatic cases can develop Kala-Azar, become dormant (ID), develop PKDL (IP) or recover (R). The Kala-Azar cases can either recover directly, or progress through dormant and PKDL stages before finally recovering. The recovered cases eventually lose immunity and become susceptible again. Sandflies are born susceptible (SF), and can become infectious (IF) at rate λF given by (2) by biting an infected human (in IA, IK, ID, IP). The blue dotted line represents this causation.
Times are in months and the rates are per capita per month. Detailed derivation can be found in [13]; here we also list original sources for the reference. “Range” is the range of values used in sensitivity analyses. The value of fAR is determined by 1 − (fAD + fAK + fAP).
Game-theoretic component of the model
We incorporate the game-theoretic component into the above transmission dynamics in the same way as done, for example, in [43].
A game is played by susceptible individuals who decide whether or not to use ITNs. As typical in these models [44–48], we assume that players are rational, act in their own self-interest, have complete information about VL, and consider only financial costs of ITNs. The players evaluate prospective costs and benefits of their options (to use or to not use the ITNs) in relation to the actions taken by others and they choose the option that maximizes their own net payoffs (benefits minus costs).
If an individual uses an ITN, they are protected against VL, but they also incur a cost of acquiring the ITN, CITN. On the other hand, when they do not use the ITN, they risk contracting VL, i.e., and, eventually, paying the cost CVL.
More specifically, let us now focus on a single individual deciding whether to use the ITN when the rest of the population uses ITNs with probability . If the population is large enough, the decision of a single individual will not have a significant impact on the disease transmission. We may thus assume that the number of infected flies
in the population does not depend on the choice of the focal individual. As in [49–51], the risk for an unprotected susceptible individual becoming exposed (rather than staying susceptible and die of natural causes) is given by
. Similarly, the probability to become exposed if the individual uses ITN for the fraction p of the time is
. In aggregate, if an individual’s ITN usage is p, the cost is given by
(3)
We note that above, the number of infected sandflies in the endemic equilibrium,
depends on
, the average ITN use in the population. Consequently, C(p) depends on
as well.
In (3), we made several implicit assumptions. First, we assumed that the cost CVL is the expected cost after getting exposed to VL. In our diagram in Fig 1, an exposed individual becomes asymptomatic with probability and then goes through various stages of VL, including PKDL (or recovers with probability fAR without ever experiencing any symptoms and incurring any costs). To fully estimate the value of CVL, all of the possible stages and costs at every stage would have to be accounted for; here we simplify the model by assuming CVL ≈ 100 to be in line with [13]. Second, we assumed that CITN is only the cost of ITN acquisition, and it does not involve the cost of ITN usage. A person that is using ITN, however infrequently, has to pay the acquisition cost, but once the ITN is acquired, we are assuming no more costs. Finally, we ignore the effect of ITN on the mortality of the sandflies; i.e., we assume that ITNs prevent transmissions between humans and sandflies, but otherwise do not affect the sandfly population.
Analysis
Equilibria of the disease dynamics
Here we show only a summary of the analysis; see S1 File for more details. There are two possible equilibria, disease-free equilibrium, DFE0 and endemic equilibrium, EE*. The disease-free equilibrium is given by
(4)
i.e.,
and SF = nFN while all other compartments are zero.
When the ITNs usage in the population is , the reproduction number,
, i.e., the average number of new infections caused by a single-infected individual in an otherwise susceptible population [14] is given by
(5)
where
(6)
and
(7)
(8)
(9)
(10)
(11)
Here, TI can be understood as an effective time spent as an infectious individual and TX, for X ∈ {E, IA, IK, ID, IP, R} is the average time spent in the corresponding compartment X.
The disease-free equilibrium is locally asymptotically stable if R0(p) < 1 and the endemic equilibrium is stable if R0(p) > 1 [52].
The endemic equilibrium is given by
(12)
and derived and expressed in closed forms in the Section Detailed calculations. For further analysis, it is important that
(13)
where
(14)
is the average time it takes to return to a susceptible compartment (given no death) after getting exposed to VL. In (14), TR is given by
(15)
By (5) and (13), is decreasing in
. This is also shown in Fig 2. Consequently, the risk of contracting VL,
also decreases with
as shown in Fig 3.
The proportion of infected sandflies in the equilibrium, is a decreasing function of the ITN population coverage
. The parameter values are as shown in Table 1. The matlab code used to generate the figures is in S2 File.
The risk of contracting VL, is a decreasing function of the ITN population coverage . The parameter values are as shown in Table 1.
Minimum ITN usage needed for VL elimination
The disease-free equilibrium is stable when R0(p) < 1. We need to find the smallest pHP ∈ [0, 1] such that when p ≥ pHP VL is in DFE. It follows from (5) that R0(p) = (1 − p)R0(0), where
(16)
is the basic reproduction number (when no one uses ITNs). As a result, the minimum ITN coverage necessary to achieve DFE is given by
(17)
Game-theoretic analysis
All individuals try to minimize C(p). Because
(18)
the minimum is achieved either for p = 0 or p = 1. Therefore, the minimum value of C(p) is either
at p = 0 or CITN at p = 1.
If , it is in an individual’s best interest to use an ITN every night. Otherwise, they should not buy or use an ITN at all. Nash equilibrium occurs when these two costs at p = 0 and p = 1 are equal, i.e. when
(19)
By (19), we must have
(20)
On the other hand, by (13),
(21)
and solving it for
yields
(22)
where
is given by (20).
Results
For the parameter values as in Table 1, the reproduction number is R0 ≈ 9.1. At the same time, the minimum level of ITN use by the susceptible population in order for VL to be eliminated is pHP ≈ 0.967 while the Nash equilibrium, i.e., the level which is optimal from the individuals’ perspective and at which no individual benefits from switching its ITN usage strategy is pNE ≈ pHP − 10−5. At the Nash equilibrium, the reproduction number is R0(pNE) ≈ 1.0002. This means that VL would be very close to being eliminated. These results agree with model prediction from [13]. Overall, our model also shows a good fit to the original KalaNet data as demonstrated in Table 3, again in line with [13].
Figs 4–6 show the sensitivity of R0, pHP and pHP − pNE on various parameters.
The parameter ranges are as in Table 1. Only parameters with sensitivity index over 7.5% are shown.
The parameter ranges are as in Table 1. Only parameters with sensitivity index over 7.5% are shown.
The parameter ranges are as in Table 1. Only parameters with sensitivity index over 7.5% are shown.
We can see from Figs 4 and 5 that as the duration of the sandfly feeding cycle, β−1, increases, R0 and pHP decreases. On the other hand, R0 and pHP are increasing in iF (the probability of VL transmission from a biting infected sandfly to humans) and iA (the probability of VL transmission from asymptomatic VL cases to sandfly). Both of these parameters are also increasing in (the lifespan of sandflies),
(sojourn times of asymptomatic infections) and nF (number of sandflies per person). The influence of other parameters is relatively negligible.
It follows from (22) that pHP−pNE is quite small. This is also illustrated in Fig 3 by the fact that the risk of contracting VL is essentially a step-wise function with “jump” occurring at pHP. Fig 6 shows that the sensitivity of pHP − pNE on various parameter values is quite small; i.e., the difference is small regardless of parameter values. We note that the difference increases most with the cost of ITN use (CITN), the duration of the feeding cycle (β−1) and the ITN usage amongst asymptomatic cases (pA) or the probability of VL transmission from Kala-Azar cases to sandflies (iK). Also, the difference is decreasing in the cost of VL (CVL), the probability of VL transmission of asymptomatic individuals to sandflies (iA) and the lifespan of sandflies ().
We specifically investigated how pHP, the level of ITN usage needed for VL elimination, depends on ITN usages amongst the infected population. We set pA = pK = pP = pD, varied these from 0 to 1 and calculated pHP from (17). The results are shown in Fig 7. It follows that pHP is staying quite high (> 90%) unless the ITN usage amongst the infected population itself is over 90%.
We assume that pA = pK = pP = pD varies in [0, 1] while other parameters are as in Table 1. The dotted line represents y = x. The two curves intersect almost exactly at 0.9; i.e., if more than 90% of the population, including symptomatic and asymptomatic VL cases, used ITNs, VL could be eliminated.
Conclusions and discussion
In this paper we investigated a model of VL transmission. We built on the model developed in [17] and incorporated the human behavior component motivated by game theory [43].
To our knowledge, the presented model and [13] are the only two models of VL which also incorporate the game-theoretical component. As demonstrated in [55], that addition provides more insight and better predictions than standard compartmental epidemiological models. Compared to other models of VL such as [17, 20, 21] the current model allows us to predict not only what ITN coverage is needed to eliminate VL, but also what coverage can be realistically achieved.
The presented model was significantly simpler than [13] which built on a comprehensive, detailed and complex model of [21] informed by results of three different tests for VL. In contrast, our model builds on a much simpler model from [17]. Yet, the outcomes of both of the models were very similar, both qualitatively and quantitatively. Moreover, as shown in Table 3, both models also agree with data. More specifically, both models predict that to eliminate VL, 96%+ of the susceptible population should be using ITN. Both models are also in agreement that increasing the time between bites and reducing the number and/or the lifespan of sandflies are the most important control measures beyond the use of ITNs. These facts demonstrate that one does not need the most detailed model to make reasonably good predictions. This is in agreement with [22] who advocates for the use of simpler epidemiological models, especially for modeling diseases affecting developing countries in order to increase the tractability of the models.
Our model also predicts that if the whole population, including symptomatic, asymptomatic and dormant VL cases, use ITNs, then the reproduction number drops below 1 and, consequently, VL can be eliminated. This is, seemingly, in contradiction with the results of the KalaNet trial during which up to 91% of the population used ITNs without a significant reduction of VL cases [12]. However, as shown for example in [56], there is often a significant time-lag between the reduction of the reproduction number and the noticeable drop of disease cases. It is therefore entirely possible that if the 91% use of ITNs in the population was sustained, the VL case reduction would follow.
From the policy-making perspective, similar to [13], even our current simpler model suggests that instead of abandoning the use of ITNs completely, the ITN use should be combined with other intervention methods, including vector control.
There are several potential directions for future research. One can incorporate the fact that ITNs do not offer complete protection as the sandflies can bite the people outside of the ITNs. One can also incorporate the seasonality if the sandflies dynamics. Finally, the decisions to use ITNs could be modeled dynamically using the imitation dynamics as done in [57] and, for example, in [58–62].
Supporting information
S1 File. This file contains detailed calculations of mathematical formulas presented in the analysis section.
https://doi.org/10.1371/journal.pone.0311314.s001
(PDF)
S2 File. This is the code we used to generate figures and perform the uncertainty and sensitivity analysis for the manuscript.
https://doi.org/10.1371/journal.pone.0311314.s002
(M)
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