Challenges in modeling complexity of neglected tropical diseases: a review of dynamics of visceral leishmaniasis in resource limited settings

The following protocol was established for this review. We searched Google Scholar, PubMed and Web of Knowledge to find articles focusing on “mathematical modeling of vector-borne diseases” (VBD), “Visceral Leishmaniasis”, “Kala-azar”, “risk factors of Leishmaniasis”, and “dynamics of Leishmaniasis”. The goal was to identify factors that have been modeled and studied for understanding VL dynamics and inform modeling strategies that needs to be undertaken in order to systematically address challenges for controlling (eventually eliminating) VL from resource limited regions. For the theoretical literature, we included epidemiological and public health studies that addressed important technicalities of modeling the dynamics of vector borne infectious diseases, including, economic models, uncertainty and sensitivity analysis, and optimization models. We also considered studies that evaluate impact of socio-economic conditions on patterns of tropical diseases.

Our search criterion was: articles that contain visceral leishmaniasis (VL) AND VBD model AND risk factors of VL. Our eligibility criteria were articles that: (i) were published in peer reviewed journals and focused on countries affected with neglected tropical diseases; (ii) aimed to model risk factors for similar (to VL) vector-borne diseases; (iii) employed mechanistic models to understand VL dynamics; and (iv) establish critical risk factors from cross-sectional and longitudinal empirical VL (in South Asia) studies. A scheme of the selection in outlined in a flowchart in the Additional file 1: Figure S4. To identify data-supported risk factors, we reviewed the literature based on searches using the term visceral leishmaniasis with the subheading risk factors, and the geographic terms India or Bangladesh or Nepal. Articles published from 1985 through September, 2016 in English were included. We included all studies that explicitly addressed factors associated with risk of VL in South Asia, the objectives, design, outcome measures and analyses were clearly described, and judged to be adequate based on the methods and data presented. A formal comparison for studies on risk factors was impossible due to scarce data from different time points, and non-comparable study designs and implications. We excluded studies that did not incorporate disease transmission dynamics such as studies with cost-effectiveness, and decision tree type models. We also excluded opinion papers, review articles, qualitative VL reports and modeling studies comparing different regions. More details on how many articles were retrieved by the search and how many were excluded on each topic were given in first paragraph of Results section.

The depth and complexity of the socio-economic challenges of a neglected tropical disease dynamics at the grassroots level may seem disparate when considering them individually. Thus, to better comprehend the nature of obstacles in the transmission process, we classified some of the key issues into the six categories viz., atmosphere, access, availability, awareness, adherence and accedence (6 A-s). We note that these 6 As in turn can be grouped into either inculcation or infrastructure (2 I-s) (Fig. 1). Endemicity of VL in the region is fueled by the deficiency in the 2 I-s, which in turn stems from the sheer poverty in the affected area. Ultimately, it is the poverty which cannot bear the cost of prevention, and leads to a vicious cycle of VL dynamics there. In this review we focus on the state of Bihar, which holds around 80% of India’s VL cases [17], and its neighboring countries of Nepal and Bangladesh where the disease dynamics are similar. We proceed to describe the relevance of this 6 A-s. Some key features defining the 6 A-s are represented in Fig. 1.

Despite the incalculable harm and countless challenges leishmaniasis inflicts on populations around the globe, only a handful of publications address the problems from a mathematical modeling perspective. In fact, a recent review by Rock et al., which tabulates all mathematical models of VL, found only twenty-four articles using mathematical models for VL, several of which used the same base model structure [46]. Of these twenty-four articles, only seven addressed VL in the Indian subcontinent. Arguably, one of the greatest modeling challenges is the limited understanding of the leishmania pathogen, the sandfly vector, and how disease manifests in humans. Dye et al. [47] spear-headed the application of mathematical models to leishmania dynamics. The authors developed a simple discrete-time model with Susceptible, Infected, and Resistant humans to study the mechanism behind inter-epidemic periods observed in VL cases between 1875 and 1950 in Assam, India. Counter to the existing theory of the time, the model demonstrated that the observed inter-epidemic patterns could be explained by intrinsic factors in leishmania transmission. This modeling effort also stressed the significance of PKDL and sub-clinical infections in determining whether a region will have endemic or epidemic leishmaniasis. A few years later, Hasibeder et al. [48] published a compartmental delay-differential equation model for canine leishmaniasis. This model accounts for two types of dogs: those that will develop symptoms, and those that will remain asymptomatic, following infection by a sandfly. The model also explicitly describes the infection dynamics of the sandfly vector and considers a fixed delay representing the extrinsic incubation period. The authors take a heuristic approach to derive a formula for the basic reproduction number \(R_0\), the number of secondary sandfly infections resulting from a single infected sandfly, in an otherwise fully susceptible population. Although this model addresses two important aspects of the natural history of the disease that may be extended to human VL, namely the asymptomatic human and infected vector populations, the model does not consider the asymptomatic population to be an infectious reservoir, assumes constant human and vector population sizes, and omits the effects of seasonality. The model does, however, introduce heterogeneous biting, determined by a dog’s “occupation”. The mathematics developed in [48] was applied to age-structured serological data for the dog population in Gozo, Malta in [49], and provided estimates for \(R_0\). This modeling work was extended in [50] to include zoonotic transmission, that is, humans, dogs, and sandflies, were explicitly modeled. Dye conducted a sensitivity analysis to determine which of three control measures would be most effective in decreasing disease prevalence. Their results suggest vector control and vaccination of the human or dog population would be more effective than treating or killing infected dogs. However, in their model, they assumed that any treatment of dogs results in full recovery and the efficacy of vaccination is extremely high. These assumptions may not be completely realistic.

More recently, Stauch et al. developed a more comprehensive model of VL for the Indian subcontinent [51], and later extended it to include drug-resistant and drug sensitive L. donovani parasites, with a focus on Bihar, India [52]. The model proposed by Stauch et al. [51] extended the basic susceptible-infectious-recovered (SIR) model structure for the human population by segmenting the infected stage into five distinct stages according to an individual’s infection status determined by the results of three diagnostic markers. These diagnostic markers were (1) PCR, the earliest infection marker, (2) DAT, which measures antibody response, and (3) LST, which can detect cellular immunity. The model also includes treatment of symptomatic VL cases, treatment failure, relapse characterized as PKDL, PKDL treatment, and HIV-co-infection (described in their Additional file 1: Figure S4). The sandfly population is modeled using an SEI (Susceptible-Latent-Infectious) model. Treatment of VL is divided into first and second-line treatment, and treatment-induced mortality caused by drug-toxicity is considered. The model was fit to data from the KalaNet trial using Maximum Likelihood. The authors explored several intervention strategies, including treatment, active case detection, and vector control. Although the authors warn that their model assumes homogeneous transmission, ignoring possible clustering of cases within affected households, their modeling approach and parameter estimation argues that the large asymptomatic reservoir precludes the ability for a treatment-only control program to attain the desired target of less than 1 case per 10,000 individuals annually. Vector-based control is much more promising, but the authors estimate it can only reasonably reduce VL incidence to 18.8 cases per 10,000. Consequently, the authors emphasize the need for active case detection, effective treatment of PKDL, and vector control to achieve VL elimination.

Based on the model in Stauch et al. [51], and following up on their finding that treatment of VL does little to reduce transmission, Stauch et al. investigated the uncertainty in their parameter estimates and explored the efficacy of different vector-based control measures [53]. They estimated that \(R_0\) for VL is approximately 4.71 in India and Nepal, and that reducing the sandfly population by 79% via reduction of breeding sites, or reducing the sandfly population by 67% through increasing sandfly mortality, are both sufficient to eliminate the L. donovani parasite in the human population. The authors argue that recent evaluations of IRS (indoor residual spraying) efficacy suggest that elimination should be possible, with the caveat that the situation may change if insecticide resistance emerges. However, vector management using LLIN’s (long-lasting insecticide-treated nets) or EVM (environmental management) would not be sufficient to achieve elimination. The authors emphasize the need to study infection rates, the parasite dynamics in both the human and vector population, animals serving as alternate hosts or potentially infectious reservoirs, and the contribution of the asymptomatic population. Furthermore, Stauch et al. suggest extensions of the deterministic modeling framework to include heterogeneity in population and seasonality.

Stauch et al. [52] extended the model from their previous study [51] to include both drug-resistant and drug-sensitive parasites. The authors considered five mechanisms by which the fitness of the resistant strain may differ from the sensitive strain: (1) increase probability of symptoms, (2) increase parasite load, (3) increase infectivity of asymptomatic humans, (4) increase transmission from symptomatic and asymptomatic host to vector, (5) increase transmission from vector to host. Simulations of this extended model indicate that a treatment failure rate over 60% is required to explain observations in Bihar. Furthermore, observations in Bihar cannot be explained without assuming an increase in fitness in resistant parasites. The authors explain that it is more likely that the necessary additional fitness is transmission-related rather than disease-related. Unfortunately, their results also suggest that once a more fit resistant parasite has been introduced, that parasite will eventually exclude the sensitive parasite, even in the absence of treatment.

The work of Mubayi et al. is the first to use a rigorous, and dynamic model to estimate underreporting of VL cases at the district level in Bihar, India [3]. The authors designed a staged-progression model, composed of a system of nonlinear, coupled, ordinary differential equations. In a typical SIR type epidemic model, the inbuilt assumption is that an individual stay in each infection category for an exponentially distributed waiting time. The stage-progression model considers a series of same infection category (for examples, \(I_1, I_2,\ldots ,I_n\) for infectious category I), each with same average waiting rate. It exploits the fact that the sum of n independent exponential distributions with rate parameter \(\lambda\) is a gamma distribution with shape parameter n and scale parameter \(1/\lambda\) (\(\Gamma (n,1/\lambda )\)), which helps in capturing the observed variability in waiting time in a epidemiological category such as incubation period, infectious period, and treatment duration. Furthermore, the authors address the differences between public and private health care facilities in their treatment and reporting practices by assuming a fraction of infected individuals p seek treatment in public health care facilities, and the remaining proportion seek treatment in private clinics. Building an empirical distribution for this parameter p and deriving a relationship between model parameters and monthly reported incidence data allowed the authors to estimate the degree of underreporting for each district for the years 2003 and 2005. This model analysis informed by incidence data revealed that districts previously designated as low-risk areas for VL are actually likely to be high-risk: the true burden masked by underreporting.

ELmojtaba et al. presented a more classical approach to modeling VL, with a focus on Sudan, in [54‐56]. Because leishmaniasis in the Sudan is zoonotic, the authors included the dynamics of an animal (rodents and dogs) reservoir in [54]. This baseline model is extended in [55] to address parasite diversity, and in [56] to explore the potential impact of mass vaccination in the presence of immigration.

More recently, Sevá et al. [57] developed a mathematical model for human and canine zoonotic VL in Brazil. The focus of this study was to test the efficacy of existing canine-based VL prevention and control methods: insecticide-impregnated dog collars, culling, and vaccination. By optimizing each of these control strategies in an ordinary differential equation model, while accounting for their respective costs, the authors were able to recommend a 90% coverage of the dog population with insecticide-impregnated collars as a control strategy that is easy to adopt and could, over time, eliminate VL in the region. All of these modeling efforts (summarized in Table 1) have either contributed to our understanding of VL or highlighted the need for better data to construct and validate future models of VL. However, there are currently no models, to the best of our knowledge, that attempt to link socio-economic factors, like the 6 A’s discussed in “Identified VL risk factors: challenges for leishmaniasis transmission dynamics in resource-limited regions” section, to VL disease transmission.

In this section, we provide some examples of published mathematical modeling studies (summarized in Table 1) where researchers have attempted to incorporate some of the factors mentioned above and studied their role in the transmission dynamics of infectious and physiological diseases. Existing models of visceral leishmaniasis, though limited in number as compared to diseases like malaria and dengue, have incorporated some of the biological complexity, contributing to a more developed understanding of the disease. However, to formulate applicable control measure recommendations with cost-estimation, models which can simultaneously incorporate the discussed risk-factors explicitly would be necessary. Many of the techniques to incorporate these factors individually can be drawn from the literature regarding heavily studied diseases like HIV, malaria, and tuberculosis [58‐60].

The results of models addressing adherence to treatment, adaptive human behavior, and resource constraints emphasize the need to bridge the gap between socio-economic factors and existing mathematical modeling frameworks. Models of other vector borne diseases have captured some of the critical factors for visceral leishmaniasis (“Models of infectious diseases incorporating socio-economic risk actors” section). However, there is need for a more comprehensive set of frameworks that can incorporate local VL risk factors at multiple scales in the same dynamical model. The 6 A’s should be systematically incorporated into VL model frameworks to assess the sensitivity of VL dynamics to these six socio-economic factors. A schematic diagram depicting one such inter-dependent modeling framework is given in Fig. 3. Failing to address factors that result in significant changes in disease dynamics may result in models that do not effectively inform public health policy. Likewise, models that do not acknowledge resource constraints may lead to infeasible control policies.