Academic Editor: Jose R. Vasconcelos

Visceral leishmaniasis (VL), one of the deadliest parasitic diseases in the world, causes more than 50,000 human deaths each year and afflicts millions of people throughout South America, East Africa, South Asia, and Mediterranean Region. In 2015 the World Health Organization classified VL as a neglected tropical disease (NTD), prompting concentrated study of the VL epidemic using mathematical and simulation models. This paper reviews literature related to prevalence and prevention control strategies. More than thirty current research works were reviewed and classified based on VL epidemic study methods, including modeling approaches, control strategies, and simulation techniques since 2013. A summarization of these technical methods, major findings, and contributions from existing works revealed that VL epidemic research efforts must improve in the areas of validating and verifying VL mathematical models with real-world epidemic data. In addition, more dynamic disease control strategies must be explored and advanced simulation techniques must be used to predict VL pandemics.

Visceral leishmaniasis (VL), or kala-azar, is a protozoan disease that, second only to malaria in numbers of fatalities, afflicts millions of people worldwide [

VL is one of the most widespread human diseases, with more than 20

The use of mathematical models to describe and predict epidemic transmissions has become a recent trend in disease research area. Mathematical models intuitively exhibit complex VL transmission processes, and they measure variables and system parameters to reveal VL spreading dynamics and related dominating factors. Rapid advancements in computer technologies have resulted in computer-aided simulation that helps mathematical models directly predict future VL prevalence. Using results from model analysis, parametric estimation, and simulation experiments, researchers can study and anticipate disease transmission dynamics and identify disease control strategies to fight a VL pandemic. Consequently, an increasing number of studies have focused on mathematical modeling and corresponding analysis for VL disease dynamics. Approaches used in these studies can be generally categorized as system dynamic models, including ordinary differential equation (ODE) or partial differential equation (PDE) models, as well as statistic models, or machine learning models. The main contributions and results are concentrated in precise prediction tested by validation, determining the key parameters by sensitive analysis and analyzing the bifurcation point of the disease reproduction number.

A well-defined mathematical model can be used to develop disease control strategies that are ascertained by solving the mathematical model or using numerical experiments. Numerical control strategies are robust and reliable approaches because potential bias from empirical data is not included. Conversely, implementation of real-life control strategies can be cost prohibitive, irreversible, and difficult to apply in a large-scale format, especially for developing countries that lack public health resources. However, computer-aided simulations that compare possible control strategies derived from a mathematical model can be carried out, and they are relatively inexpensive and can be performed repeatedly to examine system sensitivity and determine optimal control parameters. Almost half of corresponding research used mathematical modeling approaches to study potential disease control strategies, including dog culling, use of insecticidal dog collars, vector controls, and insecticide spraying strategies. Using optimal control, parametric analysis, or stochastic control methods, research results provided well-developed guidelines for disease control centers to prevent or mitigate a VL pandemic.

The rest of this paper is organized into comprehensive sections.

Pigott et al. collected and summarized prevalence data reports of CL and VL from 1960 to 2012 [

According to the WHO neglected tropical diseases (NTD) report in 2007, VL is identified as one of the NTDs [

By observing the VL epidemiological situations for each country in

Conversely, reported VL cases in several African countries have shown significant increase since 2006, as shown in

Therefore, VL is still a serious disease which threatens people lives and health especially in the developing countries. To strain the transmission of VL, WHO and health organizations in VL afflicted countries should apply effective prevention and control strategies. In the next section, this paper will review several existing strategies against VL.

Since 1995, researchers have focused on ZVL when investigating the intervention and prevention of VL transmission because many ZVL control strategies are related to animals. Tesh categorized former ZVL control strategies into three main classes: early detection of human cases, destruction or treatment of infected dogs, and vector control [

In 2014, Werneck considered the effectiveness of control strategies on the basic reproduction number _{0} [_{0} to decrease to less than 1, meaning that the number of infected individuals will eventually decrease to zero. However, the author did not find the relevant data to support these control strategies. He pointed out some potential implementation difficulties for these strategies, such as costs issues associated with the continual uses of tropical insecticides-collars. Werneck also voiced concern that the effectiveness of using insecticide collars in the large community (like Brazil) may not work so well, since the insecticide collars strategy has the relatively short-term effect and consequent need.

Due the high cost of indoor residual spraying, insecticide-treated nets (ITNs) were introduced as an alternative control strategy for ZVL [

In 2016, Özbel et al. analyzed the geographical distribution, ecological aspects, and species habits of VL vectors (sand flies) in Bangladesh [

As the most effective control strategy for infectious diseases, the successful vaccination on VL is long-awaited for the VL afflicted countries. The experiment on VL vaccination was started in 1990s; researchers tried to utilize the proteins from

For now, even though many contributions have been done for the VL controls and prevention, an effective, feasible, and economical control strategy is still an ongoing effort. The current VL control strategies and corresponding deficiencies are summarized in

In 1996 Dye first introduced a 4-equation ODE susceptible-latent-infectious-removed (SLIR) model to describe the VL epidemic [

Since WHO's designation of VL as an NTD in 2015 [_{0}. Their research showed that VL equilibrium is highly related to a critical model parameter, _{c}, as the epidemic threshold value for _{0}. Similarly, Subramanian et al. proposed a compartment-based ODE model of VL transmission to explain disease transmissions in symptomatic VL, asymptomatic VL, and PKDL-infection classes [

Biswas simplified the 12-equation ODE model from Zhao et al. to an 8-equation ODE model by dividing the nonhuman populations (dog and vector) into susceptible and infected population groups [

Although many ODE models describe VL epidemics and transmission, the development of a novel dynamic model is an active area of research in the investigation of complicated transmission behaviors of VL under various situations and the development of improved mitigation and control strategies. Bi et al. introduced a two-dimensional PDE model based on an existing ODE model [

VL attracted significant epidemiology research attention; abundant statistical data were collected and reported on the current VL pandemic worldwide by scholars and researchers. Many researchers realized the importance of data utilization in VL model development. Disease data is generally utilized in three ways: use of reported data to build statistical models, use of historical data to predict future prevalence, and use of existing data to calibrate model parameters in mathematical epidemic models.

The primary objective of building a VL statistical model is to statistically identify key parameters in the VL transmission process and determine relationships between the number of parameters and the number of infected population. Werneck et al. used consolidated census tracts to analyze VL disease prevalence data from different regions of Brazil [

Thompson et al. studied relationships between climate and VL epidemics by establishing the statistical regression model [

Ecological niche modeling (ENM), stemming from the genetic algorithm [

Parameter estimation is another essential application when validating VL mathematical models using real-world data since the use of assumed system parameters in the model may reduce model reliability. Bi et al. summarized age structures of VL infections in various regions [

The most generalized control strategy in a VL mathematical model is the parameter control strategy, which assumes that the key parameters in the model are adjustable. When the parameters are adjusted, the model outputs become dependent variables; therefore, the parameter adjustment process can be considered a corresponding real-world control strategy. In 2002 Courtenay et al. introduced the numerical control strategy to the field of VL mathematical modeling [

Lev Pontryagin established the optimal control strategy in the 1950s [

Simulation comparison is the most common method of VL mathematic control modeling in simulation. _{h}, the infected human _{h}, and the total population of sand flies _{f}) of the control strategy, which reduced disease prevalence to 80%. This simulation proved the effectiveness of the combined control strategies.

Efficacy comparison between control strategies is another type of simulation comparison. Ribas et al. compared the human prevalence influence using vector control, insecticidal collars, dog culling, dog vaccines, and dog treatment [

Spatial simulation is a simulation estimation method that provides spatial information throughout the model behaviors. Using GIS, spatial simulation can exhibit VL prevalence information from various regions, as shown in

Although globally reported, VL-confirmed cases have decreased since 2011; VL prevalence has not improved significantly worldwide except in South Asia (i.e., India and Bangladesh). However, VL outbreaks have increased in Ethiopia, Somalia, and Kenya since around 2008. Moreover, public health agencies in underdeveloped African countries such as Chad and the Central African Republic do not have resources and capabilities to collect and report VL incidences. However, because these countries are near regions severely afflicted with VL, such as Sudan and South Sudan, the number of global VL cases reported from WHO may be underestimated.

This paper reviewed current VL epidemiological research ranging from VL epidemic control strategies to VL mathematical models and related optimal control strategies. The research demonstrated how to use numerical methods such as modeling and sensitivity analysis, as well as equilibrium/stability studies and simulation experiments, to assist mitigation and prevention strategies for a worldwide VL pandemic. Governments and health organizations can utilize the modeling and simulation results to predict or estimate impacts of various control strategies.

Despite significant research efforts using mathematical models for the VL epidemic, research gaps still exist and many areas of study remain unexplored.

Future work, thorough VL epidemic research using mathematical or statistical models, ought to consider the four following aspects:

building more sophisticated mathematical models to explain underlying infectious disease transmission dynamics,

including real-world data to aid model validation and verifications,

exploring possible disease control/mitigation strategies to increase understanding of model maneuverability and robustness,

using numerical simulation experiments as a predictive tool to verify the feasibility of model and control strategies.

Moreover, future work in these four aspects of VL mathematical modeling must utilize modern analytical tools. The disadvantage of current modeling is the limited diversity of model types. A majority of existing VL mathematical models are ODE models, which are widely used but produce limited predicted results without details. Therefore, more statistical, machine learning, and PDE models are needed to build sophisticated, comprehensive mathematical models of VL. Statistical and machine learning models can more advantageously utilize real-world data to ensure model prediction accuracy, while use of a PDE model can enrich predicted results with age, gender, socioeconomic group, ethics, and spatial information. For the second aspect, the inclusion of real-world data, most test data currently used to validate and verify underlying mathematical models are estimated or assumed, consequently limiting the mathematical model to reflect only data from previous VL epidemic episodes. Future research efforts should utilize recent epidemic data with temporal and spatial data during the modeling phase, making the modeling process increasingly dynamic and reflecting real-time data while predicting possible trends of an ongoing epidemic. The current primary disadvantage of the third aspect, exploring possible control strategies, is that the control strategies lack of applicability in the real world. In fact, the most effective control strategies suggested by the mathematical models may not be operable or they may be too cost prohibitive to be implemented. Operable control strategies should be carefully quantized, such as specific consideration of the optimal level of canine culling in a particular time frame or the level of insecticide spraying in each area affected by VL. For the fourth aspect, current studies using numerical simulation experiments frequently provide insufficient information from simulation results. Most simulations of VL models can only predict the trend of VL infections. Future research should focus on spatial simulation and agent-based simulation as well as the study of the interactions between multiple regions or environments.

In conclusion, the use of mathematical models to study, analyze, and predict VL epidemics and to explore effective and implementable control strategies remains an active and study-worthy area of future research. However, research results from more comprehensive studies that use modern analytical tools will help public health organizations understand and prevent the VL disease.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Research tree for this paper.

Distributions of confirmed and borderline VL cases from 1960 to 2012.

Reported VL cases from 2006 to 2016 [

Reported VL cases in severely afflicted countries from 2006 to 2016 [

Reported VL cases in vulnerable countries from 2006 to 2016 [

System diagram of ZVL transmission model [

Infection rate distribution based on human age in various countries [

Simulation of dog culling [

Efficacy comparison of control strategies [

Spatial simulation of predicted VL rates in 2010, Brazil [

Current VL control strategies and corresponding deficiencies.

Strategy | Category | Deficiency | Reference |
---|---|---|---|

Early detection | Human control | Doesn't affect the parasite transmission | [ |

Culling dogs | Dog control | Opposition of dog owners & Hard to detect infected dogs | [ |

Dog treatments | Dog control | Expensive | [ |

Canine vaccination | Dog control | Expensive & Drug resistant | [ |

Spraying insecticide | Vector control | Expensive & Drug resistant | [ |

Immunological screening of seropositive dogs | Dog control | Expensive & Needs high level technique support | [ |

Insecticide collars | Dog control | Expensive & May not work in large community | [ |

Insecticide-treated nets | Vector control | Damaged and untreated nets have low effectiveness | [ |

Ecological control | Vector control | Needs more time to be applied in the real world | [ |

Vaccination control | Human control | Not available at the current time | [ |

Recent papers on mathematical modeling of VL.

Paper | Published Year | Real Data Involved | Control Strategies | Transmission Models | Simulation |
---|---|---|---|---|---|

Ribas et al. [ | 2013 | No | Yes | Yes | No |

Zhao et al. [ | 2016 | No | Yes | Yes | Yes |

Subramanianet al. [ | 2015 | Testing model | No | Yes | Yes |

Biswas et al. [ | 2017 | Parameter estimated | Yes | Yes | Yes |

Shimozako et al. [ | 2017 | Testing model | No | Yes | Yes |

Le Rutte et al. [ | 2017 | Testing model | No | Yes | Yes |

Costaet al. [ | 2013 | No | No | Yes | No |

Sevá et al. [ | 2016 | No | No | Yes | No |

Zouet al. [ | 2017 | No | No | Yes | Yes |

Agustoet al. [ | 2017 | No | Yes | Yes | Yes |

Bi et al. [ | Not yet | Parameter estimated | No | Yes | Yes |

de Araújo et al. [ | 2013 | Yes | No | No | No |

Karagiannis-Voules et al. [ | 2013 | Yes | No | No | Yes |

Miller et al. [ | 2014 | Yes | Yes | Yes | No |

Biswas et al. [ | 2017 | Testing model | Yes | Yes | Yes |

Shimozako et al. [ | 2017 | Testing model | Yes | Yes | Yes |

Stauch et al. [ | 2014 | Testing model | No | Yes | Yes |

Zamir et al. [ | 2017 | No | No | Yes | Yes |

Boukhalfaet al. [ | 2017 | No | No | Yes | Yes |

Gorahavaet al. [ | 2015 | No | Yes | Yes | No |

Rock et al. [ | 2016 | No | Yes | Yes | Yes |