Modeling Spatial and Temporal Heterogeneities of Cholera Dynamics: Comparison
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Cholera remains a significant public health burden in many countries and regions of the world, highlighting the need for a deeper understanding of the mechanisms associated with its transmission, spread, and control. Mathematical modeling offers a valuable research tool to investigate cholera dynamics and explore effective intervention strategies. In particular, many modeling studies have been devoted to the spatial and temporal heterogeneities of cholera dynamics.

  • mathematical modeling
  • cholera
  • disease transmission

1. Introduction

Cholera is an infectious disease caused by the bacterium Vibrio cholerae (or, V. cholerae) [1]. The main sources of the pathogen are contaminated water and food. The infection can spread rapidly in populations without safe drinking water and adequate sanitation and hygiene, and those with limited medical resources [2]. The major symptom of cholera is profuse watery diarrhea, which could result in rapid dehydration. Other symptoms may include vomiting, extreme thirst, abdominal pain, kidney failure, and drop in blood pressure. In the most severe cases, cholera can lead to death within days if not treated [3].
Although cholera is an old disease, with seven pandemics already recorded in human history, the global burden of cholera remains high at present. This is largely due to lack of access to basic drinking water and sanitation in many places of the world. It was estimated that more than 2 billion people worldwide drink water from sources that may be faecally contaminated, and 2.4 billion people do not have basic sanitation facilities, exposing them to cholera and other waterborne infections [4]. From a report published in 2015 [5], it was found that approximately 1.3 billion people were at risk for cholera in endemic countries and regions, and about 1.3–4.0 million people were infected with cholera annually, including an estimated 95,000 deaths. More recently, Yemen experienced the worst cholera outbreak in modern history that started in late 2016 and peaked in 2017, with more than 2.5 million cumulative cases reported as of November 2021 [6]. In October 2017, the World Health Organization (WHO) launched a global strategy for ending cholera, with an aim of reducing cholera deaths by 90% and eliminating cholera in 20 of the 47 countries currently affected by the disease by 2030 [7]. Ultimately, the improvement of water, sanitation and hygiene (WASH) infrastructure is the fundamental, long-term solution for cholera control [2].
Mathematical modeling for infectious diseases dates back to Bernoulli in the 18th century [8], and has since become indispensable in epidemiological research [9]. Models offer a powerful theoretical tool to understand infection and transmission mechanisms, to predict future progression of epidemics, to compare different intervention strategies, and to provide useful guidelines for outbreak management [10].

2. Spatial and Temporal Heterogeneities

2.1. Multi-Group and Multi-Patch Models

Transmission of cholera, like that of many other infectious diseases, is complicated by spatial heterogeneity that involves different ecological and geographical environments, population sizes, mobility and contact patterns, and socio-economic and demographic structures. Mukandavire et al. [11] performed a modeling study for the 2008–2009 Zimbabwe cholera outbreak, where basic reproduction numbers were estimated and relative contributions from direct and indirect transmission routes were compared for the 10 provinces in Zimbabwe. The results were highly heterogeneous, an indication that the underlying transmission pattern varied substantially throughout the country. Similarly, the study in [12] generated a range of reproduction numbers for different administrative departments in Haiti during the 2010 cholera outbreak. In addition, an investigation of the Yemen cholera outbreak during 2016–2017 [13] revealed that the transmission modes and infection risk differ significantly in the northwest, southwest, and east regions of the country. Although relatively simple mathematical models were used in these studies, the findings confirmed that spatial heterogeneity plays an important role in cholera transmission and spread. Consequently, there is a need for more detailed quantitative investigation regarding the spatial effects, especially the movement of human hosts and the dispersal of pathogenic vibrios, on cholera epidemics and endemicity. Meta-population models [14][15] have been commonly used in epidemiological studies to incorporate spatial heterogeneity from the hosts and environments. A standard approach is based on multi-group modeling [16][17][18], where the entire population is divided into a number of groups that possess different characteristics. Each group is connected to other groups, and infection can take place between individuals within the same group or from different groups. The multi-group formulation is analogous to the Lagrangian approach in fluid dynamics since it labels individual hosts of different groups and explicitly tracks disease transmission for individuals [19]. A multi-group cholera model was proposed and analyzed in [20] which considered only the indirect, environment-to-human transmission route. The authors in [21] extended the homogeneous cholera model presented in [22] to a multi-group setting, and found that the overall infection risk for the entire population represents a combination of the transmission risk for each individual group. Another multi-group model, applicable to cholera transmission, was proposed in [23] with both direction and indirect transmission pathways represented in a general incidence form. Other cholera-modeling studies based on the multi-group framework include, for example, [12][24][25]. Another popular meta-population approach, called multi-patch modeling [26][27][28][29], divides the entire population into a set of patches, where each patch is often associated with a different location. This method describes the movement of the hosts and/or pathogens between patches, with a focus on the pathogen transmission within each patch. This type of formulation is related to the Eulerian approach in fluid dynamics as it labels locations and explicitly tracks disease transmission for each location. A multi-patch cholera model was developed in [30] where the movement of the pathogenic bacteria between different patches was considered and only the indirect transmission route was included. This modeling framework was subsequently extended to predict the spatial evolution of the Haiti cholera outbreak [31][32]. Another modeling study for Haiti cholera outbreak was performed in [12], where the between-patch epidemic spread was based on a gravity model that depends on the population size and distance between regional centroids. In another multi-patch model [33], the movement of both the human hosts and environmental vibrios was incorporated, and both the direct and indirect transmission pathways were included. A sharp threshold condition was established at R0=1 for the entire system to distinguish disease extinction (R0<1) and disease persistence (R0>1). A cholera model that couples the multi-patch structure with a time-periodic environment was proposed and analyzed in [34]. Some other cholera models were proposed in [35][36] where the human hosts from different patches do not directly communicate with each other and, instead, are connected through a common environmental water reservoir. Their model structure can be regarded as a star network where the hub (or, center) corresponds to the shared water source and the leaves (or, vertices) correspond to the host patches. The basic reproduction number is determined by the direct transmission in each patch and the total indirect transmission through the water source from all patches. In most of these studies, it was found that the basic reproduction number and the outbreak size would be higher for the coupled meta-population system than those for the disconnected, individual groups or patches, indicating that increased spatial heterogeneity may lead to increased disease transmission risk. In some cases, it was also found that the connection between population groups or patches would allow cholera to persist, whereas such disease persistence may not be possible in any isolated individual population [35].

2.2. Reaction-Diffusion PDE Models

Partial differential equations (PDEs) of the reaction–diffusion type are extensively used in epidemiological modeling (e.g., [37][38][39][40][41][42]). Fick’s law [43] can be generally applied to construct a reaction–diffusion model. Often, based on an epidemic system of ordinary differential equations (ODEs), diffusion terms can be added to model the spatial spread of the disease. A diffusion process represents random movement and dispersal of hosts and/or pathogens over a spatial domain, normally without a directional preference. The underlying ODE model typically describes homogeneous dynamics of disease transmission, whereas the reaction–diffusion PDE model incorporates spatial movement, generally associated with location-dependent diffusion rates, into the epidemiological process and emphasizes the spatial heterogeneity of population dynamics [44][45] related to disease transmission and spread. A reaction–diffusion model, derived from the continuous limit of a multi-patch ODE system, was presented in [30] to account for the epidemic spreading of cholera. The spatial dispersal of V. cholerae was modeled as a diffusion process, and only the environment-to-human transmission route was considered. This model was extended in [46] to include the movement of human hosts. Another cholera model was developed in [47] where the human hosts undergo a diffusion process while the vibrios remain stationary. In [48], a PDE cholera model was proposed that represents the spatial diffusion of both the pathogens and human hosts, while incorporating both the direct and indirect transmission routes. This work was later extended in [49] to include a convection process for the pathogenic bacteria, such as the movement of the vibrios from the upstream to the downstream along a river. These cholera models and some other extensions were mathematically analyzed in a rigorous way in [50][51][52]. Additionally, the work in [53] incorporated seasonal fluctuation into the spatiotemporal dynamics of cholera. For all the aforementioned PDE-based cholera studies, the spatial domain is restricted to either a one-dimensional (1D) space or a symmetric two-dimensional (2D) space that is equivalent to a 1D domain. These simplified, 1D reaction–diffusion models may be practically meaningful when the spread of cholera is associated with a fluvial system. For example, the suspected source of the 2010–2012 Haiti cholera outbreak outbreak was Artibonite River, the longest and most important river in Haiti, and the initial spread of the disease was believed to follow the river [54]. More sophisticated PDE models of the reaction–diffusion type that involve multi-dimensional spatial domains have also been developed for cholera dynamics; see, e.g., [55][56][57][58]. These modeling studies have focused on the mathematical analysis of the PDE systems. All these PDE studies contribute to the body of knowledge in mathematical modeling of cholera. On the other hand, most of these studies are intentionally theoretical, and it remains a challenge to apply such reaction–diffusion models to fit data from real-world cholera outbreaks. Particularly, the diffusion coefficients, which generally take different values for different population groups and spatial locations, are difficult to calibrate. Thus far, there is very little published work regarding the outbreak simulation and practical data fitting of reaction–diffusion cholera models, even for the simplified cases with 1D spatial domains and constant diffusion coefficients. The challenge associated with reaction–diffusion modeling is not only for cholera, but also for many other infectious diseases. More efforts along this direction are needed to facilitate the real-world applications of these PDE epidemic models and to make such models better appreciated by the public health community.

2.3. Seasonal Variation and Climate Change

The transmission of cholera is inherently related to the environment. Many environmental factors, such as floods, droughts, precipitations, and water temperature and salinity, are seasonal and can significantly impact cholera dynamics [59][60][61][62]. For example, it has been observed that cholera becomes a seasonal disease in many endemic places and infection peaks typically occur in the rainy or monsoon season on an annual basis [63][64]. Furthermore, historical cholera data indicate that climate change, which leads to rises in sea levels and global temperatures, may influence the temporal fluctuations of cholera and increase the frequency and duration of cholera outbreaks [65][66]. Most cholera models based on ODE systems utilize constant parameters for simplicity, and these models may not be able to reflect the seasonal and climatic behavior of cholera dynamics. To overcome this difficulty, non-autonomous ODE systems with time-dependent parameters can be used. In particular, temporal periodicity may be applied to the contact rate, recovery rate, and pathogen growth rate, among other parameters, to represent regular seasonal oscillations of the infection dynamics. Simple numerical tests were conducted in [67] for three hypothetic scenarios with periodic model parameters. A more general cholera model [68] incorporated periodicity into both the incidence and pathogen functions to represent seasonal oscillations in a generic manner. This model was extended in [69] to a stochastic system based on a Markov process, where it was shown that the probability of a cholera outbreak is periodic in time. Another study [70] discussed the intra-annual seasonality and variability of cholera dynamics based on a mathematical model that incorporates both asymptomatic and symptomatic infections. The authors of [71] studied the seasonality of cholera dynamics and the fluctuations of the aquatic reservoir in endemic areas driven by rainfall and temperature, and fitted their model to the historical cholera dataset of the Bengal region in the Indian subcontinent. Another cholera study [72] incorporated seasonal environmental drivers, including river flow, temperature and chlorophyll concentration, into a spatially explicit model and showed that such drivers may generate dual-peak cholera prevalence patterns. A mathematical model presented in [73] showed that climate variability played a vital role in modulating the size of cholera outbreaks in Bangladesh. In addition, the authors in [74] reviewed several mathematical models and provided quantitative evidence for the influence of climate change on cholera dynamics. In general, the seasonal patterns and temporal variations of cholera epidemics and endemicity are complex, involving the interplay between many environmental and climatic factors. Mathematical models based on periodic systems (i.e., systems of differential equations with time-periodic parameters) are capable of simulating and predicting regular seasonal oscillations of cholera outbreaks, but these may not represent the full picture of the intra- and inter-annual dynamics of cholera. Particularly, the effects of climate change are typically not periodic and thus cannot be resolved through purely periodic models. Instead, the use of time-dependent but non-periodic model parameters would be more appropriate in this case, though the models may become difficult to analyze and may involve non-trivial data fitting procedures to calibrate the parameters. Furthermore, as pointed out in [71], stochasticity played an important role in the occurrence of some abnormally large cholera outbreaks in the Bengal region, while regularity of inter-annual cholera dynamics was found in other times (with periodicity roughly corresponding to the dominant frequency of El Niño). This indicates that a combination of periodicity and stochasticity into a single modeling framework may better explain the various environmental and climatic drivers and provide deeper insight into the temporal dynamics of cholera.

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