Global stability for cholera epidemic models

Cholera is a water and food borne infectious disease caused by the gram-negative bacterium, Vibrio cholerae. Its dynamics are highly complex owing to the coupling among multiple transmission pathways and different factors in pathogen ecology. Although various mathematical models and clinical studies published in recent years have made important contribution to cholera epidemiology, our knowledge of the disease mechanism remains incomplete at present, largely due to the limited understanding of the dynamics of cholera. In this paper, we conduct global stability analysis for several deterministic cholera epidemic models. These models, incorporating both human population and pathogen V. cholerae concentration, constitute four-dimensional non-linear autonomous systems where the classical Poincaré-Bendixson theory is not applicable. We employ three different techniques, including the monotone dynamical systems, the geometric approach, and Lyapunov functions, to investigate the endemic global stability for several biologically important cases. The analysis and results presented in this paper make building blocks towards a comprehensive study and deeper understanding of the fundamental mechanism in cholera dynamics.     

Historical Epidemiology of the Second Cholera Pandemic: Relevance to Present Day Disease Dynamics

Despite nearly two centuries of study, the fundamental transmission dynamic properties of cholera remain incompletely characterized. We used historical time-series data on the spread of cholera in twelve European and North American cities during the second cholera pandemic, as reported in Amariah Brigham’s 1832 A Treatise on Epidemic Cholera, to parameterize simple mathematical models of cholera transmission. Richards growth models were used to derive estimates of the basic reproductive number (R₀) (median: 16.0, range: 1.9 to 550.9) and the proportion of unrecognized cases (mean: 96.3%, SD: 0.04%). Heterogeneity in model-generated R₀ estimates was consistent with variability in cholera dynamics described by contemporary investigators and may represent differences in the nature of cholera spread. While subject to limitations associated with measurement and the absence of microbiological diagnosis, historical epidemic data are a potentially rich source for understanding pathogen dynamics in the absence of control measures, particularly when used in conjunction with simple and readily interpretable mathematical models.     

Serotype cycles in cholera dynamics

Interest in understanding strain diversity and its impact on disease dynamics has grown over the past decade. Theoretical disease models of several co-circulating strains indicate that incomplete cross-immunity generates conditions for strain-cycling behaviour at the population level. However, there have been no quantitative analyses of disease time-series that are clear examples of theoretically expected strain cycling. Here, we analyse a 40-year (1966-2005) cholera time-series from Bangladesh to determine whether patterns evident in these data are compatible with serotype-cycling behaviour. A mathematical two-serotype model is capable of explaining the oscillations in case patterns when cross-immunity between the two serotypes, Inaba and Ogawa, is high. Further support that cholera's serotype-cycling arises from population-level immunity patterns is provided by calculations of time-varying effective reproductive rates. These results shed light on historically observed serotype dominance shifts and have important implications for cholera early warning systems.     

A generalized cholera model and epidemic-endemic analysis

The transmission of cholera involves both human-to-human and environment-to-human pathways that complicate its dynamics. In this paper, we present a new and unified deterministic model that incorporates a general incidence rate and a general formulation of the pathogen concentration to analyse the dynamics of cholera. Particularly, this work unifies many existing cholera models proposed by different authors. We conduct equilibrium analysis to carefully study the complex epidemic and endemic behaviour of the disease. Our results show that despite the incorporation of the environmental component, there exists a forward transcritical bifurcation at R (0)=1 for the combined human-environment epidemiological model under biologically reasonable conditions.     

The ABCs (Antibody, B cells, and Carbohydrate epitopes) of cholera immunity: considerations for an improved vaccine

Cholera, a diarrheal disease, is known for explosive epidemics that can quickly kill thousands. Endemic cholera is a seasonal torment that also has a significant mortality. Not all nations with extensive rural communities can achieve the required infrastructure or behavioral changes to prevent epidemic or endemic cholera. For some communities, a single-dose cholera vaccine that protects those at risk is the most efficacious means to reduce morbidity and mortality. It is clear that our understanding of what a protective cholera immune response is has not progressed at the rate our understanding of the pathogenesis and molecular biology of cholera infection has. This review addresses V. cholerae lipopolysaccharide (LPS)-based immunogens because LPS is the only immunogen proven to induce protective antibody in humans. We discuss the role of anti-LPS antibodies in protection from cholera, the importance and the potential role of B cell subsets in protection that is based on their anatomical location and the intrinsic antigen-receptor specificity of various subsets is introduced.     

Linking within- and between-host dynamics in the evolutionary epidemiology of infectious diseases

Nested models (also called embedded models) explicitly link dynamical processes that occur at different scales. Recently there has been considerable interest in linking within- and between-host levels of disease dynamics in the study of pathogen evolution. Here we review the extent to which these nested models have increased our understanding of pathogen evolution. We suggest that, although such models have been useful for determining the nature of tradeoffs between epidemiological parameters and for evaluating the consequences of conflicting selection pressures at different scales, the vast majority of previous results could likely have been obtained without the use of nested models per se. Nevertheless, these models have proven very useful through their highlighting of the importance of within-host disease dynamics on pathogen evolution.     

A generalized cholera model and epidemic–endemic analysis

The transmission of cholera involves both human-to-human and environment-to-human pathways that complicate its dynamics. In this paper, we present a new and unified deterministic model that incorporates a general incidence rate and a general formulation of the pathogen concentration to analyse the dynamics of cholera. Particularly, this work unifies many existing cholera models proposed by different authors. We conduct equilibrium analysis to carefully study the complex epidemic and endemic behaviour of the disease. Our results show that despite the incorporation of the environmental component, there exists a forward transcritical bifurcation at R ₀=1 for the combined human–environment epidemiological model under biologically reasonable conditions.     

Mathematical models of HIV pathogenesis and treatment

We review mathematical models of HIV dynamics, disease progression, and therapy. We start by introducing a basic model of virus infection and demonstrate how it was used to study HIV dynamics and to measure crucial parameters that lead to a new understanding of the disease process. We discuss the diversity threshold model as an example of the general principle that virus evolution can drive disease progression and the destruction of the immune system. Finally, we show how mathematical models can be used to understand correlates of long-term immunological control of HIV, and to design therapy regimes that convert a progressing patient into a state of long-term non-progression.     

Pathogen adaptation to seasonal forcing and climate change

Many diverse infectious diseases exhibit seasonal dynamics. Seasonality in disease incidence has been attributed to seasonal changes in pathogen transmission rates, resulting from fluctuations in extrinsic climate factors. Multi-strain infectious diseases with strain-specific seasonal signatures, such as cholera, indicate that a range of seasonal patterns in transmission rates is possible in identical environments. We therefore consider pathogens capable of evolving their 'seasonal phenotype', a trait that determines the sensitivity of their transmission rates to environmental variability. We introduce a theoretical framework, based on adaptive dynamics, for predicting the evolution of disease dynamics in seasonal environments. Changes in the seasonality of environmental factors are one important avenue for the effects of climate change on disease. This model also provides a framework for examining these effects on pathogen evolution and associated disease dynamics. An application of this approach gives an explanation for the recent cholera strain replacement in Bangladesh, based on changes in monsoon rainfall patterns.     

Impact of Awareness Programs on Cholera Dynamics: Two Modeling Approaches

We propose two differential equation-based models to investigate the impact of awareness programs on cholera dynamics. The first model represents the disease transmission rates as decreasing functions of the number of awareness programs, whereas the second model divides the susceptible individuals into two distinct classes depending on their awareness/unawareness of the risk of infection. We study the essential dynamical properties of each model, using both analytical and numerical approaches. We find that the two models, though closely related, exhibit significantly different dynamical behaviors. Namely, the first model follows regular threshold dynamics while rich dynamical behaviors such as backward bifurcation may arise from the second one. Our results highlight the importance of validating key modeling assumptions in the development and selection of mathematical models toward practical application.     

Avian cholera,post‐hatching survival and selection on hatch characteristics in a long‐lived bird,the common eider Somateria mollisima

Infectious diseases can have dramatic impacts on animal population dynamics, but how they influence vital rates remains understudied. We took advantage of the appearance of an avian cholera epizootic in an arctic colony of common eiders Somateria mollissima to study variation in juvenile survival and selection on hatch characteristics in relation to this highly infectious disease. Avian cholera is one of the most important infectious diseases affecting wild birds and is thought to primarily affect adult survival. Here, we show that avian cholera was associated with a 90% decline in duckling survival, leading to almost zero recruitment. Before the cholera outbreak, there was significant stabilizing selection on hatching date and significant positive directional selection on hatching mass. During cholera outbreaks, selection on hatch characteristics was no longer significant. These results were based on a low sample of surviving ducklings in cholera years, but suggested that date and mass at hatching did no longer affect duckling survival in the presence of cholera. These effects of avian cholera on post‐hatching survival were likely not only the consequence of the disease per se, but also a consequence of an increase in predation rates that followed the emergence of avian cholera. Our results emphasize the dramatic direct and indirect impacts that infectious disease can have on vital rates, and thus population dynamics.     

On the global stability of a generalized cholera epidemiological model

In this paper, we conduct a careful global stability analysis for a generalized cholera epidemiological model originally proposed in [J. Wang and S. Liao, A generalized cholera model and epidemic/endemic analysis, J. Biol. Dyn. 6 (2012), pp. 568–589]. Cholera is a water- and food-borne infectious disease whose dynamics are complicated by the multiple interactions between the human host, the pathogen, and the environment. Using the geometric approach, we rigorously prove the endemic global stability for the cholera model in three-dimensional (when the pathogen component is a scalar) and four-dimensional (when the pathogen component is a vector) systems. This work unifies the study of global dynamics for several existing deterministic cholera models. The analytical predictions are verified by numerical simulation results.     

Disentangling extrinsic from intrinsic factors in disease dynamics: a nonlinear time series approach with an application to cholera

Alternative explanations for disease and other population cycles typically include extrinsic environmental drivers, such as climate variability, and intrinsic nonlinear dynamics resulting from feedbacks within the system, such as species interactions and density dependence. Because these different factors can interact in nonlinear systems and can give rise to oscillations whose frequencies differ from those of extrinsic drivers, it is difficult to identify their respective contributions from temporal population patterns. In the case of disease, immunity is an important intrinsic factor. However, for many diseases, such as cholera, for which immunity is temporary, the duration and decay pattern of immunity is not well known. We present a nonlinear time series model with two related objectives: the reconstruction of immunity patterns from data on cases and population sizes and the identification of the respective roles of extrinsic and intrinsic factors in the dynamics. Extrinsic factors here include both seasonality and long-term changes or interannual variability in forcing. Results with simulated data show that this semiparametric method successfully recovers the decay of immunity and identifies the origin of interannual variability. An application to historical cholera data indicates that temporary immunity can be long-lasting and decays in approximately 9 yr. Extrinsic forcing of transmissibility is identified to have a strong seasonal component along with a long-term decrease. Furthermore, noise appears to sustain the multiple frequencies in the long-term dynamics. Similar semiparametric models should apply to population data other than for disease.     

Geometrical models of the renewal of the epidermis

We give a review of the different models developed recently that describe the renewal of the epidermis. These models, based on concepts borrowed from statistical mechanics, geometry and topology, shed new light on the understanding of the organization and the dynamics of the system. We discuss in detail a topological model of the dynamics of the inner-most layer of the epidermis: the basal layer.     

Progress towards development of a cholera subunit vaccine

Cholera, an enteric disease that can reach pandemic proportions, remains a world-wide problem that is positioned to increase in incidence as changes in global climate or armed conflict spawn the conditions that enhance transmission to humans and, thus, precipitate epidemic cholera. An effective subunit cholera vaccine that can provide protective immunity with one parenteral immunization would be a major advantage over the existing oral vaccines that can require two doses for optimal protection. The existing vaccines are clearly effective in some settings, but are less so in others, especially with respect to specific groups such as young (2-5 years) children. In our efforts to develop a cholera subunit vaccine, we focused on two Vibrio cholerae antigens, LPS (lipopolysaccharide) and TCP (toxin co-regulated pilus), that are known to induce protective antibodies in animal models and, in the case of anti-LPS antibodies, to be associated with clinical protection of V. cholerae exposed or vaccinated individuals. This review discusses the current cholera vaccines and compares the advantages of a cholera subunit vaccine to that of the whole cell vaccines. We discuss the possible subunit antigens and prospective targeted use of a subunit cholera vaccine.     

Intestinal Colonization Dynamics of Vibrio cholerae

To cause the diarrheal disease cholera, Vibrio cholerae must effectively colonize the small intestine. In order to do so, the bacterium needs to successfully travel through the stomach and withstand the presence of agents such as bile and antimicrobial peptides in the intestinal lumen and mucus. The bacterial cells penetrate the viscous mucus layer covering the epithelium and attach and proliferate on its surface. In this review, we discuss recent developments and known aspects of the early stages of V. cholerae intestinal colonization and highlight areas that remain to be fully understood. We propose mechanisms and postulate a model that covers some of the steps that are required in order for the bacterium to efficiently colonize the human host. A deeper understanding of the colonization dynamics of V. cholerae and other intestinal pathogens will provide us with a variety of novel targets and strategies to avoid the diseases caused by these organisms.     

Analytic approximation of spatial epidemic models of foot and mouth disease

The effect of spatial heterogeneity in epidemic models has improved with computational advances, yet far less progress has been made in developing analytical tools for understanding such systems. Here, we develop two classes of second-order moment closure methods for approximating the dynamics of a stochastic spatial model of the spread of foot and mouth disease. We consider the performance of such ‘pseudo-spatial’ models as a function of R₀, the locality in disease transmission, farm distribution and geographically-targeted control when an arbitrary number of spatial kernels are incorporated. One advantage of mapping complex spatial models onto simpler deterministic approximations lies in the ability to potentially obtain a better analytical understanding of disease dynamics and the effects of control. We exploit this tractability by deriving analytical results in the invasion stages of an FMD outbreak, highlighting key principles underlying epidemic spread on contact networks and the effect of spatial correlations.     

From host immunity to pathogen invasion: the effects of helminth coinfection on the dynamics of microparasites

Concurrent infections with multiple parasites are ubiquitous in nature. Coinfecting parasites can interact with one another in a variety of ways, including through the host's immune system via mechanisms such as immune trade-offs and immunosuppression. These within-host immune processes mediating interactions among parasites have been described in detail, but how they scale up to determine disease dynamic patterns at the population level is only beginning to be explored. In this review, we use helminth-microparasite coinfection as a model for examining how within-host immunological effects may influence the ecological outcome of microparasitic diseases, with a specific focus on disease invasion. The current literature on coinfection between helminths and major microparasitic diseases includes many studies documenting the effects of helminths on individual host responses to microparasites. In many cases, the observed host responses map directly onto parameters relevant for quantifying disease dynamics; however, there have been few attempts at integrating data on individual-level effects into theoretical models to extrapolate from the individual to the population level. Moreover, there is considerable variability in the particular combination of disease parameters affected by helminths across different microparasite systems. We develop a conceptual framework identifying some potential sources of such variability: Pathogen persistence and severity, and resource availability to hosts. We also generate testable hypotheses regarding diseases and the environmental contexts when the effects of helminths on microparasite dynamics should be most pronounced. Finally, we use a case study of helminth and mycobacterial coinfection in the African buffalo to illustrate both progress and challenges in understanding the population-level consequences of within-host immunological interactions, and conclude with suggestions for future research that will help improve our understanding of the effects of coinfection on dynamics of infectious diseases.     

Integrating data mining and transmission theory in the ecology of infectious diseases

Our understanding of ecological processes is built on patterns inferred from data. Applying modern analytical tools such as machine learning to increasingly high dimensional data offers the potential to expand our perspectives on these processes, shedding new light on complex ecological phenomena such as pathogen transmission in wild populations. Here, we propose a novel approach that combines data mining with theoretical models of disease dynamics. Using rodents as an example, we incorporate statistical differences in the life history features of zoonotic reservoir hosts into pathogen transmission models, enabling us to bound the range of dynamical phenomena associated with hosts, based on their traits. We then test for associations between equilibrium prevalence, a key epidemiological metric and data on human outbreaks of rodent‐borne zoonoses, identifying matches between empirical evidence and theoretical predictions of transmission dynamics. We show how this framework can be generalized to other systems through a rubric of disease models and parameters that can be derived from empirical data. By linking life history components directly to their effects on disease dynamics, our mining‐modelling approach integrates machine learning and theoretical models to explore mechanisms in the macroecology of pathogen transmission and their consequences for spillover infection to humans.     

Dependent population dynamics between chironomids (nonbiting midges) and Vibrio cholerae

Vibrio cholerae, the causative agent of cholera, is a natural inhabitant of the aquatic ecosystem. Chironomid (nonbiting midges) egg masses were recently found to harbour V. cholerae non-O1 and non-O139, providing a natural reservoir for the cholera bacterium. Chironomid populations and the presence of V. cholerae in chironomid egg masses were monitored. All V. cholerae isolates were able to degrade chironomid egg masses. The following virulence associated genes were detected in the bacterial isolates: hapA (100%), toxR (100%), hlyA (72%) and ompU (28%). The chironomid populations and the V. cholerae in their egg masses followed the phenological succession and interaction of host-pathogen population dynamics. A peak in the chironomid population was followed by a peak in the V. cholerae population. If such a connection is further substantiated for the pathogenic serogroups of V. cholerae in endemic areas of the disease, it may lead to a better understanding of the role of chironomids as a host for the cholera bacterium.