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.     

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.     

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.     

Multiple Transmission Pathways and Disease Dynamics in a Waterborne Pathogen Model

Multiple transmission pathways exist for many waterborne diseases, including cholera, Giardia, Cryptosporidium, and Campylobacter. Theoretical work exploring the effects of multiple transmission pathways on disease dynamics is incomplete. Here, we consider a simple ODE model that extends the classical SIR framework by adding a compartment (W) that tracks pathogen concentration in the water. Infected individuals shed pathogen into the water compartment, and new infections arise both through exposure to contaminated water, as well as by the classical SIR person–person transmission pathway. We compute the basic reproductive number (ℛ₀), epidemic growth rate, and final outbreak size for the resulting “SIWR” model, and examine how these fundamental quantities depend upon the transmission parameters for the different pathways. We prove that the endemic disease equilibrium for the SIWR model is globally stable. We identify the pathogen decay rate in the water compartment as a key parameter determining when the distinction between the different transmission routes in the SIWR model is important. When the decay rate is slow, using an SIR model rather than the SIWR model can lead to under-estimates of the basic reproductive number and over-estimates of the infectious period.     

Long‐term endemism of two highly divergent lineages of the amphibian‐killing fungus in the Atlantic Forest of Brazil

The recent global spread of the amphibian‐killing fungus [Batrachochytrium dendrobatidis (Bd)] has been closely tied to anthropogenic activities; however, regional patterns of spread are not completely understood. Using historical samples, we can test whether Bd was a spreading or endemic pathogen in a region within a particular time frame, because those two disease states provide different predictions for the regional demographic dynamics and population genetics of Bd. Testing historical patterns of pathogen prevalence and population genetics under these predictions is key to understanding the evolution and origin of Bd. Focusing on the Atlantic Forest (AF) of Brazil, we used qPCR assays to determine the presence or absence of Bd on 2799 preserved postmetamorphic anurans collected between 1894 and 2010 and used semi‐nested PCRs to determine the frequency of rRNA ITS1 haplotypes from 52 samples. Our earliest date of detection was 1894. A mean prevalence of 23.9% over time and spatiotemporal patterns of Bd clusters indicate that Bd has been enzootic in the Brazilian AF with no evidence of regional spread within the last 116 years. ITS1 haplotypes confirm the long‐term presence of two divergent strains of Bd (BdGPL and Bd‐Brazil) and three spatiotemporally broad genetic demes within BdGPL, indicating that Bd was not introduced into southeast Brazil by the bullfrog trade. Our data show that the evolutionary history and pathogen dynamics of Bd in Brazil is better explained by the endemic pathogen hypothesis.     

On the role of long incubation periods in the dynamics of acquired immunodeficiency syndrome (AIDS)

In this study, we investigate systematically the role played by the reproductive number (the number of secondary infections generated by an infectious individual in a population of susceptibles) on single group populations models of the spread of HIV/AIDS. Our results for a single group model show that if R 1, the disease will die out, and strongly suggest that if R > 1 the disease will persist regardless of initial conditions. Our extensive (but incomplete) mathematical analysis and the numerical simulations of various research groups support the conclusion that the reproductive number R is a global bifurcation parameter. The bifurcation that takes place as R is varied is a transcritical bifurcation; in other words, when R crosses 1 there is a global transfer of stability from the infection-free state to the endemic equilibrium, and vice versa. These results do not depend on the distribution of times spent in the infectious categories (the survivorship functions). Furthermore, by keeping all the key statistics fixed, we can compare two extremes: exponential survivorship versus piecewise constant survivorship (individuals remain infectious for a fixed length of time). By choosing some realistic parameters we can see (at least in these cases) that the reproductive numbers corresponding to these two extreme cases do not differ significantly whenever the two distributions have the same mean. At any rate a formula is provided that allows us to estimate the role played by the survivorship function (and hence the incubation period) in the global dynamics of HIV. These results support the conclusion that single population models of this type are robust and hence are good building blocks for the construction of multiple group models. Our understanding of the dynamics of HIV in the context of mathematical models for multiple groups is critical to our understanding of the dynamics of HIV in a highly heterogeneous population.     

Stability of differential susceptibility and infectivity epidemic models

We introduce classes of differential susceptibility and infectivity epidemic models. These models address the problem of flows between the different susceptible, infectious and infected compartments and differential death rates as well. We prove the global stability of the disease free equilibrium when the basic reproduction ratio R₀ ￡ 1{\mathcal{R}_0 \leq 1} and the existence and uniqueness of an endemic equilibrium when ${\mathcal{R}_0 >1 }${\mathcal{R}_0 >1 } . We also prove the global asymptotic stability of the endemic equilibrium for a differential susceptibility and staged progression infectivity model, when ${\mathcal{R}_0 >1 }${\mathcal{R}_0 >1 } . Our results encompass and generalize those of Hyman and Li (J Math Biol 50:626–644, 2005; Math Biosci Eng 3:89–100, 2006).     

A dynamic theory of coordination of discrete movement

The concepts of pattern dynamics and their adaptation through behavioral information, developed in the context of rhythmic movement coordination, are generalized to describe discrete movements of single components and the coordination of multiple components in discrete movement. In a first step we consider only one spatial component and study the temporal order inherent in discrete movement in terms of stable, reproducible space-time relationships. The coordination of discrete movement is captured in terms of relative timing. Using an exactly solvable nonlinear oscillator as a mathematical model, we show how the timing properties of discrete movement can be described by these pattern dynamics and discuss the relation of the pattern variables to observable end-effector movement. By coupling several such component dynamics in a fashion analogous to models of rhythmic movement coordination we capture the coordination of discrete movements of two components. We find the tendency to synchronize the component movements as the discrete analogon of in-phase locking and study its breakdown when the components become too different in their dynamic properties. The concept of temporal stability leads to the prediction that remote compensatory responses occur such as the restore synchronization when one component is perturbed. This prediction can be used to test the theory. We find that the discrete analogon to antiphase locking in rhythmic movement is a tendency to move sequentially, a finding that can also be subjected to empirical test.     

Updated Global Burden of Cholera in Endemic Countries

BackgroundThe global burden of cholera is largely unknown because the majority of cases are not reported. The low reporting can be attributed to limited capacity of epidemiological surveillance and laboratories, as well as social, political, and economic disincentives for reporting. We previously estimated 2.8 million cases and 91,000 deaths annually due to cholera in 51 endemic countries. A major limitation in our previous estimate was that the endemic and non-endemic countries were defined based on the countries’ reported cholera cases. We overcame the limitation with the use of a spatial modelling technique in defining endemic countries, and accordingly updated the estimates of the global burden of cholera.Conclusion/SignificanceThe global burden of cholera remains high. Sub-Saharan Africa accounts for the majority of this burden. Our findings can inform programmatic decision-making for cholera control.     

Transfer of cholera toxin genes from O1 to non‐O1/O139 strains by vibriophages from California coastal waters

Aims: Vibrio cholerae is an important bacterial pathogen that causes global cholera epidemic. Although they are commonly found in coastal waters around the world, most environmental isolates do not contain cholera toxin genes. This study investigates vibriophages in southern California coastal waters and their ability to transfer cholera toxin genes. Methods and Results: Lytic phages infecting V. cholerae were isolated from Newport Bay, California, between May and November, while none was found in winter. Some of the phage isolates can infect multiple environmental V. cholerae strains and El Tor strains. All phages contained double‐stranded DNA. Transduction experiments using kanamycin‐resistant gene marked CTXΦ demonstrated that some environmental vibriophages can transfer CTXΦ genes from O1 El Tor strain to environmental non‐O1/O139 V. cholerae via generalized transduction. Conclusions: Vibriophages are important components of the natural aquatic ecosystem. They play an important role in influencing the dynamics and evolution of V. cholerae in the environment. Significance and Impact of the Study: This study demonstrates the significance of vibriophages in the coastal environment and transduction as one of the mechanisms of pathogenicity evolution among environmental V. cholerae.     

Evolution of endemic and sylvatic lineages of dengue virus

Recent disease outbreaks have raised awareness of tropical pathogens, especially mosquito-borne viruses. Dengue virus (DENV) is a widely studied mammalian pathogen transmitted by various species of mosquito in the genus Aedes, especially Aedes aegypti and Aedes albopictus. The prevailing view of the research community is that endemic viral lineages that cause epidemics of DENV in humans have emerged over time from sylvatic viral lineages, which persist in wild, non-human primates. These notions have been examined by researchers through phylogenetic analyses of the envelope gene (E) from the four serotypes of DENV (serotypes DENV-1 to DENV-4). In these previous reports, researchers used visual inspection of a phylogeny in order to assert that sylvatic lineages lead to endemic clades. In making this assertion, these researchers also reasserted the model of periodic sylvatic to endemic disease outbreaks. Since that study, there has been a significant increase in data both in terms of metadata (e.g., place and host of isolation) and genetic sequences of DENV. Here, we re-examine the model of sylvatic to endemic shifts in viral lineages through a phylogenetic tree search and character evolution study of metadata on the tree. We built a dataset of nucleotide sequences for 188 isolates of DENV that have metadata on sylvatic or endemic sampling along with three orthologous sequences from West Nile virus as the outgroup for the phylogenetic analysis. In contrast to previous research, we find that there are several shifts from endemic to sylvatic lineages as well as sylvatic to endemic lineages, indicating a much more dynamic model of evolution. We propose that a model that allows oscillation between sylvatic and endemic hosts better captures the dynamics of DENV transmission.     

Competitive exclusion in a vector-host model for the dengue fever

 We study a system of differential equations that models the population dynamics of an SIR vector transmitted disease with two pathogen strains. This model arose from our study of the population dynamics of dengue fever. The dengue virus presents four serotypes each induces host immunity but only certain degree of cross-immunity to heterologous serotypes. Our model has been constructed to study both the epidemiological trends of the disease and conditions that permit coexistence in competing strains. Dengue is in the Americas an epidemic disease and our model reproduces this kind of dynamics. We consider two viral strains and temporary cross-immunity. Our analysis shows the existence of an unstable endemic state (‘saddle’ point) that produces a long transient behavior where both dengue serotypes cocirculate. Conditions for asymptotic stability of equilibria are discussed supported by numerical simulations. We argue that the existence of competitive exclusion in this system is product of the interplay between the host superinfection process and frequency-dependent (vector to host) contact rates. Received 4 December 1995; received in revised form 5 March 1996     

A logistic delay differential equation model for Chagas disease with interrupted spraying schedules

This work studies a mathematical model for the dynamics of Chagas disease, a parasitic disease that affects humans and domestic mammals throughout rural areas in Central and South America. It presents a modified version of the model found in Spagnuolo et al. [A model for Chagas disease with controlled spraying, J. Biol. Dyn. 5 (2011), pp. 299–317] with a delayed logistic growth term, which captures an overshoot, beyond the vector carrying capacity, in the total vector population when the blood meal supply is large. It studies the steady states of the system in the case of constant coefficients without spraying, and the analysis shows that for given-averaged parameters, the endemic equilibrium is stable and attracting. The numerical simulations of the model dynamics with time-dependent coefficients are shown when interruptions in the annual insecticide spraying cycles are taken into account. Simulations show that when there are spraying schedule interruptions, spraying may become ineffective when the blood meal supply is large.     

Whether the Weather Drives Patterns of Endemic Amphibian Chytridiomycosis: A Pathogen Proliferation Approach

The pandemic amphibian disease chytridiomycosis often exhibits strong seasonality in both prevalence and disease-associated mortality once it becomes endemic. One hypothesis that could explain this temporal pattern is that simple weather-driven pathogen proliferation (population growth) is a major driver of chytridiomycosis disease dynamics. Despite various elaborations of this hypothesis in the literature for explaining amphibian declines (e.g., the chytrid thermal-optimum hypothesis) it has not been formally tested on infection patterns in the wild. In this study we developed a simple process-based model to simulate the growth of the pathogen Batrachochytrium dendrobatidis (Bd) under varying weather conditions to provide an a priori test of a weather-linked pathogen proliferation hypothesis for endemic chytridiomycosis. We found strong support for several predictions of the proliferation hypothesis when applied to our model species, Litoria pearsoniana, sampled across multiple sites and years: the weather-driven simulations of pathogen growth potential (represented as a growth index in the 30 days prior to sampling; GI₃₀) were positively related to both the prevalence and intensity of Bd infections, which were themselves strongly and positively correlated. In addition, a machine-learning classifier achieved ∼72% success in classifying positive qPCR results when utilising just three informative predictors 1) GI₃₀, 2) frog body size and 3) rain on the day of sampling. Hence, while intrinsic traits of the individuals sampled (species, size, sex) and nuisance sampling variables (rainfall when sampling) influenced infection patterns obtained when sampling via qPCR, our results also strongly suggest that weather-linked pathogen proliferation plays a key role in the infection dynamics of endemic chytridiomycosis in our study system. Predictive applications of the model include surveillance design, outbreak preparedness and response, climate change scenario modelling and the interpretation of historical patterns of amphibian decline.     

Climate change accelerates local disease extinction rates in a long‐term wild host–pathogen association

Pathogens are a significant component of all plant communities. In recent years, the potential for existing and emerging pathogens of agricultural crops to cause increased yield losses as a consequence of changing climatic patterns has raised considerable concern. In contrast, the response of naturally occurring, endemic pathogens to a warming climate has received little attention. Here, we report on the impact of a signature variable of global climate change – increasing temperature – on the long‐term epidemiology of a natural host–pathogen association involving the rust pathogen Triphragmium ulmariae and its host plant Filipendula ulmaria. In a host–pathogen metapopulation involving approximately 230 host populations growing on an archipelago of islands in the Gulf of Bothnia we assessed changes in host population size and pathogen epidemiological measures over a 25‐year period. We show how the incidence of disease and its severity declines over that period and most importantly demonstrate a positive association between a long‐term trend of increasing extinction rates in individual pathogen populations of the metapopulation and increasing temperature. Our results are highly suggestive that changing climatic patterns, particularly mean monthly growing season (April‐November) temperature, are markedly influencing the epidemiology of plant disease in this host–pathogen association. Given the important role plant pathogens have in shaping the structure of communities, changes in the epidemiology of pathogens have potentially far‐reaching impacts on ecological and evolutionary processes. For these reasons, it is essential to increase understanding of pathogen epidemiology, its response to warming, and to invoke these responses in forecasts for the future.     

Global dynamics of cholera models with differential infectivity

A general compartmental model for cholera is formulated that incorporates two pathways of transmission, namely direct and indirect via contaminated water. Non-linear incidence, multiple stages of infection and multiple states of the pathogen are included, thus the model includes and extends cholera models in the literature. The model is analyzed by determining a basic reproduction number R₀ and proving, by using Lyapunov functions and a graph-theoretic result based on Kirchhoff’s Matrix Tree Theorem, that it determines a sharp threshold. If R₀?1, then cholera dies out; whereas if R₀>1, then the disease tends to a unique endemic equilibrium. When input and death are neglected, the model is used to determine a final size equation or inequality, and simulations illustrate how assumptions on cholera transmission affect the final size of an epidemic.     

Strange limits of stability in host-parasitoid systems

The classical Nicholson-Bailey model for a two species host-parasitoid system with discrete generations assumes random distributions of both hosts and parasitoids, randomly searching parasitoids, and random encounters between the individuals of the two species. Although unstable, this model induced many investigations into more complex host-parasitoid systems. Local linearized stability analysis shows that equilibria of host parasitoid systems within the framework of a generalized Nicholson-Bailey model are generally unstable. Stability is only possible if host fertility does not exceede ⁴=54.5982 and if superparasitism is unsuccessful. This special situation has already been discovered by Hassell et al. (1983) in their study of the effects of variable sex ratios on host parasitoid dynamics. We discuss global behaviour of the Hassell-Waage-May model using KAM-theory and illustrate its sensitivity to small perturbations, which can give rise to radically different patterns of the population dynamics of interacting hosts and parasitoids.     

Global stability of partially closed food-chains with resources

We consider the existence and global stability of aq-member equilibrium (1≤q≤n) in partially closed food-chains of lengthn having an abiotic component as resource. We observe that such existence demands bounds of resource supply rate and these bounds are weighted sums of interaction coefficients. Particular results of global sector-stability of partially feasible equilibria of simple food-chains obeying Lotka-Volterra dynamics are shown. Lastly the elasticity of such food-chains when a new species is introduced at the highest trophic level is investigated.     

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.