Modelling and analysis of the effects of malnutrition in the spread of cholera

Although cholera has existed for ages, it has continued to plague many parts of the world. In this study, a deterministic model for cholera in a community is presented and rigorously analysed in order to determine the effects of malnutrition in the spread of the disease. The important mathematical features of the cholera model are thoroughly investigated. The epidemic threshold known as the basic reproductive number and equilibria for the model are determined, and stabilities are investigated. The disease-free equilibrium is shown to be globally asymptotically stable. Local stability of the endemic equilibrium is determined using centre manifold theory and conditions for its global stability are derived using a suitable Lyapunov function. Numerical simulations suggest that an increase in susceptibility to cholera due to malnutrition results in an increase in the number of cholera infected individuals in a community. The results suggest that nutritional issues should be addressed in impoverished communities affected by cholera in order to reduce the burden of the disease.     

An age-structured model for the spread of epidemic cholera: Analysis and simulation

Occasional outbreaks of cholera epidemics across the world demonstrate that the disease continues to pose a public health threat. Traditional models for the spread of infectious diseases are based on systems of ordinary differential equations. Since disease dynamics such as vaccine efficacy and the risk for contracting cholera depend on the age of the humans, an age-structured model offers additional insights and the possibility of studying the effects of treatment options. The model investigated is given as a system of hyperbolic (first-order) partial differential equations in combination with ordinary differential equations. First, using a representation from the method of characteristics and a fixed point argument, we prove the existence and uniqueness of a solution to our nonlinear system. Then we present a finite difference approximation to the model and study the effect of high and low rates of shedding of cholera vibrios on the dynamics of the spread of the disease. The simulations demonstrate the explosive nature of cholera outbreaks that is observed in reality. The contrast of results for high and low rates of shedding of vibrios suggest a possible underlying cause for this effect.     

Threshold dynamics for a cholera epidemic model with periodic transmission rate

A cholera epidemic model with periodic transmission rate is presented. The basic reproduction number is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the cholera eventually disappears if the basic reproduction number is less than one. And if the basic reproduction number is greater than one, there exists a positive periodic solution which is globally asymptotically stable. Numerical simulations are provided to illustrate analytical results.     

Cholera model incorporating media coverage with multiple delays

In order to investigate the impact of awareness programs and time delays on the cholera outbreaks, we propose a cholera epidemic model, incorporating awareness programs by media as a separate class and two time‐delay factors. The bifurcation theory is applied to explore the variety of dynamics of this model for various combinations of the delays when R₀>1. Moreover, we analyze the direction, stability, and period of the bifurcating periodic solutions arising through Hopf bifurcation by using the normal form concept and the center manifold theory. Finally, we present numerical simulations to verify the main theoretical results.     

Why to consider environmental pollution in cholera modeling?

We propose a deterministic model to study the impact of environmental pollution on the dynamics of cholera. We consider both human to human and human‐environment‐human transmission modes in our model. We obtain the expression for the basic reproduction number of the proposed model. The study of our model reveals that environmental pollution plays a significant role in the spread of cholera and should not be ignored. Although various dimensions of cholera has been studied using mathematical models but scanty efforts have been made to understand impact of environmental pollution on this disease. Through this study, we try to bridge this gap.     

Cholera dynamics with Bacteriophage infection: A mathematical study

Mathematical modeling of waterborne diseases, such as cholera, including a biological control using Bacteriophage viruses in the aquatic reservoirs is of great relevance in epidemiology. In this paper, our aim is twofold: at first, to understand the cholera dynamics in the region around a water body; secondly, to understand how the spread of Bacteriophage infection in the cholera bacterium V. cholerae controls the disease in the human population. For this purpose, we modify the model proposed by Codeço, for the spread of cholera infection in human population and the one proposed by Beretta and Kuang, for the spread of Bacteriophage infection in the bacteria population [1, 2]. We first discuss the feasibility and local asymptotic stability of all the possible equilibria of the proposed model. Further, in the numerical investigation, we have found that the parameter ϕ, called the phage adsorption rate, plays an important role. There is a critical value, ϕ_c, at which the model possess Hopf-bifurcation. For lower values than ϕ_c, the equilibrium E^* is unstable and periodic solutions are observed, while above ϕ_c, the equilibrium E^* is locally asymptotically stable, and further shown to be also globally asymptotically stable. We investigate the effect of the various parameters on the dynamics of the infected humans by means of numerical simulations.     

Mathematical analysis and application of a cholera transmission model with waning vaccine-induced immunity

In this paper, a cholera infection model with vaccination is investigated, in which hyperinfectious and hypoinfectious vibrios, both human-to-human and environment-to-human transmission pathways, and waning vaccine-induced immunity are considered. The basic reproduction number is calculated and verified to be a threshold determining the global dynamics of the model. In addition, an application is demonstrated by investigating the cholera outbreak in Somalia, and the relevant control measures in the short term are given by elasticity and sensitivity analysis.     

Dynamic behavior of a delay cholera model with constant infectious period

In this paper, a delay cholera model with constant infectious period is investigated. By analyzing the characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium of the model is established. It is proved that if the basic reproductive number $\mathcal{R}_0>1$, the system is permanent. If $\mathcal{R}_0<1$, by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the disease-free equilibrium. If $\mathcal{R}_0>1$, also by means of an iteration technique, sufficient conditions are obtained for the global asymptotic stability of the endemic equilibrium. Numerical simulations are carried out to illustrate the main theoretical results.     

Ergodic stationary distribution and practical application of a hybrid stochastic cholera transmission model with waning vaccine-induced immunity under nonlinear regime switching

Considering the effect of stochasticity including white noise and colored noise, this paper aims to study a hybrid stochastic cholera epidemic model with waning vaccine-induced immunity and nonlinear telegraph perturbations. First, we derive a critical value ${?}_0^C$ related to the basic reproduction number ${?}_0$ of the deterministic model. The key aim of this paper is to generalize the θ-stochastic criterion method proposed by the recent work (Han et al. in Chaos Solit Fract 140:110238, 2020) to eliminate nonlinear telegraph perturbations. Next, via constructing several θ-stochastic Lyapunov functions and using the generalized method, we further prove that the stochastic model have a unique ergodic stationary distribution under ${?}_0^C>1$. Results show that the prevention and control of cholera epidemic depend on low transmission rate and small telegraph perturbations. Finally, the corresponding numerical simulations are performed to illustrate our analytical results and a practical application on the Somalia cholera outbreak is shown at the end of this paper.     

A possible counterexample to well posedness of entropy solutions and to Godunov scheme convergence

A particular case of initial data for the two-dimensional Euler equations is studied numerically. The results show that the Godunov method does not always converge to the physical solution, at least not on feasible grids. Moreover, they suggest that entropy solutions (in the weak entropy inequality sense) are not well posed.

     

Bounding graph diameters of solvable groups

We define a graph associated with a group G by letting nontrivial degrees be the vertices, and placing an edge between distinct degrees if they are not relatively prime. Using results in the literature, it is not difficult to show that when G is solvable and the graph is connected, its diameter is at most 4. Recent results suggest that this bound might be obtained. We show that in fact this diameter is at most 3, which is best possible.     

Global analysis of a periodic epidemic model on cholera in presence of bacteriophage

We propose and analyze a recurrent epidemic model of cholera in the presence of bacteriophage. The model is extended by general periodic incidence functions for low‐infectious bacterium and high‐infectious bacterium, respectively. A general periodic shedding function for two infected class (phage‐positive and phage‐negative) and a generalized contact and intrinsic growth function for susceptible class are also considered. Under certain biological assumptions, we derive the basic reproduction number (R₀) in a periodic environment for the proposed model. We also observe the global stability of the disease‐free equilibrium, existence, permanence, and global stability of the positive endemic periodic solution of our proposed model. Finally, we verify our results with specific functional form. Copyright © 2016 John Wiley & Sons, Ltd.     

Some results on Riccati equations,Floquet theory and applications

In this paper, we present two new results to the classical Floquet theory, which provides the Floquet multipliers for two classes of the planar periodic system. One of these results provides the Floquet multipliers independently of the solution of system. To demonstrate the application of these analytical results, we consider a cholera epidemic model with phage dynamics and seasonality incorporated.     

Global dynamics of an age-structured cholera model with both human-to-human and environment-to-human transmissions and saturation incidence

In this paper, an age-structured cholera model with both human-to-human and environment-to-human transmissions and saturation incidence is proposed. In the model, we consider the infection age of infectious individuals and the biological age of pathogen in the environment. It is verified that the global dynamics of the model is completely determined by the basic reproduction number. Asymptotic smoothness is verified as a necessary argument. By analyzing corresponding characteristic equations, we discuss the local stability of each of feasible steady states. Uniform persistence is shown by using the persistence theory for infinite dimensional dynamical system. The global stability of each of feasible steady states is established by using suitable Lyapunov functionals and LaSalle’s invariance principle. Numerical simulations are carried out to illustrate the theoretical results.     

On equivariant slice knots

We suggest a method to detect that two periodic knots are not equivariantly concordant, using surgery on factor links. We construct examples which satisfy all known necessary conditions for equivariant slice knots- Naik's and Choi-Ko-Song's improvements of classical results on Seifert forms and Casson-Gordon invariants of slice knots - but are not equivariantly slice.

     

Some results in Floquet theory,with application to periodic epidemic models

In this paper, we present several new results to the classical Floquet theory on the study of differential equations with periodic coefficients. For linear periodic systems, the Floquet exponents can be directly calculated when the coefficient matrices are triangular. Meanwhile, the Floquet exponents are eigenvalues of the integral average of the coefficient matrices when they commute with their antiderivative matrices. For the stability analysis of constant and nontrivial periodic solutions of nonlinear differential equations, we derive a few results based on linearization. We also briefly discuss the properties of Floquet exponents for delay linear periodic systems. To demonstrate the application of these analytical results, we consider a new cholera epidemic model with phage dynamics and seasonality incorporated. We conduct mathematical analysis and numerical simulation to the model with several periodic parameters.     

MODELS FOR COOPERATION AND CONSPIRACY IN FISHERIES: CHANGING THE RULES OF THE GAME

Fisheries regulation is considered necessary to counteract the effects of competitive forces which can lead to a “tragedy of the commons”. Yet management initiatives have often failed because they did not take into account competitive responses of fishing enterprises. In particular, open access fisheries provide strong incentives for the development of excessive harvesting capacity. This in turn leads to harvesting that is concentrated in space and time, with adverse effects on both the resource and markets. A coalition of fishermen, such as a fishermen's cooperative, has interests similar to those of a sole owner, and thus would be expected to produce more efficient behaviour. In practice, however, fishermen's cooperatives seldom persist. Game theory is used to explore relationships between the coalition structure of the industry, economic variables, and regulation. The models are based loosely on a purse seine fishery for herring. The results suggest that the potential to form stable coalitions is affected by changes in price and harvest. Changes in regulation also affect stability of coalitions. When interpreted in the light of historical changes in the herring fishery, these results suggest that industry may not accept regulations which do not permit formation of stable coalitions.     

Global dynamics of a Cholera model with age-of-immunity structure and reinfection

To understand V.Cholera transmission dynamics, in this paper, a mathematical model for the dynamics of cholera with reinfection is formulated that incorporates the duration time of the recovery individuals (age-of-immunity). The basic reproduction number $\Re_0$ for the model is identified and the threshold property of $\Re_0$ is established. By applying the persistence theory for infinite-dimensional systems, we show that the disease is uniformly persistent if the reproductive number $ \Re_0>1$. By constructing a suitable Lyapunov function, the global stability of the infection-free equilibrium in the system is obtained for $\Re_0<1$; the unique endemic equilibrium of the system is globally asymptotically stable for $\Re_0>1$.     

Dynamics of an ecological model living on the edge of chaos

We present a new ecological model, which displays “edge of chaos” (EoC) in parameter space. This suggests that ecological systems are not chaotic, instead, their dynamics can be characterized as short-term recurrent chaos. The system’s dynamics is unpredictable and admits bursts of short-term predictability. We also provide results, which suggest that fully developed chaos will rarely be observed in natural systems.     

Consumer heterogeneity and the shape of purchase rate functions

This study suggests that non-monotonic purchase rates, frequently observed in empirical studies of consumer purchase timing, can be an artifact of consumer heterogeneity. We use a theoretical purchase timing model with consumer heterogeneity to develop market scenarios under which non-monotonic rates may obtain. The results suggest that non-monotonic rates are more likely to observe in product markets that are either highly concentrated or where the heavy buyer segment is large. In such markets, therefore, one should not rely on non-monotonic rate curves to predict household purchase incidence. © 1998 John Wiley & Sons, Ltd.