Prasenjit Das

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.

STUDY OF A CARRIER DEPENDENT INFECTIOUS DISEASE — CHOLERA

This paper analyzes an epidemic model for carrier dependent infectious disease — cholera. Existence criteria of carrier-free equilibrium point and endemic equilibrium point (unique or multiple) are discussed. Some threshold conditions are derived for which disease-free, carrier-free as well as endemic equilibrium become locally stable. Further global stability criteria of the carrier-free equilibrium and endemic equilibrium are achieved. Conditions for survival of all populations are also determined. Lastly numerical simulations are performed to validate the results obtained.

Study of an S-I epidemic model with nonlinear incidence rate: Discrete and stochastic version

Abstract Discrete and stochastic version of a susceptible-infective model system with nonlinear incidence rate is investigated. We observe that the discrete system converges to a unique equilibrium point for certain effective transmission rate of the disease and beyond which stability of the system is disturbed. Stochastic analysis suggests that the model system is globally asymptotically stable in probability for certain strengths of white noise. Numerical simulations are also performed to validate the results.

ANALYSIS OF A DISEASE TRANSMISSION MODEL OF HEPATITIS C

This study analyzes a model of hepatitis C with acute infectious, chronic infectious and the recovery or immune classes. Stability characters of disease-free and endemic proportionate equilibrium points are discussed. The role of immune system on the long-term survival of the susceptible population is derived. It has been shown that chronic infected populations persist whenever acute infected class persists and conversely. Lastly, the criterion for robustness of the system is established under stochastic perturbations. Numerical simulations are also performed to validate the results obtained.

EFFECT OF DELAY ON THE MODEL OF AMERICAN CUTANEOUS LEISHMANIASIS

This paper investigates the dynamics of transmission of a model of American cutaneous leishmaniasis (ACL). The model consists of infectious incidental hosts (humans), reservoir hosts (forest rodents, dogs, etc.) and vectors. A delay in the time between infection and infectiousness for the reservoir hosts is incorporated. In the absence of delay, it is observed that if R0 > 1, then endemic equilibrium becomes stable and if R0 < 1 then infection-free equilibrium becomes stable. The length of delay is estimated for preserving stability. Stability switching behavior of the system is also indicated. Lastly the results are verified through numerical simulations as far as possible.

Qualitative Analysis of a Cholera Bacteriophage Model

Cholera still remains as a severe global threat and is currently spreading in Africa and other parts of the world. The role of lytic bacteriophage as an intervention of cholera outbreaks is investigated using a mathematical model. Dynamics of cholera is discussed on basis of the basic reproduction number 𝑅0. Conditions of Hopf bifurcation are also derived for a positive net growth rate of Vibrio cholerae. Stability analysis and numerical simulations suggest that bacteriophage may contribute to lessening the severity of cholera epidemics by reducing the number of Vibrio cholerae in the environment. Hence with the presence of phage virus, cholera is self-limiting in nature. By using phage as a biological control agent in endemic areas, one may also influence the temporal dynamics of cholera epidemics while reducing the excessive use of chemicals. We also performed stochastic analysis which suggests that the model system is globally asymptotically stable in probability when the strengths of white noise are less than some specific quantities.

HIV/AIDS Model with Delay and the Effects of Stochasticity

We present a deterministic HIV/AIDS model with delay. We then extend the model by adjoining terms capturing stochastic effects. The intensity of the fluctuations in the stochastic system is analytically evaluated using Fourier transform methods. We carry out simulations to assess differences in the dynamical behavior of the deterministic and stochastic models. Simulation results show that they are no significant differences in the behavior of the two models.

GLOBAL DYNAMICS OF A MALARIA MODEL WITH PARTIAL IMMUNITY AND TWO DISCRETE TIME DELAYS

Asymptotic properties of a malaria model with partial immunity and two discrete time delays are investigated. The time delays represent latent period and partial immunity period in the human population. The results obtained show that the global dynamics are completely determined by the values of the reproductive number. Using a suitable Lyapunov function the endemic equilibrium is shown to be globally asymptotically stable under certain conditions. Moreover, we show that when the partially immune humans are assumed to be noninfectious, the disease is uniformly persistent if the corresponding reproductive number is greater than unity.

A STUDY OF SCHISTOSOME TRANSMISSION DYNAMICS AND ITS CONTROL

This article concentrates on the study of delay effect on a model of schistosomiasis transmission with control measures such as predation or harvesting and chemotherapy. In the presence of predation or harvesting and chemotherapy, system admits multiple endemic equilibria. Mathematical analysis shows that they are opposite in nature regarding stability. One may observe switching phenomena for the unstable equilibrium by incorporating delay. The disease may be highly endemic if there is no control measure, which is obvious from the model analysis. Results obtained in this paper are also verified through numerical simulations.

HOPF BIFURCATION AND PERMANENCE ANALYSIS OF AN S-I EPIDEMIC MODEL WITH DELAY

A Susceptible-Infective (S-I) model is considered in this paper with time delay. The death rate is assumed to be density dependent for this model. Conditions are derived under which there can be no change in stability. Using the discrete time delay as a bifurcation parameter, we show that the model undergoes a Hopf bifurcation. The permanence conditions of the system have also been derived. Results are verified by computer simulation.