Abstract: In this paper, we analyse the stability of the SEIR model
of malaria with infective immigrants which was recently formulated
by the authors. The model consists of an SEIR model for the human
population and SI Model for the mosquitoes. Susceptible humans
become infected after they are bitten by infectious mosquitoes and
move on to the Exposed, Infected and Recovered classes respectively.
The susceptible mosquito becomes infected after biting an infected
person and remains infected till death. We calculate the reproduction
number R0 using the next generation method and then discuss about
the stability of the equilibrium points. We use the Lyapunov function
to show the global stability of the equilibrium points.
Abstract: Cholera is a disease that is predominately common in
developing countries due to poor sanitation and overcrowding
population. In this paper, a deterministic model for the dynamics of
cholera is developed and control measures such as health educational
message, therapeutic treatment, and vaccination are incorporated in
the model. The effective reproduction number is computed in terms
of the model parameters. The existence and stability of the
equilibrium states, disease free and endemic equilibrium states are
established and showed to be locally and globally asymptotically
stable when R0 < 1 and R0 > 1 respectively. The existence of
backward bifurcation of the model is investigated. Furthermore,
numerical simulation of the model developed is carried out to show
the impact of the control measures and the result indicates that
combined control measures will help to reduce the spread of cholera
in the population.
Abstract: We present a deterministic model which describes
the dynamics of tuberculosis in Algerian population where the
vaccination program with BCG is in place since 1969 and where
the WHO recommendations regarding the DOTS (directly-observed
treatment, short course) strategy are in application. The impact
of an intervention program, targeting recently infected people
among all close contacts of active cases and their treatment to
prevent endogenous reactivation, on the incidence of tuberculosis,
is investigated. We showed that a widespread treatment of latently
infected individuals for some years is recommended to shift from
higher to lower equilibrium state and thereafter relaxation is
recommended.
Abstract: We used mathematical model to study the
transmission of dengue disease. The model is developed in which
the human population is separated into two populations, pregnant and
non-pregnant humans. The dynamical analysis method is used for
analyzing this modified model. Two equilibrium states are found and
the conditions for stability of theses two equilibrium states are
established. Numerical results are shown for each equilibrium state.
The basic reproduction numbers are found and they are compared by
using numerical simulations.
Abstract: Plasmodium vivax malaria differs from P. falciparum malaria in that a person suffering from P. vivax infection can suffer relapses of the disease. This is due the parasite being able to remain dormant in the liver of the patients where it is able to re-infect the patient after a passage of time. During this stage, the patient is classified as being in the dormant class. The model to describe the transmission of P. vivax malaria consists of a human population divided into four classes, the susceptible, the infected, the dormant and the recovered. The effect of a time delay on the transmission of this disease is studied. The time delay is the period in which the P. vivax parasite develops inside the mosquito (vector) before the vector becomes infectious (i.e., pass on the infection). We analyze our model by using standard dynamic modeling method. Two stable equilibrium states, a disease free state E0 and an endemic state E1, are found to be possible. It is found that the E0 state is stable when a newly defined basic reproduction number G is less than one. If G is greater than one the endemic state E1 is stable. The conditions for the endemic equilibrium state E1 to be a stable spiral node are established. For realistic values of the parameters in the model, it is found that solutions in phase space are trajectories spiraling into the endemic state. It is shown that the limit cycle and chaotic behaviors can only be achieved with unrealistic parameter values.
Abstract: The most Malaria cases are occur along Thai-Mynmar border. Mathematical model for the transmission of Plasmodium falciparum and Plasmodium vivax malaria in a mixed population of Thais and migrant Burmese living along the Thai-Myanmar Border is studied. The population is separated into two groups, Thai and Burmese. Each population is divided into susceptible, infected, dormant and recovered subclasses. The loss of immunity by individuals in the infected class causes them to move back into the susceptible class. The person who is infected with Plasmodium vivax and is a member of the dormant class can relapse back into the infected class. A standard dynamical method is used to analyze the behaviors of the model. Two stable equilibrium states, a disease-free state and an epidemic state, are found to be possible in each population. A disease-free equilibrium state in the Thai population occurs when there are no infected Burmese entering the community. When infected Burmese enter the Thai community, an epidemic state can occur. It is found that the disease-free state is stable when the threshold number is less than one. The epidemic state is stable when a second threshold number is greater than one. Numerical simulations are used to confirm the results of our model.