Abstract: In this work, we consider a deterministic model for
the transmission of leptospirosis which is currently spreading in the
Thai population. The SIR model which incorporates the features of
this disease is applied to the epidemiological data in Thailand. It is
seen that the numerical solutions of the SIR equations are in good
agreement with real empirical data. Further improvements are
discussed.
Abstract: The effect of a time delay on the transmission on
dengue fever is studied. The time delay is due to the presence of an
incubation period for the dengue virus to develop in the mosquito
before the mosquito becomes infectious. The conditions for the
existence of a Hopf bifurcation to limit cycle behavior are
established. The conditions are different from the usual one and they
are based on whether a particular third degree polynomial has
positive real roots. A theorem for determining whether for a given
set of parameter values, a critical delay time exist is given. It is
found that for a set of realistic values of the parameters in the model,
a Hopf bifurcation can not occur. For a set of unrealistic values of
some of the parameters, it is shown that a Hopf bifurcation can occur.
Numerical solutions using this last set show the trajectory of two of
the variables making a transition from a spiraling orbit to a limit
cycle orbit.
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.