Abstract: An important technique in stability theory for
differential equations is known as the direct method of Lyapunov. In
this work we deal global stability properties of Leptospirosis
transmission model by age group in Thailand. First we consider the
data from Division of Epidemiology Ministry of Public Health,
Thailand between 1997-2011. Then we construct the mathematical
model for leptospirosis transmission by eight age groups. The
Lyapunov functions are used for our model which takes the forms of
an Ordinary Differential Equation system. The globally
asymptotically for equilibrium states are analyzed.
Abstract: The main aim of this study is to describe and introduce a method of numerical analysis in obtaining approximate solutions for the SIR-SI differential equations (susceptible-infectiverecovered for human populations; susceptible-infective for vector populations) that represent a model for dengue disease transmission. Firstly, we describe the ordinary differential equations for the SIR-SI disease transmission models. Then, we introduce the numerical analysis of solutions of this continuous time, discrete space SIR-SI model by simplifying the continuous time scale to a densely populated, discrete time scale. This is followed by the application of this numerical analysis of solutions of the SIR-SI differential equations to the estimation of relative risk using continuous time, discrete space dengue data of Kuala Lumpur, Malaysia. Finally, we present the results of the analysis, comparing and displaying the results in graphs, table and maps. Results of the numerical analysis of solutions that we implemented offers a useful and potentially superior model for estimating relative risks based on continuous time, discrete space data for vector borne infectious diseases specifically for dengue disease.
Abstract: One of the important tropical diseases is
Chikunkunya. This disease is transmitted between the human by the
insect-borne virus, of the genus Alphavirus. It occurs in Africa, Asia
and the Indian subcontinent. In Thailand, the incidences due to this
disease are increasing every year. In this study, the transmission of
this disease is studied through dynamical model analysis.
Abstract: The Swine flu outbreak in humans is due to a new
strain of influenza A virus subtype H1N1 that derives in part from
human influenza, avian influenza, and two separated strains of swine
influenza. It can be transmitted from human to human. A
mathematical model for the transmission of Swine flu is developed in
which the human populations are divided into two classes, the risk
and non-risk human classes. Each class is separated into susceptible,
exposed, infectious, quarantine and recovered sub-classes. In this
paper, we formulate the dynamical model of Swine flu transmission
and the repetitive contacts between the people are also considered.
We analyze the behavior for the transmission of this disease. The
Threshold condition of this disease is found and numerical results are
shown to confirm our theoretical predictions.
Abstract: Dengue, a disease found in most tropical and
subtropical areas of the world. It has become the most common
arboviral disease of humans. This disease is caused by any of four
serotypes of dengue virus (DEN1-DEN4). In many endemic
countries, the average age of getting dengue infection is shifting
upwards, dengue in pregnancy and infancy are likely to be
encountered more frequently. The dynamics of the disease is studied
by a compartmental model involving ordinary differential equations
for the pregnant, infant human and the vector populations. The
stability of each equilibrium point is given. The epidemic dynamic is
discussed. Moreover, the numerical results are shown for difference
values of dengue antibody.
Abstract: Dengue virus is transmitted from person to person
through the biting of infected Aedes Aegypti mosquitoes. DEN-1,
DEN-2, DEN-3 and DEN-4 are four serotypes of this virus. Infection
with one of these four serotypes apparently produces permanent
immunity to it, but only temporary cross immunity to the others. The
length of time during incubation of dengue virus in human and
mosquito are considered in this study. The dengue patients are
classified into infected and infectious classes. The infectious human
can transmit dengue virus to susceptible mosquitoes but infected
human can not. The transmission model of this disease is formulated.
The human population is divided into susceptible, infected, infectious
and recovered classes. The mosquito population is separated into
susceptible, infected and infectious classes. Only infectious
mosquitoes can transmit dengue virus to the susceptible human. We
analyze this model by using dynamical analysis method. The
threshold condition is discussed to reduce the outbreak of this
disease.
Abstract: A new strain of Type A influenza virus can cause the
transmission of H1N1 virus. This virus can spread between the
people by coughing and sneezing. Because the people are always
movement, so this virus can be easily spread. In this study, we
construct the dynamical network model of H1N1 virus by separating
the human into five groups; susceptible, exposed, infectious,
quarantine and recovered groups. The movement of people between
houses (local level) is considered. The behaviors of solutions to our
dynamical model are shown for the different parameters.
Abstract: Mathematical models can be used to describe the
transmission of disease. Dengue disease is the most significant
mosquito-borne viral disease of human. It now a leading cause of
childhood deaths and hospitalizations in many countries. Variations
in environmental conditions, especially seasonal climatic parameters,
effect to the transmission of dengue viruses the dengue viruses and
their principal mosquito vector, Aedes aegypti. A transmission model
for dengue disease is discussed in this paper. We assume that the
human and vector populations are constant. We showed that the local
stability is completely determined by the threshold parameter, 0 B . If
0 B is less than one, the disease free equilibrium state is stable. If
0 B is more than one, a unique endemic equilibrium state exists and
is stable. The numerical results are shown for the different values of
the transmission probability from vector to human populations.
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.