Interpolasi menggunakan splin trigonometri dengan satu parameter bentuk
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Date
2010
Authors
Razali, Noor Khairiah
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Abstract
Infectious diseases are a major health problem throughout the world. Each year,
millions of people die from infectious diseases such as measles, tuberculosis,
malaria, HIV/AIDS, and so on. Measles is a severe and highly contagious viral
infection of the respiratory tract, though its most prominent symptom is a skin rash.
Its virus spreads through direct contact with an infected person. Usually, measles is
not considered as a fatal disease in most developed countries. However, some reports
claim that measles is the most contagious of all infectious diseases. Measles
epidemics occur when the virus spreads rapidly through a susceptible population. In
this dissertation, we compare two models; seasonally forced nonlinear SEIR model,
and seasonally nonlinear SEIR model. We derive the seasonally nonlinear models by
following works done by previous reports of Liu et.al as stated in the bibliography to
determine the steady states and basic reproductive rates. We incorporate seasonality
by allowing the contact or transmission ra.te , 13 to vary over the course of the year .
Periodic contact rate implies the model with seasonal forcing. We used the simplest
possible form of the sinusoidal contact rate, f3(t) = 130(1 + 8 cos 21ft), which has
been widely used in studies of the dynamics of the epidemic models, also been used
by the numerous researchers likes Ira B.Schwartz. We then analyze the stability of
the equilibrium and the steady states. The "epidemic threshold" is the point at which
the percentage of unvaccinated people is high enough to risk an epidemic. The model
then was evaluated by using Mathematica. From the results, we can see clearly that
IX
the domain is positively invariant. Moreover, solutions exist for all positive time as
no solution path leave through any boundary and initial value problems have unique
solution that exists on maximal intervaL Clearly, the domain always have a diseasefree
equilibrium given by E = I = R = 0 so that S = 1 where S, E, I, and R are
susceptible, exposed, infectives, and recovered respectively. Basic reproductive rate
is denoted by Q. By linearization, if Q > 1, the disease-free equilibrium is locally
asymptotically stable. But then, if Q :$ 1, the disease-free equilibrium is globally
asymptotically stable in the domain, D, since the origin is the only positively
invariant subset of the set, and all path in the origin will approach the origin.