Mathematical
Study of an Sir Epidemic Model with Nonmonotone Saturated Incidence Rate and
White Noise
Ranjith Kumar G1*,
Lakshmi Narayan K2, Ravindra Reddy B3
1Department
of Mathematics, ANURAG Group of Institutions. Hyderabad
2Department
of Mathematics, VIGNAN Institute of Tech. & Sci., Hyderabad
3Department of Mathematics, JNTUH
College of Engg., Jagityal, Karimnagar
*Corresponding Author E-mail: ranjithreddy1982@gmail.com
ABSTRACT:
This
paper contemplates an SIR epidemic model with non-monotone saturated incidence
rate for both deterministic and stochastic models. The stability of
disease-free and endemic equilibrium points of the deterministic model have
been dealt with first. As far as the stochastic version goes, the global
stability of endemic equilibrium is proved under suitable conditions on the
strength of the intensity of the white noise perturbation. Furthermore, we find
some numerical examples that attest to the analytical findings.
KEYWORDS:
Non-monotone Saturated Incidence rate, stochastic stability, Endemic
equilibrium, Lyapunov Function, White noise.
INTRODUCTION:
Epidemic models
have long been of interest to mathematicians. The first model of an epidemic
was mooted by Bernoulli in 1760. He used this model to evaluate the
effectiveness of variolation of healthy people with the smallpox virus. In 1911
Sir Ronald Ross used one form of mathematical modeling to explore the
effectiveness of various intervention strategies for Malaria. In 1927, Kermack
and McKendrick came out with an SIR mathematical model extensively in analyzing
the spread and control of infectious diseases qualitatively and quantitatively.
After Kermack-McKendrick model, different epidemic models have been formulated
and studied. A detailed history of mathematical epidemiology and the rationale
for SIR epidemic models are available in the classical books of Bailey, Murray
and Anderson and May.
It is one of the
most important issues that the dynamical behaviours are changed by the
different incidence rate in epidemic system. The non-linear incidence rate of
saturated mass action have been deployed by Liu et al, Ruan and Wang, Capasso
and Serio [1, 2, 7, 8, 10] and many others to arrive at certain persuasive
mathematical models that account for phenomena not easily amenable to a black
and white explanation.
Recent studies
have demonstrated that the non-linear incidence rate [3,4] is indeed a primary
determinant that induces periodic oscillations in epidemic models. These
periodic outbreaks are very important to predict and control the spread of
infectious diseases. This paper looks the non-linear incidence rate of the form
.
On the other
hand, if the environment is randomly varying, the population is subject to a
continuous spectrum of disturbances. That is to say, population systems are
often subject to and indeed vulnerable to environmental noise; to put it
bluntly, due to environmental fluctuations, parameters involved in epidemic
models are not absolute constants, and they fluctuate around some average
values. Keeping these factors in mind, more and more investigators began to
devote their attention to stochastic epidemic models describing the randomness
and stochasticity, and the stochastic epidemic models [5, 6] can provide an
additional degree of realism if compared to their deterministic counterparts.
In this paper we
study the effect of environmental fluctuations on the model (1.1).
we consider the
following model with non-monotonic saturated incidence rate.
5.
CONCLUSION:
In this paper,
we studied epidemic model with non-monotone saturated incidence rate for both deterministic
and stochastic approaches considering the white noise perturbation around the
endemic equilibrium state. In terms of the basic reproduction number R0,
our main results point that when R0<1, the disease-free
equilibrium is locally stable. When R0>1, the endemic equilibrium
exists and is locally stable. The model supplied in the paper has shown that
our stochastic model is globally asymptotically stable when the intensities of
white noise are less than certain threshold parameters. From the analytical and
numerical results, it is safe to conclude that the main factor that affects the
stability of the stochastic model is the intensity of white noise.
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