1. INTRODUCTION:
Mathematical modeling, that concerns itself with population dynamics of infectious diseases, has been playing a prime role on account of its significance enabling a deep understanding of epidemiological patterns and disease control over a long period (Hethcote[4]). Epidemic models have long been employed to describe rapid outbreaks of diseases that occur in a period of less than one year, while endemic models have been employed for studying diseases over longer periods, during which time there is a renewal of susceptible by births or recovery from temporary immunity. An important aspect of the mathematical study of epidemiology is the formulation of the incidence rate function. The bilinear and standard incidence rates have frequently figured in classical epidemic models [1,4,6,7,10,11,12].
Capasso and Serio (1973) [2] introduced a
saturated incidence rate of the form
. Pathak et al. (2010) [8] proposed a modified
saturated incidence rate of the form 
and also considered
transmission rate 
which exhibits a
saturation effect to explain the presence of a maximal value pertaining
to the frequency of contact established by an individual over a period with
members of a given population distributed due to sharing the space
demographically which will culminate for societal existence.
Epidemiological models have become popular
in recent times from mathematicians and researchers for their role in
treatment functions within the realm of possibility offered by dynamic
equations. The key point to bear in mind is that as the count of
individuals vulnerable to infections increases, society’s resources need
to be expedited to counter the virulative spread of infection. Treatment
besides keeping an individual in isolation or quarantine is a vital means to
minimize the spread of diseases such as measles, AIDS, tuberculosis and flu
(Wang [5]). Classical epidemic models work on the assumption that the
treatment rate of infectives is directly proportional to the number of the
infectives. This does not keep compliance because the resources for
treatment are scant but they should be large enough. In fact, every community
is expected to have the required levels of mechanism for treatment. If
it is too expansive, the community ends up with needless burden uncalled for.
If it is not large enough, the community stands the risk of a massive
outbreak of the disease due to rapid infection. Thus, it is important to make
out a viable means to tackle the disease. In paper
[3], Wang and Ruan fostered to adopting an ideal and uniform treatment
which has been found to be facilitative when the number of infectives is
exceedingly high and simulates for an optimal bearing capacity. In 2006, Wang
[5] proposed a treatment function 
defined by
where r > 0 is a constant and represents the capacity of treatment for infectives. This implies that rate at which treatment given is proportional to the number of the infectives when the capacity for treating each and every infected individual becomes difficult, and if that isn’t the case, then the model assumes maximal capacity. This occurs for instance in situation when patients need to be hospitalized whereas the hospital is not accommodative in terms of number of beds available. This stands true in cases where the supply of medicine is either falling short or the prescribed medicine is not readily in stock.
This paper illustrates the following SIRS epidemic model which considers a modified saturated incidence rate under treatment:
(1)
where 
and
pertain to
the numbers of susceptible, infective, and recovered individuals at time
respectively.
refers
to the rate of recruitment of population, 
signifies the
proportionality constant, while 
apparently mark the saturation
parameters, 
shows the
natural death rate of the population,
points to the natural
recovery rate at which individuals who are down with an infection recover,
while 
reports the rate at
which those who have recuperated revert to the former condition and become
vulnerable again.
and if 
system (2) allows a
unique endemic equilibrium
where
(3)
is a manifold of the system
(2), which is attracting in the first octant. 
we have
is a solution of (4) and for
any 
the
general solution of (4) is 
as 
which implies the
conclusion.
and the endemic equilibrium
, we
rescale (5) 
of system (6) is
the disease-free equilibrium 
of system (6) is the endemic
equilibrium 
and
, where
is
it follows that 
is determined by
into
, we obtain
if 
when
The
positive nature of the solutions concerned, 
are
linear, and 
as 
if
as
then
then
the system (5) has a unique disease free equilibrium
which is globally
asymptotically stable.
which is locally stable.
Since the system does not have nontrivial periodic orbits in the positive
quadrant, 
must be globally stable if 
, we obtain
and 
with 
and 
with 
and 
then 
is
asymptotically stable. 
with 
and is locally
asymptotically stable (Fig.1). Fig.2 reveals how there is absence of nontrivial
periodic orbits in positive quadrant with the system remaining globally
asymptotically stable.
and is
locally stable (Fig.3) 
with all
other parameter values remaining the same as they have been for Fig.1 while for
the second system 
decreases as 
which
shows that 
decreases as 
increases.
there exist no
positive equilibrium, in which circumstance disease free equilibrium is
rendered globally asymptotically stable, i.e., the disease falls away. But when
the
unique endemic equilibrium is globally stable under some parametric conditions.
In addition, treatment rate and availability impact the control of the disease.
Any disease stands banished only if the appropriate treatment increases and is
made available. Numerical results affirm the analytical results obtained
through the equations.