Modeling and Analysis of an Epidemic Model with Non-monotonic Incidence Rate under Treatment

An epidemic model with non-monotonic incidence rate under a limited resource for treatment is proposed to understand the effect of the capacity for treatment. We have assume that treatment rate is proportional to the number of infective when it is below the capacity and is a constant when the number of infective is larger than the capacity. Existence and stability of the disease free and endemic equilibrium are investigated for both the cases. Some numerical simulations are given to illustrate the analytical results.


Introduction
The incidence in an epidemiological model is the rate at which susceptible become infectious.The form of the incidence rate that is used in the classical Kermack Mckendrick model (1927) is the simple mass action λS I where S and I denote the number of susceptible and infectious, respectively, λ is called the infection coefficient.The standard incidence is λS I|N, where N is the total population size and λ is called the daily contact rate.Another kind of incidence is the saturation incidence λS I/(c + S ) where c is a constant.When the number of susceptible S is large compared to c that incidence is approximately λI.This kind of incidence was proposed by Anderson and May (1979), Lourdes and Matias (1991).Many researchers (see Hethcote and Levin, 1988;Esteva and Matias, 2001;Liu et al., 1986;Liu et al., 1987) have proposed transmission laws in which the nonlinearities are more than quadratic.Ruan and Wang (2003) studied an epidemic model with a specific nonlinear incident rate λI 2 S /(1 + αI 2 ) and presented a detailed qualitative and bifurcation analysis of the model.They derived sufficient conditions to ensure that the system has none, one, or two limit cycles and showed that the system undergoes a Bogdanov-Takens bifurcation at the degenerate equilibrium which include a saddle-node bifurcation, a Hopf bifurcation, and homoclinic bifurcation.
A more general incidence rate λI p S /(1 + αI q ) were proposed by many researchers and authors (see, Liu et al., 1986;Derrick and Van den Driessche,1993;Hethcote andVen den Driessche, 1991, Alexander andMoghadas, 2004).Xiao and Ruan, 2007 proposed an epidemic model with non-monotonic incidence rate λIS /(1 + αI 2 ).Treatment plays an important role to control or decrease the spread of diseases such as flue, tuberculosis, and measles (see Feng and Thieme, 1995;Wu and Feng, 2000;Hyman and Li, 1998).In classical epidemic models, the treatment rate is assumed to be proportional to the number of the infectious, which is almost impossible in real perspective because in that case the resources for treatment should be quite large.In fact, every country or society should have a suitable capacity for treatment.If it is too large, the country or society pays for unnecessary cost.If it is too small, the country or society has the risk of the outbreak ¢ www.ccsenet.org/jmrISSN: 1916-9795 of a disease.Wang (2006) proposed a treatment function: where K 1 = rI 0 .This type of treatment function is more realistic because in every hospital, the number of beds is limited and also they have a certain capacity of medicines.In our proposed model we have considered an epidemic model with non monotonic incidence rate under the treatment.Thus our model becomes (2) where S(t), I(t), R(t) denote the number of susceptible, infective, recovered individuals, respectively; a is the recruitment rate of the population, d is the natural death rate of the population, λ is the proportionality constant, m is the natural recovery rate of the infective individuals, β is the rate at which recovered individuals lose immunity and return to susceptible class, α is the parameter measures of the psychological or inhibitory effect.In our work we take the treatment function T (I), defined by This means that the treatment rate is proportional to the infective when the number of infective is less or equal to some fixed value I 0 and the treatment is constant when the number of infective crosses the fixed value I 0 .In practical view, the above form of treatment function is justified where patients have to be hospitalized and the number of beds is limited or the medicines are not sufficient.
Part I: SIR model with 0 ≤ I ≤ I 0 .

Equilibrium states and their stability
In this case the system (1)-(3) reduces to The system of equations ( 6)-( 8) always has the disease free equilibrium E 0 (a/d, 0, 0) for any set of parameter values.The endemic equilibrium is the solution of From the third equations we get R = {(m + r)/(d + β)}I and from the second equation S = (d + m + r)(1 + αI 2 )/λ.Now substituting R and S in the first equation, we get We define the basic reproductive number as follows From the equation ( 9) we see that if R 0 ≤ 1, there is no positive solution as in that case coefficient of I 2 , I and constant term are all positive, but if R 0 > 1, then by Descartes rule there exists a unique positive solution of (9) and consequently there exists unique positive equilibrium E * (S * , I * , R * ), called endemic equilibrium.
To investigate the stability of the system, we first prove that S (t) + I(t) + R(t) = a/d is invariant manifold of the system (6) -( 8), which is attracting the first octant. Let where Thus N(t) → a/d, as t → ∞.So the limit set of system ( 6) -( 8) is on the plane S + I + R = a/d.Thus the reduced system is Now to test the local stability of the above system we rescale the system by and obtain where Here E 0 (0, 0) is the disease free equilibrium and the unique positive equilibrium (x * , y * ) of the system ( 14)-( 15) is the endemic equilibrium E * of the model ( 6)-( 8).(x * , y * ) exists if u − K < 0 and is given by uvx Therefore, The jacobian matrix corresponding to E 0 (0, 0) is Here So, whenever E * exists, E 0 turns to an unstable saddle point.Now when (K − u) > 0 i.e.R 0 > 1, we discuss the stability of endemic equilibrium (x * , y * ).
Jacobian matrix corresponding to (x * , y * ) is ) 2 is determined by the sign of ¢ www.ccsenet.org/jmrISSN: 1916-9795 We have uvx therefore P 1 > 0, and hence det (M 1 ) is positive for any set of parameters.
Therefore, the positive equilibrium (x * , y * ) is either a node, a focus or a center.The eigen values of The fact that det (M 1 ) > 0 implies that The stability of the (x * , y * ) depends on the sign of the Trace and determinant of the jacobian matrix: So the sign of Trace (M 1 ) is determined by After some algebraic calculation using ( 18) and ( 19) we get, u 3 vP 2 = P 3 x * − P 4 , where Therefore P 3 and P 4 are positive for any set of parameters with K > u.So when (x * , y * ) exists, the condition for the local stability of (x * , y * ) becomes x * < P 4 /P 3 .
The above discussion can be stated through a theorem.
Theorem 2.1.(i) When the basic reproductive number R 0 ≤ 1, there exist no positive equilibrium of the system ( 14) -( 15), and in that case the only disease free equilibrium (0, 0) is a stable node.
(ii) When R 0 > 1, there exists a unique positive equilibrium of the system ( 14) -( 15), and in that case (0, 0) is an unstable saddle point.Also the condition for which the unique positive equilibrium will be locally stable is x * < P 4 /P 3 .
Global Stability.To investigate the global stability of the disease free equilibrium it is sufficient to show that (I(t), R(t)) → (0, 0).From here, it is clear that S (t) → a/d.Now from positivity of the solutions, I(t) and R(t) satisfy the differential inequality given by Here i(t), r(t) are linear, and (i(t), r(t)) Since I(t) ≤ i(t) and R(t) ≤ r(t), (I(t), R(t)) → (0, 0) as t → ∞ by simple comparison argument.Hence disease free equilibrium is globally stable.Now to investigate whether system (12) − (13) admits limit cycle or not, we take Dulac function D(I, R) = (1 + αI 2 )/λI, then February, 2010 hence the system ( 12)-( 13) has no limit cycle in the positive quadrant, so we reach the theorem 2.2.
Theorem 2.2.If R 0 < 1, then the disease free equilibrium E 0 (a/d, 0, 0) of the system (12) -( 13) is globally stable.But when R 0 > 1, system (12)-( 13) have unique positive equilibrium and further when x * < P 4 /P 3 that unique positive equilibrium must be locally stable.Again since the system have no limit cycle in the positive quadrant, E * (x * , y * ) must be globally stable under the condition R 0 > 1 and x * < P 4 /P 3 .
Part II.SIR model with I > I 0 .

Equilibrium states and their stability
In this case the model reduces to Since S + I + R = a/d is invariant manifold of the system ( 22)-( 24), the model reduces to we get where or, If u 1 + c > K, (29) has no positive solution, but if u 1 + c < K, it has either two positive roots or no positive root.By theory of equation has all of its roots real if G 2 + 4H 3 < 0 and H < 0, where Comparing equation ( 29) to (30) we have To investigate the local stability of the positive equilibrium ( x, ȳ) of the system ( 27)-( 28), we consider the jacobian matrix ¢ www.ccsenet.org/jmr Sign of det(M 2 ) is determined by Again (1 So, the sufficient condition for which P 5 > 0 is So the sign of Trace (M 2 ) is determined by After some algebraic calculation using ( 29) and ( 35) we get Therefore the sufficient condition for which P 6 < 0 is So we reach the theorem 3.1 Theorem 3.1.When K > u 1 + c, the system ( 27)-( 28) has two positive equilibrium ( x1 , ȳ1 ) and ( x2 , ȳ2 ), where x1 , x2 are two positive solutions of the equation ( 29) under the parametric restriction given by ( 31), moreover when the conditions (34) and ( 36) are satisfied at some equilibrium point, that equilibrium point must be asymptotically stable.

Simulation and Discussion
Case I. 0 ≤ I ≤ I 0 : If we choose the parameters as follows: 2, then we get the unique positive equilibrium point (18.18827, 5.76243, 6.050551).Here the basic reproductive number R 0 = 29.03226> 1.For the above choice of parameters we see that all the three components S (t), I(t), R(t) approach to their steady state values as time goes to infinity, the disease becomes endemic (see figure 2).

Journal of Mathematics Research
February, 2010 condition for local stability i.e. x * < P 4 /P 3 is satisfied here.We have drawn figures for both the system (S (t), I(t), R(t) and (x(T ), y(T )) to verify our result (see figures 4 and 5 ).
In our model parameter α describes the psychological or inhibitory effect.From (11), we see that the steady state value I * of the infective decreases as α increases.To verify the result we have plotted figure6 for different values of α, keeping all other values of the parameters same as for figure2.
Figure 13 shows that (19.0878, 10.2004) is a stable node, also figure 12 shows that the corresponding equilibrium point (48.96, 8.36, 4.47) of the system ( 22)-( 24) is a stable node.Figure 14 shows the dependence of the steady state value I * of I(t) on the parameter K 1 and we see that I * decreases as K 1 increases.

< Figure 14-16 >
We see that basic reproductive number plays an important role to control the disease.When R 0 ≤ 1, there exists no positive equilibrium, and in that case the disease free equilibrium is globally stable, that is the disease dies out.But when R 0 > 1, the unique endemic equilibrium is globally stable under some parametric condition.Also we see that the treatment rate plays a major role to control the disease.From figure 14(b), we can see that when the value of the parameter K 1 crosses a definite value 1.27, the disease dies out.Figure15 and 16 show that number of susceptible and recovered increases as the value of the parameter K 1 increases.