Backward Bifurcation of Sir Epidemic Model with Non- Monotonic Incidence Rate under Treatment

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1 OSR Journal of Mathematics (OSR-JM) e-ssn: , p-ssn: X. Volume, ssue 4 Ver. (Jul-Aug. 4), PP -3 Backwar Bifurcation of Sir Epiemic Moel with Non- Monotonic ncience Rate uner Treatment D. Jasmine an E. C. Henry Amirtharaj, PG & Research Department of Mathematics Bishop Heber College (Autonomous), Trichy. Abstract: Treatment is of great importance in fighting against infectious iseases. Backwar bifurcation of SR epiemic moel with treatment rate is propose an analyze by Wang W. We have reinvestigate the above moel by consiering a backwar bifurcation of SR epiemic moel with non-monotone incience rate uner treatment. t is assume that the treatment rate is proportional to the number of patients as long as this number is below a certain capacity an it becomes constant when that number of patients excees this capacity. Mathematical moels have become important tools in analyzing the sprea an control of infectious iseases. t is shown that this kin of treatment rate leas to the existence of multiple enemic equilibria where the basic reprouction number plays a big role in etermining their stability. Moreover, numerical calculations are support to analyze the global stability of unique enemic equilibrium. Keywors: Backwar Bifurcation, Basic reprouction number, Enemic, Epiemic, ncience rate, Treatment function.. ntrouction The asymptotic behavior of epiemic moels has been stuie by many researchers. Various epiemic moels have been formulate an analyze by many researchers. n the course of isease control, the treatment is an important metho to ecrease the sprea of iseases. Recently, many researchers have explore the role of treatment function T(), which represents the number of infectious iniviuals of healing, within epiemic isease moels. n classical epiemic moels, the treatment rate is assume to be proportional to the number of the infective. Assume that the capacity for the treatment of a isease in a community is a constant treatment rate r, in this case the treatment function is r. i.e., T() = r. For recent work, using an SR moel, Wang an Ruan [4 suppose that the treatment function has the form T() = r, if >, if = Where r is a constant value. They iscusse the stability of equilibria an prove the moel exhibits Boganov-Takens bifurcation, Hopf bifurcation an Homo-clinic bifurcation. Mena Lorca an Hethcote [3 also analyze an SRS moel with the same saturation incience. Non-linear incience rate of the form b p S q were investigate by Liu. et. al. [. A very general form of non-linear incience rate was consiere by Derrick an Driessche [3. A more general incience λ p S / (+α q ) was propose by many other researchers [,4,6,8,. Xiao an Ruan [5 propose an epiemic moel with non-monotonic incience rate λs / (+α ). Besies the rate an nature of incience, treatment plays an important role to control the sprea of iseases. This moel is investigate an analyze by Kar an Batabyal [. Also this moel is moifie by Gajenra. et. al. [5. As the improving of the hospital s treatment conitions, such as effective meicines, skillful techniques, the treatment rate will be increase. At last, the treatment rate of the isease won t increase until a plateau is reache. Therefore, constant recovery is a poor escription of real-worl infections. Aiming at this kin of improvement of circumstance, Wang incorporate the following piecewise linear treatment function into an SR moel. Wang [ propose a treatment function T() = r, if () K if > Where K = r for some fixe value. For classical epiemic moels, it is common that a basic reprouction number is a threshol in a sense that a isease is persistent if the basic reprouction number is greater than, an ies out if it is below. n this case, the bifurcation leaing from a isease free equilibrium to an enemic equilibrium is forwar. n recent years, papers foun backwar bifurcations ue to social groups with ifferent susceptibilities, pair formation, macro parasite infection, nonlinear inciences, an age structures in epiemic moels. n this case, the basic reprouction number oes not escribe the necessary elimination effort; rather the effort is escribe by the value of the critical parameter at the turning point. Thus, it is important to ientify backwar bifurcations to obtain threshols for the control of iseases. The treatment is an important metho to ecrease the sprea of iseases such as measles, tuberculosis an flu. n classical epiemic moels, the treatment rate of infective is assume to be proportional to the number of the infective. This is unsatisfactory because the resources for treatment shoul be quite large. n Page

2 Backwar Bifurcation of Sir Epiemic Moel with Non-Monotonic ncience Rate uner Treatment fact, every community shoul have a suitable capacity for treatment. f it is too large, the community pays for unnecessary cost. f it is too small, the community has the risk of the outbreak of a isease. Thus, it is important to etermine a suitable capacity for the treatment of a isease, we aopt a constant treatment, which simulates a limite capacity for treatment. Note that a constant treatment is suitable when the number of infective is large. n this paper, we moify it into r, ( ) k, if if,, T () where, k = r. This means that the treatment rate is proportional to the number of the infective when the capacity of treatment is not reache, an otherwise, takes the maximal capacity. This reners for example the situation where patients have to be hospitalize: the number of hospital bes is limite. This is true also for the case where meicines are not sufficient. Eviently, this improves the classical proportional treatment an the constant treatment. We will consier a population that is ivie into three types: susceptible, infective an recovere. Let S, an R enote the numbers of susceptive, infective, recovere iniviuals, respectively.. Mathematical Moel Then the moel to be stuie takes the following form: s S a S, t S ( m ) T ( ) t () R m R T ( ), t where a is the recruitment rate of the population, the natural eath rate of the population, is the proportionality constant, m is the natural recovery rate of the infective iniviuals, is the parameter measures of the psychological or inhibitory effect. t is assume that all the parameters are positive constants. Clearly, R 3 is positively invariant for system (). Since the first two equations in () are inepenent of the variable R, it suffices to consier the following reuce moel: S S a S, t (3) S ( m ) T ( ), t The purpose of this paper is to show that (3) has a backwar bifurcation if the capacity for treatment is small. We will prove that (3) has bi-stable enemic equilibrium if the capacity is low. The organization of this paper is as follows. n the next section, we stuy the bifurcations of (3). n Section 5 we present a global analysis an simulations of the moel.. Equilibrium n this section, we first consier the equilibria of (3) an their local stability. E = (A /, ) is a isease free equilibrium. An enemic equilibrium of (3) satisfies S a S S ( m ) T ( ) When <, system (4) becomes (4) 3 Page

3 Backwar Bifurcation of Sir Epiemic Moel with Non-Monotonic ncience Rate uner Treatment S a S S ( m r) (5) When >, the system (4) becomes S a S (6) S ( m ) k a Let R = ( m r). Then R is a basic reprouction number of (3). f R >, (5) amits a unique positive solution of E = (S, ) where S ( m r) S ( m r)( ) ( m r)( ) S, S a S ( m r)( ) ( m r)( ) a ( ) a ( m r) ( )( m r) Using the efinition of R, we get = R ( ) Clearly, E is an enemic equilibrium of (3) if an only if R ( ) ( R ) i. e., R R R 4 Page

4 Backwar Bifurcation of Sir Epiemic Moel with Non-Monotonic ncience Rate uner Treatment R. (7) n orer to obtain positive solution of system (6), we solve S from the first equation of (6) to obtain S, S a S a S( ) ( ) a S a( ) S ( ) When >, system (4) becomes S ( m ) k a( ) ( m ) k ( ) [( )( m ) a [ m k( ) a k (8) i.e., where A B C B m k( ) a. f B, it clear that (8) oes not have a positive solution. Let us consier the case where B<. f B 4[( )( m) a k, it is easy to obtain [ m k( ) a 4[( )( m ) a k R m r m k m a k where [ ( ) ( ) 4[( )( ) a R ( m r) is equivalent to t follows that R k( ) r ( )( m ) k p, (9) ( m r) ( m r) (or) k( ) r ( )( m ) k R. () ( m r) ( m r) Note that B < is equivalent to k( ) r R. () ( m r) t follows that B< an if an only if (9) hols. Equation (8) implies [( )( m ) a B k [( )( m ) a B B 4( )( m ) k B [( )( m ) a Then (8) has two positive solutions an where 5 Page

5 Backwar Bifurcation of Sir Epiemic Moel with Non-Monotonic ncience Rate uner Treatment ( B ) [( )( m ) a B () [( )( m ) a a( ) Set Si an Ei = (Si, i) for i =,. Then E i is an enemic equilibrium of (3) ( ) if i. Let us consier the conition uner which >. By the efinitions, we see that ( B ) [( )( m ) a This implies that ( B ) [( )( m ) a [( )( m ) a B B [( )( m ) a. (3) B [( )( m ) a. B [( s )( m ) a. (4) t follows from the efinition of B that k( ) r ( )( m ) R p. (5) ( m r) ( m r) Further, (3) implies that [ B ( )( m ). (6) By irect calculation, we see that (6) is equivalent to R ( ) p. (7) R p or R p. Hence > hols if an only if (5) an (7) are vali. Moreover, if We have. By similar arguments as above, we see that < if (5) hols, or p R P. (8) Furthermore, if R min p, p. (9) Summarizing the iscussion above, we have the following conclusions. Theorem: 4. V. Some Theorems E = (S, ) is an enemic equilibrium of (3) if an only if R. p enemic equilibrium of (3) if R P an one of the following conitions are satisfie R (i) (ii) p R. p p r ( )( m ) is equivalent to that p. Note that Theorem: 4. Enemic equilibrium E an E o not exist if. p p R Further, if R p Furthermore, E is the unique, we have the following: 6 Page

6 Backwar Bifurcation of Sir Epiemic Moel with Non-Monotonic ncience Rate uner Treatment r ( )( m ), then both E S, ) an E S, ) i) f p R p. ( ( ii) f iii) Let r ( )( m ), Then E oes not exist. Further, E exist when p an E oes not exist when R. exist when r ( )( m ), then E oes not exist but E exist if when R. R p p We consier p. f r ( )( m ), a typical bifurcation iagram is illustrate in figure, where the bifurcation from the isease free equilibrium at R = is forwar an there is a backwar bifurcation from an enemic equilibrium at R =.5, which gives rise to the existence of multiple enemic equilibrium. Further, if r ( )( m ), a typical bifurcation iagram is illustrate in Fig., where the bifurcation at R is forwar an (3) has one unique enemic equilibrium for all R. Note that a backwar bifurcation with enemic equilibrium when R is very interesting in applications. We present the following corollary to give conitions for such a backwar bifurcation to occur. Fig. The figure of infective sizes at equilibrium versus R when = 4, m =., ε =., =.8, r =.5, where (i) of theorem 4. hols. The bifurcation from the isease free equilibrium at R = is forwar an there is a backwar bifurcation from an enemic equilibrium at R =.5, which leas to the existence of multiple enemic equilibrium. Corollary: 4.3 System (3) has a backwar bifurcation with enemic equilibrium when R < if r ( )( m ), an p <. Proof: This corollary is a simple consequence of (i) of theorem 4. Example: Fix =, λ =., ε =., =, α =., k =., m =.3 an r = 6. Then p.6, p.977, p 3. an r ( )( m) 3.. Thus, (3) has a backwar bifurcation with enemic equilibrium this case (by fig3). As (the capacity of treatment resources) increases, by the efinition we see that p increases. When is so large that p >, it follows from Theorem 3. that there is no backwar bifurcation with enemic equilibrium when R <. f we increase to R p, (3) oes not have a backwar bifurcation because enemic equilibrium E an E o not exist. This means that an insufficient capacity for treatment is a source of the backwar bifurcation. 7 Page

7 Backwar Bifurcation of Sir Epiemic Moel with Non-Monotonic ncience Rate uner Treatment Fig. The iagram of, versus R when = 4,., m.,., =.8, r =.6, where (iii) of Theorem 4. hols. The bifurcation at R = is forwar an (3) has a unique enemic equilibrium for R >. Fig. 3 The figure of, an versus R that shows a backwar bifurcation with enemic equilibrium when R <, where Corollary 4.3 hols. Theorem: 4.4 E is asymptotically stable if R an R. E is asymptotically stable if. E is sale whenever it exists. For E, we have i) E is stable if either 3 ( ) a 3 ( m ) / ( m ) ( ) k () (or) 3 ( ) k ( ) a 3 ( m ) / ( m ), 4( ) a ( ) k ( ) a ( m )( m ) ( m ) () ii) E is unstable if 3 ( ) k ( ) a 3 ( m ) / ( m ), 4( ) a ( ) k ( ) a ( m )( m ) ( m ) () 8 Page

8 Backwar Bifurcation of Sir Epiemic Moel with Non-Monotonic ncience Rate uner Treatment V. Global Analysis an Simulation We begin from the global stability of the isease free equilibrium E Theorem: 5. The isease free equilibrium E is globally stable, i.e., the isease ies out, if one of the following conitions is satisfie: R an p. Proof (i) (ii) R, p < an p R implies that E oes not exist. Suppose p. t follows from the Theorem 4. that E exists or E exist only if R > p, which is possible since we have R. Let us now suppose p < an p. f r ( )( m ), since p p, it follows from the iscussions for (i), (ii) of Theorem 4. that E or E exists only if R p, which is impossible since we have R. f r ( )( m ), since p, it follows from (iii) of Theorem 4. that E an E o not exist. n summary, enemic equilibrium o not exist uner the assumptions. t is easy to verify that positive solutions of (3) are ultimately boune. Note that the nonnegative S- axis is positively invariant an that the non negative -axis repels positive solutions of (3). Since E is asymptotically stable, it follows from the Benixson Theorem that every positive solution of (3) approaches E as t approaches infinity. The limit cycles of (3) play crucial roles on the structure of ynamical behaviors of the moel. For example, if there is no limit cycle an its enemic equilibrium is unique, the unique enemic equilibrium is globally stable. For this reason, we aopt Dulac functions to obtain conitions for the nonexistence of a limit cycle in (3). We enote the right han sies of (3) by f an f. Then we get ( Df) ( Df) S then (3) oes not have a limit cycle. Theorem: 5. System (3) oes not have a limit cycle if r <. Proof By the first equation of (3), it is easy to see that positive solutions of (3) eventually enter an remain in the region a P : ( S, ) : S. Thus, a limit cycle, if it exists, must lie in the region P. Take a Dulac function D S. Then we have if. f ( Df) ( Df) a S S, it is easy to see that ( Df) ( Df) a k a ( a ks ) ( r ) S S S S S Hence (3) oes not have a limit cycle. Theorem 5. implies that there is no limit cycle in (3) if the treatment rate is less than the eath rate. We now present a ifferent conition for the nonexistence of limit cycles of (3) which is inepenent of the treatment Theorem: 5.3 System (3) oes not have a limit cycle if ( ) a ( m ). (3) Proof We consier a Dulac function D efine by, 9 Page

9 Backwar Bifurcation of Sir Epiemic Moel with Non-Monotonic ncience Rate uner Treatment, if, S D, if, S Then we have ( Df) ( Df) a S S if. Note that (3) implies that a a [ a ( ) S ( m ) S Sa/ a ( )( ) ( m )( ) a ma a a a a a a a a ma a a( a a m ) a( a a m ) [ a ( ) S ( m ) S Sa/ t follows that for ( Df) ( Df) [ a ( ) S ( m ) S S S where the omain P is consiere. Therefore, (3) oes not have a limit cycle. Fig. 4 E is globally stable when a =, = 4, =., m =., =., =.8, k =., r =.5 an α =. 3 Page

10 Backwar Bifurcation of Sir Epiemic Moel with Non-Monotonic ncience Rate uner Treatment Fig.5 One region of isease persistence an one region of isease extinction where a = 5, =, k =., =.,. m =., = an r = 6, where Corollary.3 hols. Fig.6 A bi-stable case where enemic equilibria E an E are stable when a =6, = 4, =., m.,., =.8, r =.5, where (i) of theorem 4. hols. f the conitions of theorem 4., 5. or 5.3 are satisfie, then E is one unique enemic equilibrium an is globally stable. Numerical calculations shows that the unique enemic equilibrium E is also globally stable if Theorems 5.3 an 5.4 are not satisfie (see Fig. 5). f R > p. Theorem 4. implies that E is a unique enemic equilibrium of (3). Numerical calculations shows that this unique enemic equilibrium is also globally stable. However, by irect calculations, we see that > if R > p. This means that we have more infecte members in the limite resources for treatment than those with the unlimite resources for treatment. Let the conitions of Corollary 4.3 hol. f R <, the stable manifols of the sale E split R into two regions. The isease is persistent in the upper region an ies out in the lower region (see Fig. 6). The stable manifols of the sale E split the feasible region into two parts; positive orbits in the lower part approach to the enemic equilibrium E an positive orbits in the upper part approach to the enemic equilibrium E. Thus, we have a bi-stable case. Concluing Remarks n this paper, we have propose an epiemic moel to simulate the limite resources for the treatment of patients, which can occur because patients have to be hospitalize but there are limite bes in hospitals, or there is not enough meicine for treatments. We have shown in Corollary 4.3 that backwar bifurcations occur because of the insufficient capacity of treatment. We have also shown that (3) has bi-stable enemic equilibrium because of the limite resources. This means that riving the basic reprouction number below is not enough to eraicate the isease. The level of initial infectious invasion must be lowere to a threshol so that the isease ies out or approaches a lower enemic steay state for a range of parameter 3 Page

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