# Analysis of a Dengue Disease Model with Nonlinear Incidence

Authors: Shu-Min Guo; Xue-Zhi Li; Mini Ghosh
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

A dengue disease epidemic model with nonlinear incidence is formulated and analyzed. The equilibria and threshold of the model are found. The stability of the system is analyzed through a geometric approach to stability. The proposed model also exhibits backward bifurcation under suitable conditions on parameters. Our results imply that a nonlinear incidence produces rich dynamics and they should be studied carefully in order to analyze the spread of disease more accurately. Finally, numerical simulations are presented to illustrate the analytical findings.


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## Body

## 1. Introduction

Dengue disease is transmitted in humans by mosquitoes, called vectors, who carry the disease without getting it themselves. Every year many people die from dengue disease. It has become a global health problem due to its rapid spread. In order to predict the development extent of such epidemics, it is useful to find the suitable mathematical models, which can provide insight into the dynamics of a disease and can help us to make right decisions on public health policies.However in the region of high infective prevalence such as Africa, some tropical countries, and so forth, there are lots of people infected with dengue, and so forth. and hence there are lots of infected mosquitoes. So in these endemic regions residents are frequently bitten by infected mosquitoes, thus making individuals exposed with multiple bites of mosquito which causes individuals to move to infected class with increased rate compared to people who are exposed with single effective bite of mosquito. So it is reasonable to consider nonlinear saturation incidence in place of bilinear incidence or standard incidence in the model as it also incorporates the effects of increased exposure to mosquito bites.In fact the incidence rate is very important in the transmission dynamics of the disease and the qualitative behaviour of the disease changes with the change in the incidence rate. In a recent paper by Xiao and Tang [1], interaction of the nonlinear incidence and the partial immunity for a simple SIV epidemic model is investigated and it is shown that due to the nonlinear incidence, vaccination may contribute to disease spread rather than to its elimination. A vector-host epidemic model with nonlinear incidence is also analyzed by Cai and Li [2]. Here they considered nonlinear saturating type incidence and they studied the effect of this type of incidence rate on the basic reproduction number and the equilibrium level of the infective population. Hu et al. [3] analyzed an SIR epidemic model with nonlinear incidence rate and treatment. Here it is shown that reducing the basic reproduction number below one is not enough to eliminate the disease as backward bifurcation occurs for this model. Also when the basic reproduction number is greater than unity, there are multiple endemic equilibria. The conditions for the stability of endemic equilibrium and the existence of limit cycle are discussed in this paper. A discrete-time epidemic model with nonlinear incidence rates is studied by Li et al. [4], where a very complex dynamical behaviors are demonstrated. It is shown that there are possibilities of the transcritical bifurcation, flip bifurcation, Hopf bifurcation and chaos. A vector-borne model with nonlinear incidence is analyzed by Ozair et al. [5]. Here they considered the conditions for the stability of endemic equilibrium.In recent years, dengue has become a major public health problem and is endemic in several countries. It is affecting almost one-third of the world's population. Dengue is an infection caused by a virus known as flavivirus and is spread by their vector Aedes aegypti. There are four different serotypes of flavivirus but infection with one serotype gives life-long immunity to that serotype but not to other three serotypes. So one can formulate dengue disease model as SIS or SIR model depending upon serotypes of flavivirus. In this paper we have formulated an SIR epidemic model with nonlinear incidence assuming the infection of dengue is caused by the flavivirus which is giving life-long immunity. In dengue-endemic area a person can be infected with more than one serotype of flavivirus in his/her lifetime. But in the current study we have ignored this fact as it will complicate the model and its analysis.The paper is organized as follows. In Section2 the model and some preliminary properties are presented. In Section 3 equilibria and bifurcation are studied. Section 4 is devoted to the global stability analysis of the endemic equilibrium, when it is unique. Numerical simulations are given in Section 5, we end the paper with conclusions in Section 6.


## 2. Model Formulation

The vector-host epidemic model with nonlinear incidence can be described by the following system of differential equations:(1)dS(t)dt=b1-λ1S(t)V(t)[1+αV(t)]-μ1S(t),dI(t)dt=λ1S(t)V(t)[1+αV(t)]-γI(t)-μ1I(t),dR(t)dt=γI(t)-μ1R(t),dM(t)dt=b2-λ2M(t)I(t)-μ2M(t),dV(t)dt=λ2M(t)I(t)-μ2V(t).
The total host population size at time t, given by N1(t), is partitioned into subclasses of individuals who are susceptible, infectious and recovered, with sizes denoted by S(t), I(t) and R(t), respectively. Furthermore, the host population dies at a natural death rate μ1. In addition, the host population is recruited at a rate b1. We assume that vertical transmission in the host does not occur so that all newly recruited individuals are susceptible. The per capita recovery rate of the hosts is given by γ. The recovered individuals are assumed to acquire permanent immunity and there is no transfer from the R class back to the S class. The nonlinear incidence rate is λ1S(t)V(t)[1+αV(t)], where λ1 and α are positive constants, V(t) is the number of vectors at time t who carry the pathogen. It corresponds to an increased rate of infection due to two exposures over a short time period. The single contacts lead to infection at the rate λ1S(t)V(t), whereas the new infective individuals arise from double exposures at a rate αλ1S(t)V2(t).The second component of the vector population is the number of pathogen-free (susceptible) vectors at timet, given by M(t). The total size of the vector population at time t, given by N2(t) and is subdivided into these two vector-population classes, the susceptible vectors and infectious vectors. The vector population dies at a natural death rate μ2. In addition, the vector population is recruited at a birth rate b2. Although there are evidences that the pathogen of several vector-borne diseases (e.g., West Nile fever, yellow fever and Lyme disease) can be transmitted from (female) parent to offspring in the vector population, we will assume that all newborn vectors are susceptible and vertical transmission is neglected. Susceptible vectors start carrying the pathogen after getting into contact (biting) an infective host at a rate λ2 so that the incidence of newly infected vectors is given by a mass-action term λ2M(t)I(t). In contrast to the host population, once the vectors become carriers of the microparasite, they carry it for life.We make some reasonable technical assumptions on the parameters of the model, namely thatγ>0,  μj>0 and bj>0 for j=1,2. The above systems for the host population and the vector are also equipped with initial conditions as follows: S(0)=S0, I(0)=I0, R(0)=R-0, M(0)=M0 and V(0)=V0.The total population sizeN1(t) can be determined by N1(t)=S(t)+I(t)+R(t) or from the differential equation
(2)dN1(t)dt=b1-μ1N1(t),
which is derived by adding the first three equations in (1). The total number of vectors N2(t) can be determined by N2(t)=M(t)+V(t) or from the differential equation
(3)dN2(t)dt=b2-μ2N2,
which is derived by adding the last two equations in (1).It is easily seen that both for the host population and for the vector population the corresponding total population sizes are asymptotically constant:(4)limt→∞N1(t)=b1μ1,limt→∞N2(t)=b2μ2.
So we can assume without loss of generality that N1(t)=b1/μ1,  N2(t)=b2/μ2 for all t≥0.Then the dynamics of system (1) is qualitatively equivalent to the dynamics of system given by
(5)dS(t)dt=b1-λ1S(t)V(t)[1+αV(t)]-μ1S(t),dI(t)dt=λ1S(t)V(t)[1+αV(t)]-γI(t)-μ1I(t),dV(t)dt=λ2[b2μ2-V(t)]I(t)-μ2V(t).
The values of R and M can be determined correspondingly from R=b1/μ1-S-I, and M=b2/μ2-V respectively.For biological reasons we need the solutions nonnegative. Mathematical properties of the solutions lead us to study the system (5) in the closed set
(6)Γ={(S,I,V)∈R+3∣0≤S+I≤b1μ1,0≤V≤b2μ2,S≥0,I≥0},
where R+3 denotes the non-negative cone of R3 including its lower dimensional faces. It can be verified that Γ is positively invariant with respect to (5). We denote by ∂Γ and Γ0 the boundary and the interior of Γ in R3, respectively.


## 3. Equilibria and Bifurcation Analysis

The disease-free equilibrium of system (5) is E0=(b1/μ1,0,0). The endemic equilibria E1=(S*,I*,V*) of system (5) can be deduced by the system,
(7)b1-λ1S*V*[1+αV*]-μ1S*=0,λ1S*V*[1+αV*]-γI*-μ1I*=0,λ2[b2μ2-V*]I*-μ2V*=0,
which gives,
(8)S*=b1λ1V*(1+αV*)+μ1,I*=μ2V*λ2(b2/μ2-V*),
and V* is the positive solution of the following equation,
(9)a1V*2+a2V*+a3=0,
where
(10)a1=λ1λ2b1α+μ2λ1α(γ+μ1),a2=λ1λ2b1+μ2λ1(γ+μ1)-λ1λ2b1b2αμ2=αμ1μ2(γ+μ1)(R1*-R0),a3=μ1μ2(γ+μ1)-λ1λ2b1b2μ2=μ1μ2(γ+μ1)(1-R0),R1*=λ1λ2b1+μ2λ1(γ+μ1)αμ1μ2(γ+μ1),R0=λ1λ2b1b2μ1μ22(γ+μ1).
Thus, we observe that
(11)a1>0;a2<0⟺R1*<R0;a3>0⟺R0<1.
By the Descartes’ rule of signs, it follows that there is a unique endemic equilibrium whenever a3<0 and there are two endemic equilibria whenever a3>0, a2<0 and a22-4a1a3>0.Furthermore, we observe that there is a bifurcation point whena3>0, a2<0 and a22-4a1a3=0. In fact the quantity a22-4a1a3 can be expressed in terms of new threshold R0*, where
(12)R0*=M0N0,M0=2λ12λ22b12b2αμ1μ2(γ+μ1)+2λ12λ2b1b2αμ1+P0,N0=λ1λ2b1μ22+2λ1μ23(γ+μ1)b2+λ1μ24(γ+μ1)2λ2b1b2+λ1λ2b1b2α2+P0,P0=4λ1λ2b1αμ2+4λ1αμ22(γ+μ1).
By a little algebraic calculation, we have a22-4a1a3>0 whenever R0>R0*. Taking into account the above consideration, we have the following theorem.Theorem 1.
System (7) admits a unique endemic equilibrium E when R0≥1; there are no endemic equilibria for R0<R0*; there are two endemic equilibria, E1 and E2, for R<R0<1 and R=max{R0*,R1*}.Next we analyze the stability of the disease-free equilibriumE0=(b1/μ1,0,0). The Jacobian matrix corresponding to (7) is
(13)J(S,I,V)=(-λ1V(1+αV)-μ10-λ1S(1+2αV)λ1V(1+αV)-γ-μ1λ1S(1+2αV)0λ2(b2μ2-V)-λ2I-μ2).
First, we observe that
(14)J(b1μ1,0,0)=(-μ10-λ1b1μ10-γ-μ1λ1b1μ10λ2b2μ2-μ2),
so that the eigenvalues ξ are given by the roots of the following qubic equation
(15)(μ1+ξ)[ξ2+(γ+μ1+μ2)ξ+μ2(γ+μ1)(1-R0)]=0.
Hence if R0<1, then the eigenvalues are all negatives and E0 is locally asymptotically stable. If R0>1, then two eigenvalues are negative and one is positive, so that E0 is unstable.Since there are no explicit expressions for equilibriaE1 and E2 and the existence conditions for the two equilibria are very complex, it is very difficult to prove their stability by using the existing analytic methods.Theorem1 establishes that R0=1 is a bifurcation value. In fact, across R0=1 the disease free equilibrium changes its stability properties. Now, we investigate what kind of bifurcation occurs at R0=1. In order to do that, we will make use of the result summarized below, which has been obtained in [6] and is based on the use of the center manifold theory [7].Let us consider a general system of ODEs with a parameterϕ:
(16)x˙=f(x,ϕ);f:Rn×R→Rn,f∈C2(Rn×R).
Without loss of generality, we assume that x=0 is an equilibrium for (16).Theorem 2 (see (Castillo-Chavez and Song [6])).
Assume:(A1)
A=Dxf(0,0) is the linearization matrix of system (16) around the equilibrium x=0 with ϕ evaluated at 0. Zero is a simple eigenvalue of A and all other eigenvalues of A have negative real parts.(A2)
MatrixA has a right eigenvector ω and a left eigenvector θ corresponding to the zero eigenvalue.
Letfk denotes the kth component of f and,
(17)a=∑k,i,jn‍θkωiωj∂2fk∂xi∂xj(0,0),b=∑k,i,jn‍θkωi∂2fk∂xi∂ϕ(0,0).
Then the local dynamics of system (16) around x=0 are totally determined by a and b(i)
a>0, b>0. When ϕ<0, with |ϕ|≪1, x=0 is locally asymptotically stable and there exists a positive unstable equilibrium; when 0<|ϕ|≪1, x=0 is unstable and there exists a negative and locally asymptotically stable equilibrium;(ii)
a<0, b<0. When ϕ<0, with |ϕ|≪1, x=0 is unstable; when 0<|ϕ|≪1, x=0 is locally asymptotically stable and there exists a positive unstable equilibrium;(iii)
a>0, b<0. When ϕ<0, with |ϕ|≪1, x=0 is unstable and there exists a locally asymptotically stable negative equilibrium; when 0<|ϕ|≪1, x=0 is stable and a positive unstable equilibrium appears;(iv)
a<0, b>0. When ϕ changes from negative to positive, x=0 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable.It clearly appears that, atϕ=0 a transcritical bifurcation takes place: more precisely, when a<0 and b>0, such a bifurcation is forward; when a>0 and b>0 the bifurcation at ϕ=0 is backward. We will apply Theorem 2 to show that system (7) may exhibit a backward bifurcation when R0=1. We consider the disease-free equilibrium E0=(b1/μ1,0,0) and observe that the condition R0=1 can be seen, in terms of the parameter λ1, as λ1=λ*=μ1μ22(γ+μ1)/λ2b1b2. The eigenvalues of the matrix
(18)J(E0,λ*)=(-μ10-λ*b1μ10-γ-μ1λ*b1μ10λ2b2μ2-μ2),
are given by
(19)ξ1=-μ1,ξ2=-γ-μ1-μ2,ξ3=0.
Thus ξ3=0 is a simple zero eigenvalue of the matrix J(E0,λ*) and the other eigenvalues are real and negative. Hence, when λ1=λ* (or equivalently when R0=1), the disease-free equilibrium E0 is a nonhyperbolic equilibrium: the assumption (A1) of Theorem 2 is then verified.Now we denote byω=(ω1,ω2,ω3)T, a right eigenvector associated with the zero eigenvalue ξ3=0. It follows:
(20)-μ1ω1-μ22(γ+μ1)λ2b2ω3=0,-(γ+μ1)ω2+μ22(γ+μ1)λ2b2ω3=0,λ2b2μ2ω2-μ2ω3=0,
so that
(21)ω=(-γ+μ1μ1,1,λ2b2μ22)T.
Furthermore, the left eigenvector θ=(θ1,θ2,θ3) satisfying ω·θ=1 is given by:
(22)-μ1θ1=0,-(γ+μ1)θ2+λ2b2μ2θ3=0,μ22(γ+μ1)λ2b2(θ2-θ1)-μ2θ3=0,θ3(λ2b2μ2(γ+μ1)+λ2b2μ22)=1.
Then, the left eigenvector θ turns out to be:
(23)θ=(0,μ2γ+μ1+μ2,μ22(γ+μ1)λ2b2(γ+μ1+μ2)).
We can thus compute the coefficient a and b defined in Theorem 2, that is,
(24)a=∑k,i,j=13‍θkωiωj∂2fk∂xi∂xj(E0,λ*),b=∑k,i=13‍θkωi∂2fk∂xi∂λ1(E0,λ*).
Taking into account of system (5) and considering in a and b only the nonzero derivatives for the terms (∂2fk/∂xi∂xj)(E0,λ*) and (∂2fk/∂xi∂λ1)(E0,λ*), it follows that:
(25)a=2θ1ω1ω3∂2f1∂S∂V(E0,λ*)+θ1ω32∂2f1∂V2(E0,λ*)+2θ2ω1ω3∂2f2∂S∂V(E0,λ*)+θ2ω32∂2f2∂V2(E0,λ*)+2θ3ω2ω3∂2f3∂V∂I(E0,λ*),b=θ1ω1∂2f1∂S∂λ1(E0,λ*)+θ1ω3∂2f1∂V∂λ1(E0,λ*)+θ2ω1∂2f2∂S∂λ1(E0,λ*)+θ2ω3∂2f2∂V∂λ1(E0,λ*).
In view of (21) and (23), we get
(26)a=2λ2μ1μ23(γ+μ1+μ2)[λ1λ2b1b22αa=2λ2μ1μ23(γ+μ1+μ2)-μ22(γ+μ1)(λ1b2+μ1μ2)],b=λ2b1b2μ1μ2(γ+μ1+μ2).
Observe that the coefficient b is always positive so that, according to Theorem 2, it is the sign of the coefficient a which decides the local dynamics around the disease-free equilibrium for λ1=λ*.Let us introduce(27)δ=λ1λ2b1b22αμ22(γ+μ1)(λ1b2+μ1μ2).
The coefficient a is positive if and only if δ>1. In this case, the direction of the bifurcation of system (5) at R0=1 is backward. So we summarize the above results in the following theorem.Theorem 3.
Ifδ>1, system (5) exhibits a backward bifurcation when R0=1. If δ<1, system (5) exhibits a forward bifurcation when R0=1.The local dynamics in the neighborhood of the bifurcation valueR0=1 is described in the former case by the (i) of Theorem 2 and in the later case by the (iv) of the same theorem.


## 4. Global Stability Analysis

In Section3 it has been shown that R0≥1 implies the existence and uniqueness of the endemic equilibrium E. The stability analysis of E will be here performed through the geometric approach to global stability due to Li and Muldowney [8]. The method has been summarized in the Appendix.System (5), under the assumption R0>1, satisfies conditions (H1) and (H2) in Appendix. In fact, when R0>1, then E0 is unstable. The instability of E0, together with E0∈∂Γ, implies the uniform persistence, that is, there exists a constant c>0 such that:
(28)limt→+∞infS(t)>c,limt→+∞infI(t)>c,limt→+∞infV(t)>c.
The uniform persistence is equivalent to the existence of a compact set in the interior of Γ which is absorbing for (5). Thus, (H1) is verified. Moreover, as previously shown, E is the only equilibrium in the interior of Γ, so that (H2) is verified, too. Now it remains to find conditions for which the Bendixson criterion given by Theorem A.4 is verified. Taking into account of the Jacobian matrix (13), we obtain the second additive compound matrix J[2](S,I,V),
(29)J[2]=(J11[2]λ1S(1+2αV)λ1S(1+2αV)λ2(b2μ2-V)J22[2]00λ1V(1+αV)J33[2]),
where
(30)J11[2]=-λ1V(1+αV)-γ-2μ1,J22[2]=-λ1V(1+αV)-μ1-λ2I-μ2,J33[2]=-γ-μ1-λ2I-μ2.
We consider the following function
(31)P=P(S,I,V)=diag{IV,IV,IV}.
So that
(32)Pf=diag{I′V-V′IV2,I′V-V′IV2,I′V-V′IV2},PfP-1=diag{I′I-V′V,I′I-V′V,I′I-V′V}.
Therefore
(33)B=PfP-1+PJ[2]P-1=(B11B12B21B22),
where
(34)B11=I′I-V′V-λ1V(1+αV)-2μ1-γ,B12=[λ1S(1+2αV),λ1S(1+2αV)],B21=[λ2(b2μ2-V),0]T,B22=[I′I-V′V-λ1V(1+αV)-μ1-λ2I-μ20λ1V(1+αV)I′I-V′V-γ-μ1-λ2I-μ2].Consider now the norm inR3 as
(35)|u1,u2,u3|=max{|u1|,|u2|+|u3|},
where (u1,u2,u3) denotes the vector in R3 and denote by μ the Lozinskii˘ measure with respect to this norm. It follows
(36)μ(B)≤sup{g1,g2}=sup{μ1(B11)+|B12|,μ1(B22)+|B21|},
where |B12|,  |B21| are matrix norms with respect to the l1 vector norm, and μ1 denotes the Lozinskii˘ measure with respect to the l1 norm. More specifically,
(37)μ1(B11)=I′I-V′V-λ1V(1+αV)-γ-2μ1,|B12|=λ1S(1+2αV),|B21|=λ2(b2μ2-V),μ1(B22)=I′I-V′V-μ1-λ2I-μ2.
Therefore,
(38)g1=I′I-V′V-λ1V(1+αV)-2μ1-γ+λ1S(1+2αV),(39)g2=I′I-V′V-μ1-λ2I-μ2+λ2(b2μ2-V).
Observe that system (5) provides the following equalities
(40)V′V=λ2b2Iμ2V-λ2I-μ2.
Substituting (40) into (38) and (40) into (39), respectively, we have
(41)g1=I′I-λ2b2Iμ2V+λ2I+μ2-λ1V(1+αV)-2μ1-γ+λ1S(1+2αV)≤I′I+λ2I+μ2-λ1V(1+αV)-2μ1-γ+λ1S(1+2αV),g2=I′I-λ2b2Iμ2V-μ1+λ2(b2μ2-V)≤I′I-μ1+λ2(b2μ2-V).
Thus, (36) implies
(42)μ(B)≤I′I-μ1+Λ,
where
(43)Λ=max{{b2μ2}λ2I+μ2-λ1V(1+αV)-μ1-γΛ=max+λ1S(1+2αV),λ2(b2μ2-V)}.
In view of c≤S,  I≤b1/μ1 and c≤V≤b2/μ2, we can deduce that if
(44)λ2(b2μ2-c)<μ1,λ2b1μ1+μ2+λ1b1μ1(1+2αb2μ2)<λ1c(1+αc)+μ1+γ,
then
(45)μ(B)≤I′I+Π,
where
(46)Π=max{λ2b1μ1+μ2+λ1b1μ1(1+2αb2μ2)-λ1c(1+αc)Π=max-μ1-γ,λ2(b2μ2-c)-μ1}<0.
Hence
(47)1t∫0t‍μ(B)ds≤1tlogI(t)I(0)+Π,
and the Bendixson criterion given by Theorem A.4 is thus verified. The discussions above may be summarized as follows.Theorem 4.
IfR0≥1, then system (5) admits a unique endemic equilibrium E. It is globally asymptotically stable with respect to solutions of (5) initiating in the interior of Γ, provided that inequality (44) holds true.


## 5. Simulation

The system (5) is simulated for various sets of parameters using the package XPP [9]. In Figures 1–4, (S,I) phase planes are drawn which confer the existence and the stability of different equilibria of the system (5). Here Figure 1 corresponds to the situation stated in Theorem 1, where R<R0<1 and we get two endemic equilibria E1 and E2 and one disease free equilibria E0(1500,0,0). And for δ>1, it is found that disease free equilibrium E0(1500,0,0) and the endemic equilibrium point with largest infective population E2(557.678,861.551,161.682) are stable and the equilibrium point E1(1170.057,301.662,62.589) is unstable.Figure 1
Phase plot ofI verses S showing bistability when R<R0<1 and δ>1 for the parameter values b1=240,  λ1=0.00012,  α=0.08,  μ1=0.16,  γ=0.015,  λ2=0.0001,  b2=550,  μ2=0.5.Figure 2
Phase plot ofI verses S showing stability of unique endemic equilibrium point when R0>1 and δ>1 for the parameter values b1=80,  λ1=0.0004,  α=0.04,  μ1=0.06,  γ=0.015,  λ2=0.0002,  b2=200,  μ2=0.5.Figure 3
Phase plot ofI verses S showing stability of unique feasible disease free equilibrium point when R0<R0*<1 for the parameter values b1=80,  λ1=0.00001,  α=0.04,  μ1=0.06,  γ=0.015,  λ2=0.0002,  b2=200,  μ2=0.5.Figure 4
Phase plot ofI verses S showing stability of unique feasible endemic equilibrium point and instablity of disease free equilibrium point when δ<1,  R0>1 for the parameter values b1=10,  λ1=0.0005,  α=0.01,  μ1=0.01,  γ=0.01,  λ2=0.0003,  b2=60,  μ2=0.5.Figure2 corresponds to the situation when R0>1 and also δ>1 and in this case we get unique stable endemic equilibrium point and unstable disease free equilibrium point E0. Figure 3 shows the stability of unique feasible disease free equilibrium E0(1333.3,0,0) for R0<R0*<1. For δ<1, and R0≥1 we get two feasible equilibria: the unstable disease free equilibria and a unique endemic equilibrium which is shown in Figure 4. This fact is more clear from the bifurcation Figure 5, which is obtained by considering λ1 as the critical parameter. The horizontal axis is labelled with the appropriate value of the reproduction number R0 corresponding to this bifurcation parameter λ1. For 0<R0<1 we get stable infection free equilibrium point E0 and at R0=1 we get forward bifurcation, that is, for R0>1 we get unstable disease free equilibrium and unique stable endemic equilibrium point. Figure 6 is showing the backward bifurcation for the parameter values satisfying δ>1. This diagram too is obtained by considering λ1 as the critical parameter and horizontal axis is labelled with the appropriate value of the reproduction number R0 corresponding to this bifurcation parameter λ1. It is observed that when the reproduction number R0 is between 0 to 0.190526, the infection free equilibrium alone is stable, for 0.1905261<R0<1 we have bistability where either the infection free equilibrium is stable or the equilibrium E2 is stable. Again by increasing λ1 we get R0>1 and we have unique stable endemic equilibrium point and unstable disease free equilibrium E0. The equilibrium E1 when it exists is always saddle. As backward bifurcation is quite significant in this figure, so it seems that it can arise for biologically reasonable parameter values too.Figure 5
Forward bifurcation diagram forδ<1 for the parameter values b1=10,  λ1=0.0005,  α=0.01,  μ1=0.01,  γ=0.01,  λ2=0.0003,  b2=60,  μ2=0.5.Figure 6
Backward bifurcation diagram forδ>1 for the parameter values b1=240,  λ1=0.00012,  α=0.08,  μ1=0.16,  γ=0.015,  λ2=0.0001,  b2=550,  μ2=0.5.


## 6. Conclusions

In this paper, a dengue disease epidemic model with nonlinear incidence has been analyzed. IfR0<1, the disease-free equilibrium E0 is locally asymptotically stable, that is, the disease dies out. We find the condition under which the system exhibits backward bifurcation at R0=1. The stability of two endemic equilibria in the case of backward bifurcation for R0<1 is not discussed analytically but we have shown these by numerical simulations. It is found that the endemic equilibrium with the largest infective population is stable, while the endemic equilibrium with less infective population is unstable. In the case of a unique endemic equilibrium, the global stability analysis has been performed. A generalization of the Poincaré-Bendixson criterion has been used. The sufficient conditions that we found are not completely satisfactory, as they involve the constant of uniform persistence c. How to implement the geometric approach to give better stability conditions for epidemic models with convex incidence rate appears to be an open and stimulating challenge.



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*Source: 320581-2013-02-28.xml*