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Asymptotic analysis of endemic equilibrium to a brucellosis model

  • Brucellosis is one of the worlds major infectious and contagious bacterial disease. In order to study different types of brucellosis transmission models among sheep, we propose a deterministic model to investigate the transmission dynamics of brucellosis with the flock of sheep divided into basic ewes and other sheep. The global dynamical behavior of this model is given: including the basic repro-duction number, the existence and uniqueness of positive equilibrium, the global asymptotic stability of the equilibrium. We prove the uniqueness of positive endemic equilibrium through using proof by contradiction, and the global stability of endemic equilibrium by using Lyapunov function. Especially, we give the specific coefficients of global Lyapunov function, and show the calculation method of these specific coefficients. By running numerical simulations for the cases with the basic reproduction number to demonstrate the global stability of the equilibria and the unique endemic equilibrium, re-spectively. By some sensitivity analysis of the basic reproduction number on parameters, we find that vaccination rate of sheep and seropositive detection rate of recessive infected sheep are very important factor for brucellosis.

    Citation: Mingtao Li, Xin Pei, Juan Zhang, Li Li. Asymptotic analysis of endemic equilibrium to a brucellosis model[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5836-5850. doi: 10.3934/mbe.2019291

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  • Brucellosis is one of the worlds major infectious and contagious bacterial disease. In order to study different types of brucellosis transmission models among sheep, we propose a deterministic model to investigate the transmission dynamics of brucellosis with the flock of sheep divided into basic ewes and other sheep. The global dynamical behavior of this model is given: including the basic repro-duction number, the existence and uniqueness of positive equilibrium, the global asymptotic stability of the equilibrium. We prove the uniqueness of positive endemic equilibrium through using proof by contradiction, and the global stability of endemic equilibrium by using Lyapunov function. Especially, we give the specific coefficients of global Lyapunov function, and show the calculation method of these specific coefficients. By running numerical simulations for the cases with the basic reproduction number to demonstrate the global stability of the equilibria and the unique endemic equilibrium, re-spectively. By some sensitivity analysis of the basic reproduction number on parameters, we find that vaccination rate of sheep and seropositive detection rate of recessive infected sheep are very important factor for brucellosis.


    Brucellosis is an infectious bacterial disease often spread via direct contact with infected animals or contaminated animal products [1,2]. Brucellosis can transmit to other animals through direct contacts with infected animals or indirect transmission by brucella in the environment. The disease primarily affects cattle, sheep and dogs. In the real world, the infected sheep remain the main source of brucellosis infection, and the basic ewes and other sheep (which includes stock ram and fattening sheep) are often mixed feeding together, therefore there must exist the mixed cross infection between other sheep and basic ewes [3]. Brucellosis is prevalent for more than a century in many parts of the world, and it is well controlled in most developed countries. However, more than 500,000 new cases are reported each year around the world [4,5,6,7].

    Mathematical modeling has the potential to analyze the mechanisms of transmission and the complexity of epidemiological characteristics of infectious diseases [8]. In recent years, several mathematical modeling studies have reported on the transmission of brucellosis [3,9,10,11,12,13,14,15,16,17,18,19]. However, these earlier models have mainly focused on the spread of brucellosis between sheep and human through using the dynamic model. Only Li et al [3] proposed a deterministic multi-group model to study the brucellosis transmission among sheep (which the flock of sheep were divided into basic ewes and other sheep). However, they only gave the global stability of disease-free equilibrium and the existence the endemic equilibrium, but the uniqueness and global stability of the endemic equilibrium were not shown when the basic reproduction number is larger than 1. Multi-group model is a class of highly heterogenous models with complex interactions among distinct groups, and the difficulty of global dynamics of multi-group models lies in establishing uniqueness and global stability of endemic equilibrium when basic reproduction number is larger than one [20]. In this paper, we want to study the global dynamic behavior of multi-group type model for the transmission of brucellosis among sheep which are absent from previous papers [3]. We prove the uniqueness of positive endemic equilibrium through using proof by contradiction, and the global stability of endemic equilibrium by using Lyapunov function. Especially, we give the specific coefficients of global Lyapunov function, and show the calculation method of these specific coefficients. By running numerical simulations for the cases with the basic reproduction number to demonstrate the global stability of the equilibria and the unique endemic equilibrium, respectively. By some sensitivity analysis of the basic reproduction number on parameters, we find that vaccination rate of sheep and seropositive detection rate of recessive infected sheep are very important factor for brucellosis.

    This paper is organized as follows. In Section 2, we present the dynamical model. And the mathematical analysis including the uniqueness and global stability of positive endemic equilibrium will be given in Section 3. In Section 4, some numerical simulations are given on the global stability of the equilibria and the unique endemic equilibrium. Section 5 gives a discussion about main results.

    In previous paper [3], we proposed a multi-group model with cross infection between sheep and human. In this model, So(t),Eo(t),Io(t),Vo(t) and Sf(t),Ef(t),If(t),Vf(t) represent susceptible, recessive infected, quarantined seropositive infected, vaccinated other sheep and basic ewes, respectively. W(t) denotes the quantity of sheep brucella in the environment. Sh(t),Ih(t),Yh(t) represent susceptible individuals, acute infections, chronic infections, respectively. There are some assumptions on the dynamical transmission of brucellosis among sheep and from sheep to humans, which are demonstrated in the flowchart (See Figure 1). The following ordinary differential equations can describe a multi-group brucellosis model with Figure 1:

    {dSodt=Ao(βooEo+βofEf+βoW)So+λoVo(γo+do)So,dEodt=(βooEo+βofEf+βoW)So(co+do)Eo,dIodt=coEo(αo+do)Io,dVodt=γoSo(λo+do)Vo,dSfdt=Af(βffEf+βfoEo+βfW)Sf+λfVf(γf+df)Sf,dEfdt=(βffEf+βfoEo+βfW)Sf(cf+df)Ef,dIfdt=cfEf(αf+df)If,dVfdt=γfSf(λf+df)Vf,dWdt=ko(Eo+Io)+kf(Ef+If)(δ+nτ)W,dShdt=Ah(βhoEo+βhfEf+βhW)ShdhSh+pIh,dIhdt=(βhoEo+βhfEf+βhW)Sh(m+dh+p)Ih,dYhdt=mIhdhYh. (2.1)
    Figure 1.  Transmission diagram, where g(So)=(βooEo+βofEf+βoW)So, g(Sf)=(βffEf+βfoEo+βfW)Sf and g(Sh)=(βhoEo+βhfEf+βhW)Sh, respectively [3].

    Because the last three equations are independent of the first nine equations, we can only consider the first nine equations. Rewrite system (2.1) for general form into the following model:

    {dSidt=Ai(di+γi)Si+λiVi2j=1βijSiEjβiSiW,dEidt=2j=1βijSiEj+βiSiW(di+ci)Ei,dIidt=ciEi(di+αi)Ii,dVidt=γiSi(λi+di)Vi,dWdt=2i=1ki(Ei+Ii)δW.i=1,2. (2.2)

    Adding the first four equations of (2.2) gives

    d(Si+Ei+Ii+Vi)dtAidi(Si+Ei+Ii+Vi),

    which implies that limtsup(Si+Ei+Ii+Vi)Aidi. It follows from the last equation of (2.2) that limtsupW2i=1kiAidiδ. Hence, the feasible region

    X={(S1,E1,I1,V1,S2,E2,I2,V2,W)|0Si+Ei+Ii+ViAidi,0W2i=1kiAidiδ,i=1,2.}

    is positively invariant with respect to model (2.2). Model (2.2) always admits the disease-free equilibrium P0=(S0i,0,0,V0i,0)i=1,2 in X, where S0i=Ai(λi+di)di(di+λi+γi),V0i=Aiγidi(di+λi+γi), and P0 is the unique equilibrium that lies on the boundary of X.

    According to the definition of Rc in [21,22,23] and the calculation of Ro in our previous paper [3], we can obtain the basic reproduction number of model (2.2) is

    R0=ρ(FV1)=A11+A22+(A11A22)2+4A12A212,

    where

    A11=S01d1+c1(β11+β1k1(d1+α1+c1)δ(d1+α1)),A12=S01d2+c2(β12+β1k2(d2+α2+c2)δ(d2+α2)),
    A21=S02d1+c1(β21+β2k1(d1+α1+c1)δ(d1+α1)),A22=S02d2+c2(β22+β2k2(d2+α2+c2)δ(d2+α2)),
    S01=A1(d1+λ1)d1(d1+λ1+γ1),S02=A2(λ2+d2)d2(λ2+d2+γ2).

    In our previous paper [3], for the global stability of disease-free equilibrium and the existence of the positive endemic equilibrium of system (2.2), we have following theorems.

    Theorem 3.1. If R01, the disease-free equilibrium P0 of system (2.2) is globally asymptotically stable in the region X.

    Theorem 3.2. If R0>1, then system (2.2) admits at least one (componentwise) positive equilibrium, and there is a positive constant ϵ such that every solution (Si(t),Ei(t),Ii(t),W(t)) of system (2.2) with (Si(0),Ei(0),Ii(0),W(0))Rn+× Int R2n+1+ satisfies

    min{lim inftEi(t),lim inftIi(t),lim inftW(t)}ϵ,i=1,2,...,n.

    If R0>1, then it follows from Theorem 3.2 that system (2.2) is uniformly persistent, together with the uniform boundedness of solutions of (2.2) in the interior of X, which implies that (2.2) admits at least one endemic equilibrium in the interior of X.

    Let P=(Si,Ei,Ii,Vi,W),i=1,2 be a positive equilibrium of system (2.2), we will show its uniqueness in the interior of the feasible region X.

    Theorem 3.3. System (2.2) only exists a unique positive endemic equilibrium in the region X when R0>1.

    Proof. For the positive equilibrium P of system (2.2), we have the following equations:

    {Ai(di+γi)Si+λiVi2j=1βijSiEjβiSiW=0,2j=1βijSiEj+βiSiW(di+ci)Ei=0,ciEi(di+αi)Ii=0,γiSi(λi+di)Vi=0,2i=1ki(Ei+Ii)δW=0.i=1,2.

    It is easy to obtain that

    Vi=γiSidi+λi,di(Si+Vi)=Ai(di+ci)Ei,Ii=cidi+αiEi,W=2i=1ki(ci+di+αi)di+αiEiδ,
    Si2j=1βijEj+βiSi2i=1ki(ci+di+αi)di+αiEiδ=(di+ci)Ei,i=1,2.

    Hence, the positive equilibrium of system (2.2) is equivalent to the following system

    Mi(AiniEi)2j=1ξijEjniEi=0,i=1,2. (3.1)

    where

    Mi=di+λidi(di+λi+γi),ξij=βij+βikj(cj+dj+αj)δ(dj+αj),ni=di+ci.

    Firstly, we prove that E=e,e=(e1,e2) is the only positive solution of system (3.1). Assume that E=e and E=k are two positive solutions of system (3.1), both nonzero. If ek, then eiki for some i (i = 1, 2). Assume without loss of generality that e1>k1, and moreover that e1/k1ei/ki for all i (i = 1, 2). Since e and k are positive solutions of system (3.1), we substitute them into (3.1). It is easy to obtain

    M1(A1n1e1)2j=1ξ1jejn1e1=M1(A1n1k1)2j=1ξ1jkjn1k1=0,

    so

    M1(A1n1e1)2j=1ξ1jejk1e1n1k1=M1(A1n1k1)2j=1ξ1jkjn1k1=0,
    M1(A1n1e1)2j=1ξ1jejk1e1=M1(A1n1k1)2j=1ξ1jkj.

    But (ei/e1)k1ki and M1(A1n1e1)<M1(A1n1k1); thus from the above equalities we get

    M1(A1n1e1)2j=1ξ1jejk1e1M1(A1n1e1)2j=1ξ1jkj<M1(A1n1k1)2j=1ξ1jkj.

    This is a contradiction, so there is only one positive solution Ei=e of system (3.1). So when R0>1, system (2.2) only exists a positive equilibrium P.

    In this section, we will show the global asymptotic stability of endemic equilibrium P of system (2.2) in the interior of the feasible region X.

    Theorem 3.4. Suppose that matrix [βij]1i,j2 is irreducible. Then the endemic equilibrium P of system (2.2) is globally asymptotically stable in the region X when R0>1.

    Proof. Let Li1=SiSiSilnSiSi+ViViVilnViVi+EiEiEilnEiEi, Li2=IiIiIilnIiIi and L3=WWWlnWW. For i=1,2, differentiating and using the equilibrium equations give

    dLi1dt=(1SiSi)Si+(1ViVi)Vi+(1EiEi)Ei=(1SiSi)(Ai(di+γi)Si+λiVi2j=1βijSiEjβiSiW)+(1ViVi)(γiSi(λi+di)Vi)+(1EiEi)(2j=1βijSiEj+βiSiW(di+ci)Ei)=(1SiSi)((di+γi)(SiSi)+λi(ViVi)2j=1βij(SiEjSiEj))(1SiSi)βi(SiWSiW)+γiSi(1ViVi)(SiSiViVi)+(1EiEi)(2j=1βij(SiEjSiEjEiEi)+βi(SiWSiWEiEi))=diSi(2SiSiSiSi)+λiVi(2SiViSiViSiViSiVi)+diVi(3SiSiViViSiViSiVi)+2j=1βijSiEj((1SiSi)(SiEjSiEj1)+(1EiEi)(SiEjSiEjEiEi))+βiSiW((1SiSi)(SiWSiW1)+(1EiEi)(SiWSiWEiEi))2j=1βijSiEj(2SiSiEiEi+EjEjSiEjEiSiEjEi)+βiSiW(2SiSiEiEi+WWSiWEiSiWEi).

    Using the inequality 1alna,a>0, one can obtain that

    2SiSiEiEi+EjEjSiEjEiSiEjEiEjEjEiEilnSiSilnSiEjEiSiEjEi=EjEjlnEjEj+lnEiEiEiEi,2SiSiEiEi+WWSiWEiSiWEiWWEiEilnSiSilnSiWEiSiWEi=WWlnWW+lnEiEiEiEi.

    Hence, we have

    dLi1dt2j=1βijSiEj(2SiSiEiEi+EjEjSiEjEiSiEjEi)+βiSiW(2SiSiEiEi+WWSiWEiSiWEi)2j=1βijSiEj(EjEjlnEjEj+lnEiEiEiEi)+βiSiW(WWlnWW+lnEiEiEiEi).

    Similarly, we can obtain

    dLi2dt=(1IiIi)Ii=(1IiIi)(ciEi(di+αi)Ii)=(1IiIi)(ciEiciEiIiIi)=ciEi(1+EiEiIiIiEiIiEiIi)ciEi(EiEilnEiEi+lnIiIiIiIi).
    dL3dt=(1WW)W=(1WW)(2j=1(kiEi+miIi)δW)=(1WW)(2j=1(kiEi+miIi)2j=1(kiEi+miIi)WW)2i=1kiEi(EiEilnEiEi+lnWWWW)+2i=1miIi(IiIilnIiIi+lnWWWW).

    Define the Lyapunov function

    L=2i=1υi(ai1Li1+ai2Li2+ai3L3).

    It follows that

    dLdt=2i=1υi(ai1Li1+ai2Li2+ai3L3)2i=1υi(ai12j=1βijSiEj(EjEjlnEjEj+lnEiEiEiEi)+ai1βiSiW(WWlnWW+lnEiEiEiEi)+ai2ciEi(EiEilnEiEi+lnIiIiIiIi)+ai32i=1kiEi(EiEilnEiEi+lnWWWW)+ai32i=1miIi(IiIilnIiIi+lnWWWW)).

    Considering the following equations

    {(ai1βiSiWai32i=1(kiEi+miIi))(WWlnWW)=0,(ai32i=1miIiai2ciEi)(IiIilnIiIi)=0.

    We have

    ai2=2i=1miIiciEiai3,ai3=βiSiW2i=1(kiEi+miIi)ai1.

    and

    (ai1βiSiWai2ciEiai32i=1kiEi)(lnEiEiEiEi)=0

    Let ai2=1 and take the equation ai2 and ai3 into the equation dLdt, we can obtain

    dLdt=2i=1υi(Li1+βiSiW2i=1(kiEi+miIi)(2i=1miIiciEiLi2+L3))2i=1υi(2j=1βijSiEj(EjEjlnEjEj+lnEiEiEiEi))=2i,j=1υiβijSiEj(EjEjlnEjEj+lnEiEiEiEi)

    Due to matrix [βij]1i,j2 is irreducible, hence we can calculate υ1=β21S2E1,υ2=β12S1E2 such that

    2i,j=1υiβijSiEj(EjEjlnEjEj+lnEiEiEiEi)=0.

    The equality L=0 holds only for Si=Si,Ei=Ei,Ii=Ii,i=1,2 and W=W. Hence, one can obtain that the largest invariant subset where L=0 is the singleton P using the same argument as in [24]. By LaSalle's Invariance Principle [25], P is globally asymptotically stable in the region X when R0>1.

    Remark 3.1. In this model, the host populations are divided into 2 homogeneous groups. If the host populations have n groups, we can extend our Lyapunov function into the following equation:

    L=ni=1υi(Li1+βiSiWni=1(kiEi+miIi)(ni=1miIiciEiLi2+L3))

    Furthermore we can obtain that

    dLdt=ni=1υi(Li1+βiSiWni=1(kiEi+miIi)(ni=1miIiciEiLi2+L3))ni,j=1υiβijSiEj(EjEjlnEjEj+lnEiEiEiEi)

    Hence, if the matrix [βij]1i,jn is irreducible, and according to the methods and conclusions in [24,26,27], there exist constants υi>0,i=1,2,...,n such that

    ni,j=1υiβijSiEj(EjEjlnEjEj+lnEiEiEiEi)=0.

    In an epidemic model, the basic reproduction number R0 is calculated and shown to be a threshold for the dynamics of the disease. Taking parameter values δ=3.6,A1=1976000,d1=0.6, λ1=0.4,A2=1680000,d2=0.4, λ2=0.4,α2=12,α1=12, β1=1.0×108,β11=1.8×107, β2=1.0×108,β22=2.1×107, k1=15,k2=15, γ1=0.316×0.82,γ2=0.316×0.82, c1=0.15,c2=0.15, β12=β21=1.35×107 in paper [3], we run numerical simulations with system (2.2) for R0>1 (see Figure 2) and R0<1 (see Figure 3) to demonstrate the conclusions in Theorem 3.4 and Theorem 3.1.

    Figure 2.  Numerical simulations for R0=1.9789>1 with different initial values. (a) The infectious cases with group 1. (b) The infectious cases with group 2.
    Figure 3.  Numerical simulations for R0=0.8330<1 with different initial values, where γ1=1×0.82,γ2=0.316×0.82,c1=0.3,c2=0.3,β12=β21=0. (a) The infectious cases with group 1. (b) The infectious cases with group 2.

    In order to evaluate the influence for infectious individuals over time with the key parameters (such as the efficient vaccination rate γ1,γ2, the seropositive detection rate c1,c2, and the transmission rate β1,β11,β2,β22,β12,β21). We explored these parameter space by performing an uncertainty analysis using a Latin hypercube sampling (LHS) method and sensitivity analysis using partial rank correlation coefficients (PRCCs) with 1000 samples [28]. In the absence of available data on the distribution functions, we chose a normal distribution for all selected input parameters with the same values in paper [3], and tested for significant PRCCs for these parameters of system (2.2). PRCC indexes can be calculated for multiple time points and plotted versus time, and this can allow us to assess whether significance of one parameter occur over an entire time interval during the progression of the model dynamics.

    Figure 4 show the plots of 1000 runs output and PRCCs plotted for selected parameters with respect to the number of infected individuals in group 1 and 2 for system (2.2). Figure 4 (b) and (d) show that the effects of parameters γ1,γ2,c1,c2,β1,β11, β2,β22,β12,β21 change with respect to I1 and I2 over time. In the early time, these selected parameters with PRCCS have obvious change, and finally they remain constant. In Figure 4 (b), the efficient vaccination rate γ1,γ2 and the seropositive detection rate of group 2 c2 are negatively correlated with PRCCs for I1, and the other parameters are positively correlated with PRCCs for I1. But In Figure 4 (d), the seropositive detection rate of group 2 c2 is positively correlated with PRCCs and the seropositive detection rate of group 1 c1 is negatively correlated with PRCCs, other parameters have the same correlation with PRCCs for I1. From Figure 4, we can see the efficient vaccination rate γ1,γ2 have the strong negatively correlated PRCCs (black solid and dotted lines) for I1 and I2, and the seropositive detection rate has the strong positively correlated with PRCCs for infected individuals. Hence, one can conclude that the vaccination and the seropositive detection of infected individuals are the effective control measures.

    Figure 4.  Sensitivity analysis. (a) and (c) Plots of output (1000 runs) of system (2.2) for I1 and I2. (b) and (d) PRCCs of system (2.2) for I1 and I2 with parameters γ1,γ2,c1,c2,β1,β11,β2,β22, β12,β21.

    To find better control strategies for brucellosis infection, we perform some sensitivity analysis of the basic reproduction number R0 in terms of the efficient vaccination rate (γ1,γ2) and the seropositive detection rate (c1,c2). We show the combined influence of parameters on R0 in Figure 5. Figure 5(a) depicts the influence of sheep efficient vaccination rate γ1,γ2 on R0. Though vaccinating susceptible sheep is an effective measure to decrease R0, R0 cannot become less than one even if the vaccination rate of all sheep is 100% (which is the efficient vaccination rate γ1=0.82,γ2=0.82 in Figure 5(a), under this circumstances R0=1.3207). Figure 5(b) indicates the influence of seropositive detection rate c1,c2 on R0, which shows to increase seropositive detection rate of recessive infected sheep can make R0 less than one, which means under the current control measures, increase seropositive detection rate of recessive infected sheep can control the brucellosis. Hence, we can conclude that combining the strategy of vaccination and detection is more effective than vaccination and detection alone to control brucellosis.

    Figure 5.  The combined influence of parameters on R0. (a) R0 in terms of γ1 and γ2. (b) R0 in terms of c1 and c2. Other parameters are the same values in paper [3].

    In this paper, in order to show the uniqueness and global stability of the endemic equilibrium for brucellosis transmission model with common environmental contamination, the multi-group model in paper [3] is chosen as our research objectives. Firstly, we show the basic reproduction number R0 of the model (2.2). Then, we obtain the uniqueness of positive endemic equilibrium through using proof by contradiction when R0>1. Finally, the proof of global asymptotical stability of the endemic equilibrium when R0>1 is shown by using Lyapunov function. Especially, we give the specific coefficients of global Lyapunov function, and show the calculation method of these specific coefficients. Numerical analysis also show that the global asymptotic behavior of system (2.2) is completely determined by the size of the basic reproduction number R0, that is, the disease free equilibrium is globally asymptotically stable if R0<1 while an endemic equilibrium exists uniquely and is globally stable if R0>1. With the uncertainty and sensitivity analysis of infected individuals for selected parameters γ1,γ2,c1,c2,β1,β11,β2,β22, β12,β21 using LHS/PRCC method, one can conclude that the efficient vaccination rate γ1,γ2 have the strong negatively correlated PRCCs (black solid and dotted lines in Figure 4), and the seropositive detection rate has the strong positively correlated with PRCCs for infected individuals. By some sensitivity analysis of the basic reproduction number R0 on parameters, we find that vaccination rate of sheep and seropositive detection rate of recessive infected sheep are very important factor for brucellosis.

    The project is funded by the National Natural Science Foundation of China under Grants (11801398, 11671241, 11601292) and Natural Science Foundation of Shan'Xi Province Grant No. 201801D221024.

    All authors declare no conflicts of interest in this paper.



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