The Dynamics of Epidemic Model with Two Types of Infectious Diseases and Vertical Transmission Raid

An epidemic model that describes the dynamics of the spread of infectious diseases is proposed. Two different types of infectious diseases that spread throughboth horizontal and vertical transmission in the host population are considered.Thebasic reproduction number R 0 is determined. The local and the global stability of all possible equilibrium points are achieved. The local bifurcation analysis and Hopf bifurcation analysis for the four-dimensional epidemic model are studied. Numerical simulations are used to confirm our obtained analytical results.


Introduction
Mathematical models can be defined as a method of emulating real life situations with mathematical equations to expect their future behavior.In epidemiology, mathematical models play role as a tool in analyzing the spread and control of infectious diseases.Although one of the most famous principles of ecology is the competitive exclusion principle that stipulates "two species competing for the same resources cannot coexist indefinitely with the same ecological niche" [1,2], Volterra was the first scientist who used the mathematical modeling and showed that the indefinite coexistence of two or more species limited by the same resource is impossible [3].Moreover, Ackleh and Allen [4] were the first who used the competitive exclusion principle of the infectious disease with different levels in single host population.
It is well known that one of the most useful parameters concerning infectious diseases is called basic reproduction number.It can be specific to each strain of an epidemic model.In fact the basic reproduction number of the model is defined as the maximum reproduction numbers of other strains [5][6][7].Diekmann et al. [8] had studied epidemic models with one strain, while Martcheva in [9] studied the -type of disease with multistrain.However, Ackleh and Allen [10] studied -type of disease with n strain and vertical transmission.
Keeping the above in view, in our proposed model two strains with two different types of infectious diseases are considered.Accordingly two different reproduction numbers are obtained and then competitive exclusion principle is presented.It is assumed that two different types of diseases transmission, say horizontal and vertical transmission, are used too.The horizontal transmission occurs by direct contact between infected and susceptible individuals, while vertical transmission occurs when the parasite is transmitted from parent to offspring [11][12][13].The incidence of an epidemiological model is defined as the rate at which susceptible becomes infectious.Different types of incidence rates are introduced into literatures [14][15][16][17].Finally two types of incidence rates, say bilinear mass action and nonlinear type, are used with the horizontal and vertical transmission, respectively.The local and global stability for all possible equilibria are carried out with the help of Lyapunov function and LaSalle's invariant principle [18].An application of Sotomayor theorem [19,20] for local bifurcations is used to study the occurrence of local bifurcations near the equilibria.The Hopf bifurcation [21,22] conditions are derived.Finally, numerical simulations are used to confirm our obtained analytical results and specify the control set of parameters.

Model Formulation
Consider a real world system consisting of a host population () that is divided into four compartments: () which (4) The individuals in the  1 compartment are facing death due to the disease with infection death rate  1 ≥ 0. They recover from disease and get immunity with a recovery rate  > 0.
(5) The individuals in the  2 compartment are facing death due to the disease with infection death rate  2 ≥ 0. They also recover from the disease but return back to be susceptible with recovery rate  > 0.
(6) The individuals in the  compartment are losing the immunity from the  1 disease and return back to be susceptible again with losing immunity rate 0 ≤  < 1.
(7) There is a natural death rate  > 0 for the individuals in the host population.Finally, it is assumed that both the diseases cannot be transmitted to the same individual simultaneously.
According to these assumptions the dynamics of the above real world system can be represented mathematically by the following set of differential equations: with the initial condition (0) > 0,  1 (0) > 0,  2 (0) > 0, and (0) > 0. Moreover to insure that the recruitment Λ in the susceptible compartment is always positive the following hypotheses are assumed to be holding always: Theorem 1.The closed set Ω = {(,  1 ,  2 , ) ∈ R 4 + :  ≤ Λ/} is positively invariant and attracting with respect to model (1).
Proof.Let ((),  1 (),  2 (), ()) be any solution of system (1) with any given initial condition.Then by adding all the equations in system (1) we obtain that Thus, from standard comparison theorem [20], we obtain Consequently it is easy to verify that Thus, Ω is positively invariant.Further, when (0) > Λ/, then either the solution enters Ω in finite time, or () approaches Λ/ as  → ∞.Hence, Ω is attracting (i.e., all solutions in R 4 + eventually approach, enter, or stay in Ω).Therefore, the system of equations given in model ( 1) is mathematically well-posed and epidemiologically reasonable, since all the variables remain nonnegative ∀ ≥ 0. Further since the equations of model ( 1) are continuous and have continuously partial derivatives then they are Lipschitzian.In addition to that from Theorem 1, model ( 1) is uniformly bounded.Therefore the solution of it exists and is unique.Hence, from now onward it is sufficient to consider the dynamics of model (1) in Ω.

Equilibrium Points and Basic Reproduction Number
Model (1) has four equilibrium points that are obtained by setting the right hand sides of this model equal to zero.The first equilibrium point is the disease-free equilibrium (DFE) point that is denoted by  0 = ( 0 , 0, 0, 0) with  0 = Λ/.Moreover the basic reproduction number of model (1), which is denoted by  0 , is the maximum eigenvalue of the next generation matrix (i.e., the maximum of the reproduction numbers, those computed of each disease).That is, Here  1 = ( 1  0 +  1 )/( +  1 + ) and  2 = ( 2  0 +  2 )/( +  2 + ).
The other three equilibrium points can be described as follows.
The first disease-free equilibrium point, which is located in the boundary  2 -plane, is denoted by  1 = (, 0,  2 , 0) where and here  2 =  2  0 /(+ 2 +− 2 ).Clearly  1 exists uniquely in the interior of  2 -plane provided that The second disease-free equilibrium point that is located in the boundary  1 -space is given by  2 = ( S, Ĩ1 , 0, R) where and here  1 =  1  0 /( +  1 +  −  1 ).Obviously  2 exists uniquely in the interior of positive octant of  1 -space provided that Finally, the endemic equilibrium point, which is denoted by where exists uniquely in the interior of Ω provided that the following conditions hold: Keeping the above in view, it is easy to verify with the help of condition (2) that Then directly we obtain   > 1 (  < 1) ⇔  0 > 1 ( 0 < 1).Consequently,   represent the threshold parameters for the existence of the last three equilibrium points of model (1).Moreover, it is well known that the basic reproduction number ( 0 ) is representing the average number of secondary infections that occur from one infected individual in contact with susceptible individuals.Therefore if  0 < 1, then each infected individual in the entire period of infectivity will produce less than one infected individual on average, which shows the disease will be wiped out of the population.However, if  0 > 1, then each infected individual in the entire infection period having contact with susceptible individuals will produce more than one infected individual; this leads to the disease invading the susceptible population.

Local Stability Analysis
In this section, the local stability analyses of all possible equilibrium points of model ( 1) are discussed by determining the Jacobian matrix with their eigenvalues.Now the general Jacobian matrix of model ( 1) can be written: where (,  1 ) =  1 /(1 +  1 ) − ( +  1 +  −  1 ).Therefore the local stability results near the above equilibrium points can be presented in the following theorems.
Proof.The characteristic equation of the Jacobian matrix of model ( 1) at the disease-free equilibrium can be written as So, if  0 < 1, then according to ( 6), (15) has four negative real roots (eigenvalues).Hence, the DFE is locally asymptotically stable.Further, for  0 > 1 (15) has at least one positive eigenvalue and then the DFE is a saddle point.
Theorem 4. The second disease-free equilibrium point  2 = ( S, Ĩ1 , 0, R) of model ( 1) is locally asymptotically stable provided that where P2 is given in the proof.
Proof.The characteristic equation of the Jacobian matrix of model (1) at  2 can be written as with Clearly, the eigenvalue λ 2 in the  2 -direction can be written as and thus λ 2 < 0 under the condition (19a).In addition from (20) we have  > 0 always, while  can be written as Hence,  > 0 provided that the sufficient condition (19b) holds.Further it is easy to verify that Hence, due to the Routh-Hurwitz criterion the third-degree polynomial term in (20) has roots (eigenvalues) with negative real parts.Hence  2 is locally asymptotically stable.1) is locally asymptotically stable provided that

Theorem 5. The endemic equilibrium point 𝐸
where  and  5 are given in the proof.
Proof.The characteristic equation of the Jacobian matrix of model ( 1) at  3 can be written as Here with Obviously,   > 0,  = 1, 2, 3, 5, while  4 is positive under condition (26a).Now, by using the values of   and the sufficient condition (26b), then straightforward computation gives and here  = ( 1  2 +  3 )( + ).Moreover we have where  =  +  4 ( 2 + ( + )).Therefore we obtain that Here  = ( Hence, according to condition (26b) it is easy to verify that Therefore, all the coefficients of ( 27) are positive and Hence, due to the Routh-Hurwitz criterion all the eigenvalues ( *  ,  *  1 ,  *  2 and  *  ) of the Jacobian matrix near the endemic equilibrium point  3 have negative real parts.Thus, the proof is complete.

Global Stability Analysis
This section deals with the global stability of the equilibrium points of model ( 1) using Lyapunov methods with LaSalle's invariant principle.The obtained results are presented in the following theorems.Theorem 6. Assume that DFE  0 = ( 0 , 0, 0, 0) of model ( 1) is locally asymptotically stable; then it is global asymptotically stable in Ω.
Proof.Consider V : Ω → R that is defined by Computing the derivative of this positive semidefinite function with respect to time along the solution of model ( 1) and then simplifying the resulting terms give Since the solution of model ( 1) is bounded by Since   < 1,  = 1, 2, due to the local stability condition of  0 then  V/ < 0. Also we have that  V/ = 0 on the set {(,  1 ,  2 , ) ∈ Ω :  1 =  2 = 0}, so  V/ is negative semidefinite and hence according to Lyapunov first theorem  0 is globally stable point.Now, since on this set we have if and only if  =  0 ,  = 0, thus the largest invariant set contained in this set is reduced to the disease-free equilibrium point  0 .Hence according to LaSalle's invariant principle [18],  0 is attractive point and hence it is globally asymptotically stable in Ω.

Theorem 7.
Assume that the first disease-free equilibrium point  1 = (, 0,  2 , 0) is locally asymptotically stable; then it is global asymptotically stable in Ω provided that Proof.Consider that  : Ω → R that is defined by Clearly  is continuous and positive definite function.Now by taking the derivative of  with respect to time along the solution of model ( 1), we get after simplifying the resulting terms that Hence according to local stability condition (16)  (41) Proof.Consider the function  : Ω → R that is defined by Clearly  is continuous and positive definite function.Now by taking the derivative of  with respect to time along the solution of model (1), we get after simplifying the resulting terms that Now by using the given conditions (40a)-(40d) we get that where +  1 ) ,  24 = ,  14 = . (46) Proof.Consider the function  : Ω → R that is defined by Clearly the function  is continuous and positive definite function.By taking the derivative of  with respect to time along the solution of model ( 1), we get after simplifying the resulting terms that Then by using the given conditions (45a)-(45c) we obtain that Hence, / is negative semidefinite, and / = 0 on the set {(,  1 ,  2 , ) ∈ Ω :  =  * ,  1 =  * 1 ,  2 > 0,  =  * }, so according to Lyapunov first theorem  3 is globally stable point.Further, since on this set we have if and only if  2 =  * 2 , then the largest compact invariant set contained in this set is reduced to the endemic equilibrium point  3 .Hence according to LaSalle's invariant principle [18],  3 is attractive point and hence it is globally asymptotically stable in Ω.

Bifurcation Analysis
In this section the local bifurcations near the equilibrium points of model ( 1) are investigated as shown in the following theorems with the help of Sotomayor theorem [20].Note that model (1) can be rewritten in a vector form / = (), where  = (, 1 ,  2 , )  and  = ( 1 ,  2 ,  3 ,  4 )  with   ,  = 1, 2, 3, 4, are given in the right hand side of model (1).Moreover, straightforward computation gives that the general second derivative of the Jacobian matrix ( 14) can be written: where  is any bifurcation parameter and  = ( 1 ,  2 ,  3 ,  4 )  is any eigenvector.
(3) The determinant of the Jacobian matrix at  3 , say ( 3 ), cannot be zero and hence it has no real zero eigenvalue.So there is no bifurcation near  3 .
Keeping the above in view, in the following theorem we detect of the possibility of having Hopf bifurcation.
Theorem 13.Assume that condition (26a) holds and let the following conditions be satisfied.Then model (1) undergoes Hopf bifurcation around the endemic equilibrium point when the parameter  2 crosses a critical positive value and here while  2 and  4 are given in the proof.
Proof.It is well known that, in order for Hopf bifurcation in four-dimensional systems to occur, the following conditions should be satisfied [21,22]: (1) The characteristic equation given in (28) has two real and negative eigenvalues and two complex eigenvalues, say,  * ( 2 ) =  1 ( 2 ) ±  2 ( 2 ). ( Accordingly the first two points are satisfied if and only if while the third condition holds provided that Now, straightforward computation shows the condition where with  1 ,  2 ,  3 ,  4 , and  5 given in (28) and  6 =  1 −  2  * 2 .Clearly, from condition (71a) there is unique positive root, say,  2 = β * 2 .Consequently by using Δ 2 = 0 in the characteristic equation and then doing some algebraic computation we get four roots, Now, it is easy to verify that  3 and  4 are real and negative provided that (71b).

Numerical Simulations
In this section, the global dynamics of model ( 1 It is easy to verify that for the data (82) we have  0 = 0.86 < 1, and the solution approaches to  0 = (22.22,0, 0, 0).Now in order to investigate the effect of varying one parameter value at a time on the dynamical behavior of model (1), the following results are observed.
(i) Varying of the parameters values (Λ, ,  2 , ,  1 ,  2 , ) does not affect the dynamical behavior of model (1); that is, the system still approaches to coexistence equilibrium point.

Conclusion
In  1) such that only the second disease-free equilibrium point appeared in case of  2 < 1 <  1 .However only the first disease-free equilibrium point appeared in case of  1 < 1 <  2 .Finally the coexistence of both the diseases occurred in case of 1 <  2 <  1 and the sufficient condition (12) holds.
The dynamical behavior of model ( 1) has been investigated locally as well as globally using Routh-Hurwitz criterion and Lyapunov function, respectively.The local bifurcations of model ( 1) and the Hopf bifurcation around the endemic equilibrium point are studied.Finally to understand the effect of varying each parameter on the global dynamics of system (1) and to confirm our obtained analytical results, model (1) has been solved numerically and the following results are obtained for the set of hypothetical parameters values given by (81).

Figure 1 : 2 Figure 2 :
Figure 1: Globally asymptotically stable positive equilibrium point of model (1) for the parameters set (81), started from different sets of initial point.
along with the sufficient condition (37) it obtains that / is negative definite function.Thus due to Lyapunov second theorem  1 is global asymptotically stable in Ω.
Hence according to local stability condition (19a) it obtains that / is negative definite function.Thus due to Lyapunov second theorem  1 is global asymptotically stable in Ω.
1  2 [ 3 ( +  2 ) this paper, we proposed and analyzed an epidemic model involving vertical and horizontal transmission of infection with nonlinear incidence rate.It is assumed that the rates of infections  1 ,  2 are less than the recovery rates  and , respectively.According to the diseases in model (1) the population is divided into four subclasses: susceptible individuals that are represented by (), infected individuals for -type of disease that are represented by  1 (), infected individuals for -type of disease that are represented by  2 (), and recovery individuals that are denoted by ().The boundedness and invariant of the model are discussed.The basic reproduction number of the model and the associated threshold parameter values, namely,   ,  = 1, 2, are determined.It is observed that if the basic reproduction number is less than unity then the diseases are eradicated from the model.The competitive exclusion principle occurred in model ( 4 + .(4) As the infection rate of the second disease ( 2 ) decreases keeping other parameters fixed as in (81) the solution of model (1) approaches asymptotically to the equilibrium point  2 = (27.33,17.80, 0, 15.5).However in case of increasing this parameter the model is still globally asymptotically stable in the interior of R 4 + .(5) As the mortality rate () increases keeping other parameters fixed as in (81) the solution of model (1) approaches asymptotically to the equilibrium point  2 = (13.8,5.16, 0, 2.5) and when  decreases the model is still globally asymptotically stable in the interior of R 4 + .Further, it is observed that  1 has the same effect as  on the dynamical behavior of model (1).(6) For the parameter set given in (82) the solution of model (1) approaches asymptotically to DFE.
(1)Finally, although for our selected parameters values model(1)does not undergo periodic dynamics, the model still has possibility to have periodic dynamics for other sets of parameters, especially Hopf bifurcation existing analytically.