Abstract
An epidemic model that describes the dynamics of the spread of infectious diseases is proposed. Two different types of infectious diseases that spread through both horizontal and vertical transmission in the host population are considered. The basic reproduction number 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.
1. 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 [5–7]. 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 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 [11–13]. 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 [14–17]. 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.
2. Model Formulation
Consider a real world system consisting of a host population that is divided into four compartments: which represents the number of susceptible individuals at time ; and that represent the number of infected individuals at time for -type of disease and -type of disease, respectively; finally that represents the number of recovered individuals at time , thus . Now in order to formulate the dynamics of the above system mathematically, the following assumptions have been adopted:(1)There is a constant number of the host populations entering to the system with recruitment rate .(2)There is a vertical transmission of both of the diseases; that is, the infectious host gives birth to a new infected host of rates and for the diseases and , respectively. Consequently and individuals enter into infected compartments and , respectively, and the same quantities are disappearing from recruitment in the susceptible compartment.(3)The diseases are transmitted by contact, according to the mass action law, between the individuals in the -compartment and those in compartments with nonlinear incidence rate for that is given by , in which represents the infection force rate while represents the inhibition effect of the crowding effect of the infected individuals, and linear incidence rate for that is given by , where represents the infection rate.(4)The individuals in the compartment are facing death due to the disease with infection death rate . They recover from disease and get immunity with a recovery rate .(5)The individuals in the compartment are facing death due to the disease with infection death rate . They also recover from the disease but return back to be susceptible with recovery rate .(6)The individuals in the compartment are losing the immunity from the disease and return back to be susceptible again with losing immunity rate .(7)There is a natural death rate 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 , , , and . Moreover to insure that the recruitment in the susceptible compartment is always positive the following hypotheses are assumed to be holding always:
Theorem 1. The closed set is positively invariant and attracting with respect to model (1).
Proof. Let be any solution of system (1) with any given initial condition. Then by adding all the equations in system (1) we obtain thatThus, from standard comparison theorem [20], we obtainConsequently it is easy to verify thatThus, is positively invariant. Further, when , then either the solution enters Ω in finite time, or approaches as . Hence, is attracting (i.e., all solutions in eventually approach, enter, or stay in ).
Therefore, the system of equations given in model (1) is mathematically well-posed and epidemiologically reasonable, since all the variables remain nonnegative . 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 .
3. 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 with . Moreover the basic reproduction number of model (1), which is denoted by , 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 and .
The other three equilibrium points can be described as follows.
The first disease-free equilibrium point, which is located in the boundary -plane, is denoted by whereand here . Clearly exists uniquely in the interior of -plane provided thatThe second disease-free equilibrium point that is located in the boundary -space is given by whereand here . Obviously exists uniquely in the interior of positive octant of -space provided thatFinally, the endemic equilibrium point, which is denoted by , whereexists uniquely in the interior of provided that the following conditions hold:Keeping the above in view, it is easy to verify with the help of condition (2) thatThen directly we obtain . 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 () is representing the average number of secondary infections that occur from one infected individual in contact with susceptible individuals. Therefore if , 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 , 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.
4. 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 . Therefore the local stability results near the above equilibrium points can be presented in the following theorems.
Theorem 2. The disease-free equilibrium is locally asymptotically stable when and unstable for .
Proof. The characteristic equation of the Jacobian matrix of model (1) at the disease-free equilibrium can be written asSo, if , then according to (6), (15) has four negative real roots (eigenvalues). Hence, the DFE is locally asymptotically stable. Further, for (15) has at least one positive eigenvalue and then the DFE is a saddle point.
Theorem 3. The first disease-free equilibrium point of model (1) is locally asymptotically stable provided that
Proof. The characteristic equation of the Jacobian matrix of model (1) at can be written ashere and due to condition (16). Hence both the eigenvalues and , which describe the dynamics in the direction and direction, respectively, have negative real parts. Moreover, from (17), the eigenvalue in the direction can be written asThus under the given condition (16), we have , while is always negative. Hence, is locally asymptotically stable.
Theorem 4. The second disease-free equilibrium point of model (1) is locally asymptotically stable provided thatwhere is given in the proof.
Proof. The characteristic equation of the Jacobian matrix of model (1) at can be written asherewithClearly, the eigenvalue in the direction can be written asand thus under the condition (19a). In addition from (20) we have always, while can be written asHence, provided that the sufficient condition (19b) holds. Further it is easy to verify thatHence, due to the Routh-Hurwitz criterion the third-degree polynomial term in (20) has roots (eigenvalues) with negative real parts. Hence is locally asymptotically stable.
Theorem 5. The endemic equilibrium point of model (1) is locally asymptotically stable provided thatwhere and are given in the proof.
Proof.
The characteristic equation of the Jacobian matrix of model (1) at can be written asHerewithObviously, , , while is positive under condition (26a). Now, by using the values of and the sufficient condition (26b), then straightforward computation gives and here . Moreover we havewhere . Therefore we obtain thatHere .
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 of the Jacobian matrix near the endemic equilibrium point have negative real parts. Thus, the proof is complete.
5. 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 of model (1) is locally asymptotically stable; then it is global asymptotically stable in Ω.
Proof. Consider that is defined byComputing the derivative of this positive semidefinite function with respect to time along the solution of model (1) and then simplifying the resulting terms giveSince the solution of model (1) is bounded by as , Since , due to the local stability condition of then . Also we have that on the set , so is negative semidefinite and hence according to Lyapunov first theorem is globally stable point. Now, since on this set we have if and only if ,, thus the largest invariant set contained in this set is reduced to the disease-free equilibrium point . Hence according to LaSalle’s invariant principle [18], is attractive point and hence it is globally asymptotically stable in .
Theorem 7. Assume that the first disease-free equilibrium point is locally asymptotically stable; then it is global asymptotically stable in provided that
Proof. Consider that that is defined byClearly 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 thatHence according to local stability condition (16) along with the sufficient condition (37) it obtains that is negative definite function. Thus due to Lyapunov second theorem is global asymptotically stable in .
Theorem 8. Assume that the second disease-free equilibrium point of model (1) is locally asymptotically stable; then it is global asymptotically stable in if where
Proof. Consider the function that is defined byClearly 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 thatNow by using the given conditions (40a)–(40d) we get thatHence according to local stability condition (19a) it obtains that is negative definite function. Thus due to Lyapunov second theorem is global asymptotically stable in .
Theorem 9. Assume that the endemic equilibrium point of model (1) is locally asymptotically stable; then it is global asymptotically stable in if where
Proof. Consider the function that is defined byClearly 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 thatHence, is negative semidefinite, and on the set , so according to Lyapunov first theorem is globally stable point. Further, since on this set we haveif and only if , then the largest compact invariant set contained in this set is reduced to the endemic equilibrium point . Hence according to LaSalle’s invariant principle [18], is attractive point and hence it is globally asymptotically stable in .
6. 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 and with , 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 is any eigenvector.
Theorem 10. Assume that ; then as passes through the value , model (1) near the disease-free equilibrium has (1) no saddle-node bifurcation; (2) a transcritical bifurcation; (3) no pitchfork bifurcation.
Proof.
Since ; then . Now straightforward computation shows that the Jacobian matrix of model (1) at with has zero eigenvalue ( and can be written as follows:Let be the eigenvector corresponding to . Thus giveswhere is any nonzero real number, , and .
Similarly, represents the eigenvector corresponding to eigenvalue of . Hence gives thatand here is any nonzero real number. Now, since thus , which gives .
Thus, according to Sotomayor’s theorem for local bifurcation, model (1) has no saddle-node bifurcation near DFE at .
Now since then, . Now, by substituting and in (51) we getTherefore,So, according to Sotomayor’s theorem model (1) has a transcritical bifurcation at with parameter , while the pitchfork bifurcation cannot occur.
Note that similar results as those of Theorem 10 are obtained at or .
Theorem 11. Assume that ; then model (1) near the first disease-free equilibrium point has (1) no saddle-node bifurcation; (2) a transcritical bifurcation; (3) no pitchfork bifurcation.
Proof.
There are two cases; in the first case, it is assumed that ; then straightforward computation shows that ; that is, . So (by Theorem 10) model (1) has no bifurcation and then the proof is complete.
Now in the second case it is assumed that or equivalently . So straightforward computation shows that the Jacobian matrix of model (1) at with has zero eigenvalue and can be written as follows: Let be the eigenvector corresponding to , which satisfies , so we getwhere is any nonzero real number, , and .
Similarly the eigenvector that is corresponding to the eigenvalue of satisfies , so we getand here is any nonzero real number. Now, sincethen by substituting the values of and we obtain that and hence we get that .
Thus according to Sotomayor’s theorem for local bifurcation, model (1) has no saddle-node bifurcation near at . Now sincethen . Thus by substituting and in (51) we getTherefore, .
So, model (1) has a transcritical bifurcation at with parameter , while the pitchfork bifurcation cannot occur and hence the proof is complete.
Theorem 12. Assume that condition (19b) holds and let ; then model (1) near the second disease-free equilibrium point undergoes (1) no saddle-node bifurcation; (2) a transcritical bifurcation; (3) no pitchfork bifurcation.
Proof.
From it is obtained thatand then straightforward computation shows that the Jacobian matrix of model (1) at with has zero eigenvalue and can be written as follows:Further the eigenvector that is corresponding to satisfies , so we getwhere is any nonzero real number and .
Similarly the eigenvector that is corresponding to eigenvalue of satisfies , so we getand here is any nonzero real number. Now, since , therefore , which yields . Consequently according to Sotomayor’s theorem for local bifurcation, model (1) has no saddle-node bifurcation near at .
Now sincethen . Now, by substituting and in (51) we getTherefore, . Thus model (1) undergoes a transcritical bifurcation at with parameter , while the pitchfork bifurcation cannot occur.
Moreover, the following results are obtained too:(1)Although gives , model (1) does not undergo any of the above types of bifurcation near the equilibrium point with parameter .(2)Although gives , model (1) does not undergo any of the above types of bifurcation near the equilibrium point with parameter .(3)The determinant of the Jacobian matrix at , say , cannot be zero and hence it has no real zero eigenvalue. So there is no bifurcation near .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 crosses a critical positive value ,and here and , while and 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).(3) (The transversality condition).Accordingly the first two points are satisfied if and only if while the third condition holds provided that That means .
HereNow, straightforward computation shows the condition wherewith , and given in (28) and . Clearly, from condition (71a) there is unique positive root, say, . Consequently by using in the characteristic equation and then doing some algebraic computation we get four roots,Now, it is easy to verify that and are real and negative provided that (71b).
Further for the general form of complex eigenvalues can be written asSubstituting into characteristic equation and after that calculating the derivative with respect to and then comparing the real and imaginary parts give thatMoreover, by solving the above linear system for and then we get thatThus it is easy to verify that provided that (71c) and (71d). Thus, the proof is complete.
7. Numerical Simulations
In this section, the global dynamics of model (1) is investigated numerically for different sets of initial values and different sets of parameters values. The objectives of such investigation are to determine the effect of varying the parameters values and confirm our obtained results. It is observed that, for the following biologically feasible set of hypothetical parameters valuesthe solution of model (1) approaches asymptotically to the endemic equilibrium point as shown in Figure 1, started from different sets of initial points.
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Clearly Figure 1 confirms our obtained analytical results regarding existence of a globally asymptotically stable positive equilibrium point when the parameters values are satisfying . On the other hand, model (1) for the following set of hypothetical data approaches asymptotically to the DFE as shown in Figure 2:It is easy to verify that for the data (82) we have , and the solution approaches to .
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 does not affect the dynamical behavior of model (1); that is, the system still approaches to coexistence equilibrium point.(ii)For the data (81) with , the solution of model (1) approaches asymptotically to in the interior of positive quadrant of -plane with . However for the solution of the model still approaches asymptotically to even when . Finally when the solution of model (1) approaches to coexistence equilibrium point with as shown in Figure 3.(iii)Similar results are obtained in case of varying the parameter keeping the rest of parameters values in (81) fixed. In fact for , we have , and the solution of model (1) approaches asymptotically to . However for , we have , and it is observed that the solution of model (1) approaches asymptotically to too, while when the solution of model (1) approaches asymptotically to as shown in Figure 4.(iv)Now decreasing the parameter , keeping the rest of the parameters values in (81) fixed, gives similar dynamical behavior as that of varying . Further it is observed that when , for which , the solution of model (1) approaches asymptotically to ; however when , we have , and the solution of model (1) still approaches to . Finally, for that satisfies , the solution of model (1) approaches asymptotically to . Clearly this confirmed our obtained existence conditions (10) and (12) as well as stability conditions of these points.(v)Finally, varying the parameter , keeping the rest of the parameters values in (81) fixed, showed that, for , that satisfies , the solution of model (1) approaches asymptotically to ; however when , which satisfies too, the solution of model (1) approaches asymptotically to as shown in Figure 5.
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8. Conclusion
In 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 , 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 , infected individuals for -type of disease that are represented by , 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, , 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 (1) such that only the second disease-free equilibrium point appeared in case of . However only the first disease-free equilibrium point appeared in case of . Finally the coexistence of both the diseases occurred in case of 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).(1)Model (1) approaches asymptotically to a globally asymptotically stable point .(2)Varying one of the parameters values at a time keeping other parameters fixed has no effect on the dynamical behavior of the model.(3)As the infection rate of the first disease () decreases keeping other parameters fixed as in (81) the solution of model (1) approaches asymptotically to the equilibrium point . However in case of increasing this parameter the model is still globally asymptotically stable in the interior of .(4)As the infection rate of the second disease () decreases keeping other parameters fixed as in (81) the solution of model (1) approaches asymptotically to the equilibrium point . However in case of increasing this parameter the model is still globally asymptotically stable in the interior of .(5)As the mortality rate () increases keeping other parameters fixed as in (81) the solution of model (1) approaches asymptotically to the equilibrium point and when decreases the model is still globally asymptotically stable in the interior of . Further, it is observed that 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.(7)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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.