Stability analysis of delay seirepidemic model

Khan et al/ International Journal of Advanced and Applied Sciences, 3(7) 2016, Pages: 46‐53 47 The structure of the paper is as follow: We formulate the model in Section 2. In Section 3, we present the local stability for the model. In Section 4, we find the global stability of the endemic equilibrium. The numerical solution of the model is presented in Section 5. Finally, discussion with conclusion is presented in Section 6. 2. Model constructions We consider a mathematical model of the type (Eq. 1): = Γ (1 − ) − , = − − , = − − , = − , (1) with initial conditions (Eq. 2): (0) = ≥ 0, (0) = ≥ 0, (0) = ≥ 0, (0) = ≥ 0. (2) Here, ( ) is the density of susceptible within the population, ( ) is the density of exposed individuals ( ) is the infected and ( ) is the density of recovered. shows the natural death rate of the population, represents the disease contact rate and Γ is the intrinsic growth rate. The exposed individuals infected at a rate of and move to the class of the infected. The recovery rate from infection is shown by and is the carrying capacity of the susceptible. The saturation term which measures the inhibitory effect is shown by . Here, we use the new variable , known to be the information variable that collects information for the disease present state, i.e. depending on present values and some previous values of the state variables. Many authors used this variable in their model such as D’Onofrio et al. (2007), D’Onofrio, Manfredi (2007), Buonomo et al. (2008), Kar and Mondal (2011). We take the formula (Eq. 3): ( ) = ( ( ), ( )) ( − ) (3) here ( − ) represents the delaying kernel (Wang et al., 2014), the distributed delay is , that shows for any time t the susceptible, S, the exposed people, E, and infected people, I can be affected with these variables S, E and I, possibly, at some previous times ≤ . In this work, we assume ( , ) = , and ( − ) = ( ), where T shows the average delay of the summarized information and also concerning the historical memory on the concern disease. With these assumptions, the following system is presented (Eq. 4): = Γ (1 − ) − , = − − , = − − , ( ) = ( ( ), ( )) ( − ) , = − , (4) the system (4) is in the form of the non-linear Integro-differential system. By transformation, we can write it in the form of non-linear ordinary differential equations (Eq. 5): = Γ (1 − ) − , = − − , = − − , = ( − ), = − , (5) in system (5), the last equation is independent of the rest, so we omit it, because R depends only on I. So the model becomes (Eq. 6): = Γ (1 − ) − , = − − , = − − , = ( − ), (6) where Γ, , , , , , > 0. 3. Stability analysis In this section, we present the local stability analysis of the system (6). In the subsection (3.1), we present the local stability of disease free and the subsequent section (3.2) we find the local stability of endemic equilibrium. 3.1. Disease free stability The disease free equilibrium point of system (6) is (Eq. 7). Theorem: An unstable equilibrium exists for All > 0 at . Proof: At the equilibrium point , the Jacobean matrix is (Eq. 8). The eigenvalues associated to (0,0,0,0) are, Γ, −( + ), −( + ) − . This proves the instability of the system at . Theorem: If < 1, the jacobian matrix at is locally asymptotically stable for all T>0. Khan et al/ International Journal of Advanced and Applied Sciences, 3(7) 2016, Pages: 46‐53 48 Proof: The jacobian matrix at is (Eq. 9). The eigenvalues associated to Jacobian matrix at are,−Γ, − , −( + ) − ( + ). Theorem: (i) The model (6), at = (0,0,0,0) has a trivial equilibrium point while at = ( , 0,0, ) the disease free equilibrium exists. ( , , , ) = Γ 1 − − + − + ( ) 0 − 0 0 −( + ) ( ) 0 −( + ) 0 0 0 − (7) (0,0,0,0) = Γ 0 0 0 0 −( + ) 0 0 0 −( + ) 0 0 0 − (8) ( , 0,0, ) = −Γ 0 − 0 0 −( + ) 0 0 0 −( + ) 0 0 0 − (9) (ii) Let >1, for system (6), a unique positive endemic equilibrium exists at: = ( ∗, ∗, ∗, ∗), when ( ( + ) > )where = (( ) ) ( ) is the basic reproduction number and ∗ = ∗ = ( ) ( ( ) ) , ∗ = ( ) ( − 1), ∗ = ( ) ( − 1), ∗ = ( ) ( − 1). Proof: (i): The first is easy to prove. (ii): The proof of the theorem is easy, when > 1, clearly, shows the unique endemic equilibrium point. 3.2. Local stability of endemic equilibrium In this subsection, we present the local stability of the system (6) at the endemic equilibrium point . In order to do this, in the following we state and prove the results. Theorem: The endemic equilibrium point , of the system (6) is stable locally asymptotically if > 1, provided that (2 ( + ) ≥ ( + )) and ( > ) are satisfied. Proof: The Jacobian matrix of the system (6), at the endemic equilibrium point is given by: ∗ = Γ − ∗ 0 − ∗ ∗ 0 0 −( + ) ∗ ∗ ∗ ( ∗) 0 −( + ) 0 0 0 − The Jacobian matrix ∗ gives the following equation (Eq. 10), + + + + = 0, (10) where = 1 − + 2( + ) + + , = + + 1 + + + + ( + )( + ) − − 2 ( + ) + ( + ) + 1 , = − + + − 2 1 + − ( − 2 )( + ) − ( − 2 )( + ) + ( + )( + ) − ( − 2 )( + )( + ), = − 2 ( + )( + ) + (1 + ) + (1 + ) . where = ( + ), = ( + ). The Routh-Hurtwiz conditions are satisfied with some sufficient conditions. So, the local stability of Khan et al/ International Journal of Advanced and Applied Sciences, 3(7) 2016, Pages: 46‐53 49 the endemic equilibrium point of the system (6) is locally asymptotically stable. 4. Global stability In this section, we show the global stability for the system (6). We use the method presented in (Li et al., 1996), to obtain the global stability for our model. The disease persistence occurs when the endemic equilibrium is stable globally asymptotically. In such a case, the disease permanently exists in the community. Here, we consider the subsystem of (6), = Γ (1 − ) − , = − μE − μ E, = μ E − μ I − μI, (11) Theorem: For R > 1, the endemic equilibrium point E , of the subsystem (11) is stable global asymptotically. Proof: The second additive compound matrix J∗, of subsystem (11) is (Eq. 12): J∗ = Γ − ∗ 0 − ∗ ∗ 0 −(μ + μ ) ∗ ∗ 0 μ −(μ + μ ) (12) consider the function: P = P(S, E, I) = diag( , , ), then P = diag( , , ) P = diag( , ། , ། ), And P P = ( − , − , − ), also, J∗ = PJ∗P . B = P P + PJ∗P = B B B B where B = − + Γ − , B = (0 − ), B = (0 0) , = − − ( + ) 1 + − − ( + ) consider the norm in R as: |(u, v, w)| = max(|u|, |v|, |w|) where (u, v, w), represents the vector in R and shows by lthe Lozinskii measure w. r. to the norm in (Martin, 1974). l(B) ≤ sup{g , g } = sup{l(B ) + |B |, l(B ) + |B |}, here, |B |, |B |represents the matrix norm with respect to L vector norm and l denote the Lozinskii measure with respect to the L norm . l(B ) = − + Γ − , B = | | and using = ( ) − μ − μ , and second equation of system (6) we get (Eq. 12-16): ∴ g = l(B ) + |B |, = − + Γ − + , = − ( ) + μ + μ + Γ − 2μ + , = + μ + max{μ − ( ) + Γ − 2μ + } (13) ∴ g = l(B ) + |B |, = − − (μ + μ ) − (μ + μ ) + μ + , = − ( ) − (μ + μ ) + μ + , = + μ + max { − (μ + μ ) + ( ) , 2μ − Γ − μ − + ( ) } (14) i. e. l(B) = + μ − min{− + (μ + μ ) + ( ) , ( ) + 2μ − Γ − μ − } (15)l(B) ≤ + μ − ω, ω > 0, = − (ω − μ), (15) i. e. l (B)dS ≤ log ( ) ( ) − (ω − μ), (16) i. e. lim → sup sup l (B)dS ≤ −(ω − μ) < 0. (17) Now, consider the limit system, ∗ = (S∗ − Z∗), (18) based on (19), Z(t) = e [Z(0) + ∗ e ds], (19) Khan et al/ International Journal of Advanced and Applied Sciences, 3(7) 2016, Pages: 46‐53 50 which implies that Z(t) → ∗ = Z∗, as t → ∞ (20) then, we get that E∗is globally asymptotically stable. The proof is completed. 5. Numerical results In this section, we find the numerical solution of the proposed model (6). For the values, Γ = 2, k = 5.7, β = 0.5, a = 0.01, μ = 0.4, μ = 0.2, μ = 0.2, the positive equilibrium of the system at E is E = (2.45902, 4.6606,2.3303, 2.45902). The eigenvalues of the system E are (-2.,-1.72414,-0.4,0.2). 6. Discussion and conclusion We have successfully presented and analyzed a delay SEIR epidemic (special type) model that contains the information variable, Depending on the present values of the variables of the state. We have analyzed the model and found its basic reproduction numberR . We observed that when R < 1, then disease free equilibrium is stable locally asymptotically and the disease comes to halt in the community and an unstable endemic equilibrium exists when R > 1, the disease becomes endemic and will spread in the community. Further, we observed that the value of R at E = (0,0,0,0) the model is unstable and when E = (1,0,0,0)the model becomes stable. The delay parameter T also affects the endemic equilibrium. The endemic equilibrium is locally stable when, R > 1,together with some conditions. For R > 1, we observed that the model is globally asymptotically stable. For the value of T=0.38, the endemic equilibrium is stable locally asymptotically and unstable equilibrium exists for the value of T>0.38. Moreover, we observed that when T=0.58, the endemic equilibrium becomes unstable. Finally, we have presented the numerical solutions of the model and the theoretical results are justified in the form of Figs. 1 to 4. Fig. 1 is the result of the basic reproduction numberR , when R < 1, a stable disease free equilibrium exists and when R > 1, and unstable endemic equilibrium exist. Fig 2-4, shows the behavior of the model for the suggested values. In Fig. 2, a stable equilibrium exists. In Fig. 3, a stable equilibrium exists only for the case when Γ=6 and T=0.68. An unstable equilibrium exists when T>0.38. Figs. 5-7 show the phase portrait of the model (6). Fig. 1: When R > 1, the individuals get risk Fig. 2: Plot with parameter values Γ = 2, k = 5.7, β = 0.5, α = 0.001, μ = 0.4, μ = 0.2, T = 0.38, μ = 0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 R 0 I The individuals that get risk 0 100 200 300 400 500 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 time P op ul at io n Population behavior S E I Khan et al/ International Journal of Advanced and Applied Sciences, 3(7) 2016, Pages: 46‐53 51 Fig. 3: Stable equilibrium exists when Γ = 6 and T = 0.68 and taking the rest of the parameters are same Fig.


Introduction
*In mathematical epidemiology, the constructing and study of models that describe the disease information about the spread and control the infectious disease is the most important major research area of biology. The first who introduced the model is called the "SIR", S-I-R (susceptible, infected, recovered) in 1927 by Kermack and McKendrick (1927). They described a simple SIR model in a closed population over time and incorporated the theoretical number of infected individuals with a contagious disease. The dynamic process between the susceptible and infected individuals is the disease transmission. The analysis and behavior of the SIR type models is greatly influenced by which the transmission among the infective individuals and susceptible are modeled. Often, the epidemiological models are based on the so-called mass action (Anderson et al., 1991). The models, in which realistic transmission functions are used by authors have faced some problems but consequently gained much attention (Ruan and Wang, 2003;Xiao and Ruan, 2007).
Study of mathematical models incorporating the exposed class plays important role in epidemiology due to importance of incubation period of such diseases, like Dengue fever (3 to 14) days. In epidemiology the incubation period or latency can be modeled by adding the exposed class to the model to understand its behavior and dynamics (Gubler, 1988). In Gubler (1988), the delay SIR and delay SEIR models and the comparison for the reproduction number were presented. In literature, the delay differential equations have been used for a variety of models such as, SIR, SEIR, SIS and SIRS. An S-I-S epidemic model with the constant time delay has been studied by Hethcote and Van den Driessche (1995), admitting the duration of infectiousness. Beretta et al. (2001) studied the global stability in SIR epidemic model with distributed delay which described the time taking for an individual to lose infection. Further, with the effect of time delay on the stability of an endemic equilibrium has been studied by Song and Cheng (2005). They provided the conditions in which the stability of endemic equilibrium exists for all delays. Mathematical models that describe epidemiology are widely used for the analysis of global stability of infection free and endemic equilibrium. Many authors worked on the infectious diseases such as Berezovsky et al. (2005), Esteva andMatias, (2001) Greenhalgh (1992), Hsu and Zee (2014), Yi et al. (2009), Zhang et al. (2008 and the references therein. In this work, we study an epidemic SEIR model with the information variable. The work of Kar and Mondal (2011) motivated us for current study, and incorporated the E(t) (exposed Class) to our new model, and studying the model with the new information variable. By incorporating this class, the analysis of SEIR model with information variable will be more interesting for readers and researchers.
The structure of the paper is as follow: We formulate the model in Section 2. In Section 3, we present the local stability for the model. In Section 4, we find the global stability of the endemic equilibrium. The numerical solution of the model is presented in Section 5. Finally, discussion with conclusion is presented in Section 6.

Model constructions
We consider a mathematical model of the type (Eq. 1): with initial conditions (Eq. 2): Here, ( ) is the density of susceptible within the population, ( ) is the density of exposed individuals ( ) is the infected and ( ) is the density of recovered.
shows the natural death rate of the population, represents the disease contact rate and Γ is the intrinsic growth rate. The exposed individuals infected at a rate of and move to the class of the infected. The recovery rate from infection is shown by and is the carrying capacity of the susceptible. The saturation term which measures the inhibitory effect is shown by . Here, we use the new variable , known to be the information variable that collects information for the disease present state, i.e. depending on present values and some previous values of the state variables. Many authors used this variable in their model such as D 'Onofrio et al. (2007), D 'Onofrio, Manfredi (2007), Buonomo et al. (2008), Kar and Mondal (2011). We take the formula (Eq. 3): here ( − ) represents the delaying kernel (Wang et al., 2014), the distributed delay is , that shows for any time t the susceptible, S, the exposed people, E, and infected people, I can be affected with these variables S, E and I, possibly, at some previous times ≤ . In this work, we assume ( , ) = , and where T shows the average delay of the summarized information and also concerning the historical memory on the concern disease. With these assumptions, the following system is presented (Eq. 4): the system (4) is in the form of the non-linear Integro-differential system. By transformation, we can write it in the form of non-linear ordinary differential equations (Eq. 5): in system (5), the last equation is independent of the rest, so we omit it, because R depends only on I. So the model becomes (Eq. 6): where Γ, , , , , , > 0.

Stability analysis
In this section, we present the local stability analysis of the system (6). In the subsection (3.1), we present the local stability of disease free and the subsequent section (3.2) we find the local stability of endemic equilibrium.

Disease free stability
The disease free equilibrium point of system (6) is (Eq. 7). Theorem: An unstable equilibrium exists for All > 0 at . Proof: At the equilibrium point , the Jacobean matrix is (Eq. 8).

Local stability of endemic equilibrium
In this subsection, we present the local stability of the system (6) at the endemic equilibrium point . In order to do this, in the following we state and prove the results. Theorem: The endemic equilibrium point , of the system (6) is stable locally asymptotically if > 1, provided that (2 ( + ) ≥ ( + )) and ( > ) are satisfied.

Proof:
The Jacobian matrix of the system (6) The Jacobian matrix * gives the following equation (Eq. 10), where = ( + ), = ( + ). The Routh-Hurtwiz conditions are satisfied with some sufficient conditions. So, the local stability of the endemic equilibrium point of the system (6) is locally asymptotically stable.

Global stability
In this section, we show the global stability for the system (6). We use the method presented in (Li et al., 1996), to obtain the global stability for our model. The disease persistence occurs when the endemic equilibrium is stable globally asymptotically. In such a case, the disease permanently exists in the community. Here, we consider the subsystem of (6), Theorem: For R > 1, the endemic equilibrium point E , of the subsystem (11) is stable global asymptotically. Proof: The second additive compound matrix J * , of subsystem (11) is (Eq. 12):  |(u, v, w)| = max(|u|, |v|, |w|) where (u, v, w), represents the vector in R and shows by ℓthe Lozinskii measure w. r. to the norm in (Martin, 1974).

Discussion and conclusion
We have successfully presented and analyzed a delay SEIR epidemic (special type) model that contains the information variable, Depending on the present values of the variables of the state. We have analyzed the model and found its basic reproduction numberR . We observed that when R < 1, then disease free equilibrium is stable locally asymptotically and the disease comes to halt in the community and an unstable endemic equilibrium exists when R > 1, the disease becomes endemic and will spread in the community. Further, we observed that the value of R at E = (0,0,0,0) the model is unstable and when E = (1,0,0,0)the model becomes stable. The delay parameter T also affects the endemic equilibrium. The endemic equilibrium is locally stable when, R > 1,together with some conditions. For R > 1, we observed that the model is globally asymptotically stable. For the value of T=0.38, the endemic equilibrium is stable locally asymptotically and unstable equilibrium exists for the value of T>0.38. Moreover, we observed that when T=0.58, the endemic equilibrium becomes unstable.
Finally, we have presented the numerical solutions of the model and the theoretical results are justified in the form of Figs. 1 to 4. Fig. 1 is the result of the basic reproduction numberR , when R < 1, a stable disease free equilibrium exists and when R > 1, and unstable endemic equilibrium exist. Fig  2-4, shows the behavior of the model for the suggested values. In Fig. 2, a stable equilibrium exists. In Fig. 3, a stable equilibrium exists only for the case when Γ=6 and T=0.68. An unstable equilibrium exists when T>0.38. Figs. 5-7 show the phase portrait of the model (6).

Conflict of Interests
The authors have declared that no competing interests exist.

Source of Funding
No current funding sources for this study. Greenhalgh D (1992). Some results for an SEIR epidemic model with density dependence in the death rate. Mathematical Medicine and Biology, 9(2): 67-106.