Successive Vaccination and Difference in Immunity of a Delay SIR Model with a General Incidence Rate

and Applied Analysis 3 (4) The parameters a are a convex combination constant, r = b − d > 0 is the intrinsic growth rate (b is the birth rate), and K > 0 is the carrying capacity of the population.The term (b − (arN(t)/K)) has a densitydependent per capita birth rate and the term (d+((1− a)rN(t)/K))has a density-dependent per capita death rate [39]. (5) For 0 < a < 1, the birth and death rates are consistent with the limited resources associated with density dependence. The birth rate is density independent when a = 0 and the death rate is density independent when a = 1. Thus, the spread of the disease (animals such as rodents, etc.) is assumed to be governed by the following system of logistic equations with time delay. (6) The total population is assumed to be large enough to be adequately described by a deterministic model and is divided into compartments based on the disease status [40]. (7) The successive vaccination rate θ is positive. The positive constant μ1 is the recovery rate of the infectious individuals from compartment I to S.The parameters β are the effective per capita contact rate constant of infected individuals. The parameters δ are the recovery rate of infected individuals. (8) Models are formulated as functional differential and/or integral equations when time delay is included [40]. Ours follows the former with the assumption that the I-equation satisfies a certain integral condition [41]. Models with multiple delays are not common, but few authors have in the past considered these-Beretta et al. [42]to name but a few. Since N(t) = S(t) + I(t) + R(t), thus the governing equation (1) can be rewritten as dN dt = r [1 − N K ]N, dI dt = βe1 (N − I − R) f (I (t − τ)) − [d + (1 − a) rN K ] I − μ1I − e −d 1 δI (t − ω) , dR dt = e1δI (t − ω) − [d + (1 − a) rN K ]R + θ (N − I − R) . (3) Let τ = max(ω, τ). Then (3) satisfies the following initial conditions N(0) = S0 > 0, I (0) = φ (θ) ≥ 0, ∀θ ∈ [−τ, 0] , R (0) = R0 ≥ 0. (4) In this paper, we will consider two different delays τ, ω which are important parameters on the dynamic behavior. So, the present study is continuation of the previous work τ = ω by Naresh et al. [43]. Lemma 1. All solutions of the model system (3) starting in R + are bounded and eventually enter the compact attracting set Φ = {(S, I, R) ∈ R 3 + : S (t) + I (t) + R (t) = N (t) ≤ K} . (5) Lemma 2. Let the initial data be N(0) = S0 > 0, I(0) = I0(u) ≥ 0, for all u ∈ [−τ, 0], with I0(0) > 0, R(0) = R0 ≥ 0. Then, the solution (S(t), I(t), R(t)) of the model remains positive for all time t > 0. Lemma 3 (see [44]). For the characteristic equation in the form p(λ) + q(λ)e = 0, where p and q are polynomials with real coefficients and r > 0 is the delay, suppose (a) p(λ) ̸ = 0, R(λ) > 0; (b) |q(iy)| < |p(iy)|; 0 ≤ y < ∞; (c) lim|λ|→∞,R(λ)≥0|q(λ)/p(λ)| = 0. Then R(λ) < 0 for every root λ and all r > 0. 3. Equilibrium and Stability Analysis In this section, we focus on the existence and local stability of equilibria. Let the right-hand side of equalities in model (3) be zero. Then, there are two equilibria; namely, (i) E0 = (K, 0, p), p = Kθ/[d+(1−a)r+θ]), disease-free equilibrium; (ii) E = (N, I, R), endemic equilibrium, where the values ofN, I, and R are given in Section 3.2. 3.1. Community Matrix. Firstly, after computing the Jacobian or community matrix of model (3) at point (N, I, R), the characteristic equation is given by 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 r − 2r K N − λ 0 0 βe1f (I) − (1 − a) r K I me1 − δe1 − n − λ βe1f (I) θ − (1 − a) r K R δe1 − θ − [d + (1 − a) r K N + θ] − λ 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 = 0, (6) 4 Abstract and Applied Analysis where m = β(N − I − R)f(I), n = βe1f(I) + d + ((1 − a)r/K)N + μ1. Now, we analyze the equilibria stability of system (3). Computing the Jacobian of system (3) evaluated at E0, one gets the following matrix J (E0) = ( −r − λ 0 0 0 a22 − λ 0 (d + θ) θ d + (1 − a) r + θ δe1 − θ − [d + (1 − a) r + θ] − λ ) , (7) where a22 = β (K − p)f 󸀠 (0) e −(d 1 +λ)τ − δe1 − [d + (1 − a) r + μ1] . (8)


Introduction
The current threat of some new type diseases has raised our awareness that curbing the spread of some emerging and reemerging human diseases is of public health importance such as H1N1.This emerging disease, which was first reported in Mexico, spread very quickly, due to the travel of infected persons by airplanes, trains, and buses to some other regions.It continued to spread around the world and caused about 5000 deaths.In recent years, many mathematical models have been developed for the transmission dynamics of infectious diseases such as SARS, HIV/AIDS, measles, and smallpox ( [1][2][3][4][5][6], to name a few, and the references therein).These models have provided understanding of the underlying mechanisms which influence the spread of diseases and suggested some control strategies.Moreover, to our knowledge, the first effective control strategy for the elimination of infectious diseases is obtaining immunity.It has been reported that "People's immunity to A/H1N1 flu virus is greater than previously thought after access vaccines.The WHO is working to give more nations access vaccines to fight the H1N1 flu pandemic." Besides these above studies, many authors formulated and analyzed SIR epidemic models for the control of diseases [7][8][9][10][11][12][13][14][15][16][17][18][19][20].In particular, some authors have studied the effects of vaccination on the spread of diseases [7][8][9][10][11]; others have studied the effects of treatment on the spread of diseases [12][13][14][15].Gao et al. have proposed an epidemic model with density-dependent birth pulses and seasonal prevention [16].Recently, some works have investigated permanent and temporary immunity [17][18][19][20].However, in these SIR models, an unrealistic assumption is that all the rest of infected individuals acquire immunity besides death.Measles encephalitis in adults in [21,22] shows that there is difference in immunity of infected individuals.That is, some infected individuals can acquire immunity after recovery, but some do not acquire immunity and can be infected once more.At the same time, vaccination is an important strategy for the elimination of infectious diseases [7][8][9][10][11].
Vaccinations have many types; impulsive vaccination and successive vaccination are two main policies.Successive vaccination is that people have been vaccinated at birth to protect themselves from disease; the studies can be found in 2 Abstract and Applied Analysis [23,24].Makinde in [23] studied a SIR model for the transmission dynamics of a childhood disease in the presence of a preventive vaccine and analyzed the vaccination reproductive number for disease control and eradication qualitatively.Impulsive vaccination (only at fixed time sequence we execute effectively the vaccination for the disease) is an important and effective strategy for the elimination of infectious diseases and has been studied in the literature.For example, see [10,17,[25][26][27].In above-mentioned papers, authors almost considered the vaccination of susceptible population.But, in fact, under a certain situation, the vaccine treatment also should be considered for the newborns of the susceptible, the exposed, and the removed.We find that there are few studies on the aspect of the vaccination of newborns.In this case, successive vaccination seems more reasonable than impulsive vaccination.Therefore, in this paper, a SIR model with difference in immunity and successive vaccination is considered.
As far as disease transmission is concerned, the incidence rate, defined as the rate of new infection, plays a very important role in modelling infectious diseases.Bilinear incidence rate  in [28,29] and standard incidence rates / in [30,31] have often been used in epidemic models.However, it is unreasonable to consider the bilinear incidence rate (based on the law of mass action) as the number of susceptibles is large, owing to the number of susceptibles with which every infective contact within a certain time is limited.Standard incidence rate may be a good approximation if the number of available partners is large enough but it is not possible to make more contacts when the population  is small.Combine the two previous approaches by assuming that if the number of available partners  is low, the number of actual per capita partners is proportional to , whereas if the number of available partners is large, there is a saturation effect which makes the number of actual partners constant.Considering this case, a saturation incidence rate of type () with () = /(1 + ) is being proposed in [32].More general incidence rate used in the literature is the one for which () =   /(1 +  ℎ ) [33,34], where   measures the infection force of the disease and () = 1/(1 +  ℎ ) measures the inhibitory effect caused by behavioral changes.Note that if () is decreasing when  is large, this may be interpreted as the fact that susceptibles tend to reduce their social contacts if the perceived number of infectives increases over a psychologically significant value.The above saturation incidence rate depends also on the size of the infectives  termed as infectives-dependent. Particular examples of susceptibles-dependent incidence rate are () = /(1 + ) [35].Very general incidence rates which are not linear in  are also used in Derrick and van den Driessche [36] ((, , ) = Φ(, , ), where  =  + ), Korobeinikov and Maini [37] ((, ) = ℎ 1 ()ℎ 2 ()), and Moghadas and Alexander [38] ((, ) = (1 + (, V))).
Nie et al. [19] and Ji et al. [39] respectively considered a delay SIR epidemic model with nonlinear incidence rate and density-dependent birth and death rates.Motivated by the main idea described in [6,39], in this paper, we consider a delay SIR model with difference in immunity and successive vaccination and an abstract incidence rate.The main difference between our study and those described in [6,39] is the difference in immunity and successive vaccination and an abstract incidence rate.An abstract incidence rate of type () is employed to model the spread of the disease which is propagated through the infective individuals, under a few biologically feasible assumptions upon ().
In view of above facts, we will formulate a mathematical model in Section 2. We provide the region of biologically feasible solutions in Section 3.Then, we study the existence and stability of the steady states in the next section, analyze the permanence result in Section 5, and give some numerical simulations in Section 6. Lastly, we end the paper with a brief discussion of our results in Section 6.

Model Formulation and Invariant Region
In this section, we will present a delay SIR epidemic model with a general nonlinear incidence rate.The total population () is divided into three subclasses, namely, the susceptibles (), the infectives (), and the recovered individuals ().
(1) There are no disease induced deaths, and all the newborns are susceptible.(2) () is the nonlinear incidence rate satisfying the following assumptions: (3) The force of infection at any time  is dominated by  − 1  ()((−)), where  is incubation period and 0 <  − (5) For 0 <  < 1, the birth and death rates are consistent with the limited resources associated with density dependence.The birth rate is density independent when  = 0 and the death rate is density independent when  = 1.Thus, the spread of the disease (animals such as rodents, etc.) is assumed to be governed by the following system of logistic equations with time delay.
(6) The total population is assumed to be large enough to be adequately described by a deterministic model and is divided into compartments based on the disease status [40].(8) Models are formulated as functional differential and/or integral equations when time delay is included [40].Ours follows the former with the assumption that the -equation satisfies a certain integral condition [41].
Models with multiple delays are not common, but few authors have in the past considered these-Beretta et al. [42]to name but a few.Since () = () + () + (), thus the governing equation ( 1) can be rewritten as Let  = max(, ).Then (3) satisfies the following initial conditions In this paper, we will consider two different delays ,  which are important parameters on the dynamic behavior.So, the present study is continuation of the previous work  =  by Naresh et al. [43].

Lemma 1.
All solutions of the model system (3) starting in  3 + are bounded and eventually enter the compact attracting set Lemma 2. Let the initial data be (0 Then, the solution ((), (), ()) of the model remains positive for all time  > 0.
(3) It is sufficient to show that () = 0 does not admit a purely imaginary root.In fact, if  = V (V > 0) is a root of (() = 0), then by separating the real part, one gets Together with the condition of Proposition 4, we have This is impossible.
(4) It is easy to show that () = 0 has no imaginary root whose real part is positive.Otherwise, there is an imaginary root  =  + V with  > 0. Without any loss of generality, we consider V > 0.Then, we take the real and imaginary parts of () = 0; namely, Combined with (9), we have This is a contradiction which implies that all eigenvalues roots of () have negative real parts.Therefore, the disease-free equilibrium of the system (3) is locally asymptotically stable when (9) holds.The proof is completed.
Proposition 8.For 0 <  < , () < 0 for every root  of ℎ() = 0 when From what has been discussed above, we get the following.

Existence of Endemic
Equilibrium.Thus, by Theorem 9, we may define the basic reproduction number as This threshold  0 defines the average number of secondary infections generated by a typical infectious individual in a completely susceptible population in a steady demographic state.
In Theorem 9, we have already shown that the system (3) has an infection-free steady state which is locally asymptotically stable under condition  0 < 1.The disease-free equilibrium is unstable when  0 > 1, and the system (3) has a nontrivial endemic equilibrium  * = ( * ,  * ,  * ) when  0 > 1.From (3), where  = ( − 1   − )/( + (1 − ) + ).Substituting these values of  * and  * in the second equation of (3), we get the following equation for : Obviously,  = 0 is one of the roots of (29) as (0) = 0. Therefore, to exclude that root, choose It can easily be seen that the function () is negative for large positive ; that is, Next, we determine the sign of its derivative It can easily be seen that  > .In addition, from the properties of the function (), in particular from (0) = 0 and   (0) < 0, it follows that () −   () > 0, and consequently   () < 0 for all  > 0. Therefore, for a positive root of () = 0 to exist, () has to satisfy (0) > 0; that is, Hence, one needs the requirement that  0 > 1 to ensure the existence of the endemic equilibrium.From the above analysis, we have the following theorem.

Local Stability of the Endemic
where Since all the model parameters are assumed to be nonnegative, it follows that one eigenvalue is negative; that is,  1 = −.Thus, the stability of  * depends on the roots of the quasipolynomial We note that  1 > 0 and  0 > 0, whereas  1 and  0 may be positive or negative.For  = 0, we state the following results that follow directly from (39).The endemic steady state is locally asymptotically stable if the following conditions hold: The main purpose of this paper is to study the stability behavior of  * in the case  ̸ = 0. Obviously,  ( > 0) is the root of ( 29) if and only if  satisfies Separating the real and imaginary parts, we have Eliminating  by squaring and adding ( 39) and ( 40), we obtain a polynomial in  as Suppose that the conditions hold for all  ≥ 0.Then, the infected steady state of the system (3) is locally asymptotically stable.

Permanence
In this section, we investigate a permanence result [5].The following is our main result of this paper.We will give the following result by using some techniques given in [8,11].The proof of the permanence with nonlinear incidence is a daunting task.Consequently, for simplicity and mathematical convenience, let us choose a linear incidence rate () = .The result holds with the nonlinear incidence, as shown numerically, but the algebraic proof is long and tedious, and the conditions to impose on some of the parameters may be very restrictive.Now, let us firstly give the following theorem.
In fact, let  <  be arbitrary.Choose  1 >  so large that () >  −  for  >  1 .We have the following inequality: for all  ≥  1 , which implies that lim inf Note that  may be arbitrarily small so that lim  → ∞ inf () ≥  1 .
In the rest, we now need to consider two cases: (i) () ≥  * for all large ; (ii) () oscillates about  * for all large .

Numerical Analysis
Since it is important to visualize the dynamical behavior of the model, the model system (3) is integrated numerically with the help of MATLAB 7.0 using the following set of parameters.
(3) The results of numerical simulation are displayed graphically in Figures 3 and 4. In Figure 3, the variation of the infective population and recovered population is shown with time for different values of the removal rate constant from groups  to ,  1 .It is found that both the infective population and the recovered population decrease as  1 increases.Figure 4 depicts the variation of infective population and recovered population, respectively, with time for the different successive vaccination rate, .As  increases, the infective population decreases whereas the recovered population increases.

Discussion
In this paper, we will consider two different delays which are important parameters on the dynamic behavior.So, the present study is continuation of the previous work by [43].Furthermore, from biological epidemic point of view, we investigate successive vaccination and difference in immunity in our system.From mathematical point of view, we study the stability of disease-free equilibrium and the existence of endemic equilibrium for different delay and consider the permanence of the system in the new paper.
In Theorems 9, 10, 11, and 13 corresponding to their corollaries, when the effect of the successive vaccination rate and the transfer rate from the infectious group to the susceptible group after treatment is strong, that is,  >  * and  1 >  * 1 , the basic reproduction number  0 being unity is a strict threshold for the control of the disease; the disease will be extinct or otherwise will tend to break out and persist.The other results are displayed graphically from our numerical simulation.We show the variation of the infective population and recovered population with time for different values of  1 .It is found that both the infective population and the recovered population decrease as  1 increases.The infective population decreases whereas the recovered population increases as the successive vaccination rate increases, , respectively.

( 7 )
The successive vaccination rate  is positive.The positive constant  1 is the recovery rate of the infectious individuals from compartment  to .The parameters  are the effective per capita contact rate constant of infected individuals.The parameters  are the recovery rate of infected individuals.
[20]1 represents the survival probability of individuals in the population after time [20].It is also assumed that  1 ≤  in [−, 0], where  is the death rate and  1 is the death rate in the time interval [−, 0].