key: cord-0913928-67vwr9tl
authors: Feng, Xiao-mei; Liu, Li-li; Zhang, Feng-qin
title: Dynamical Behavior of SEIR-SVS Epidemic Models with Nonlinear Incidence and Vaccination
date: 2022-04-09
journal: Acta Math Appl Sin
DOI: 10.1007/s10255-022-1075-7
sha: 42620c9c1c34f733142c65946a4982cbb1148ef0
doc_id: 913928
cord_uid: 67vwr9tl

For some infectious diseases such as mumps, HBV, there is evidence showing that vaccinated individuals always lose their immunity at different rates depending on the inoculation time. In this paper, we propose an age-structured epidemic model using a step function to describe the rate at which vaccinated individuals lose immunity and reduce the age-structured epidemic model to the delay differential model. For the age-structured model, we consider the positivity, boundedness, and compactness of the semiflow and study global stability of equilibria by constructing appropriate Lyapunov functionals. Moreover, for the reduced delay differential equation model, we study the existence of the endemic equilibrium and prove the global stability of equilibria. Finally, some numerical simulations are provided to support our theoretical results and a brief discussion is given.

Mathematical models have been used extensively to study the transmission dynamics of infectious diseases and design or evaluate control measures on the spread of diseases. It is well known that vaccination is an effective strategy against infectious diseases. Most epidemic models are compartmental models described by systems of ordinary differential equations (ODEs). It is generally assumed that individuals from a compartment move to another compartment are homogeneous in ODE models. For instance, vaccinated individuals have the same immunity periods regardless of individual heterogeneity. However, it is pointed out that for some infectious diseases such as mumps vaccinated individuals always lose their immunity at different rates depending on the inoculation time [3] . Therefore, in modeling, it should be more reasonable to introduce vaccination age (i.e., the time individuals spend in the vaccinated class) to describe different immunity waning rates. If we introduce the vaccination age, it will be formulated as an age-dependent epidemic model.

The earlier age-structured epidemic models were proposed by Kermack and Mckendrick [18, 19] and the mathematical theory was developed in the 80-90's of last century [16, 34] . Since then, many research studies about age-structured epidemic models (such as infection age [4, 14, 26, 27, 36] , latency age [8, 33] , vaccination age [17, 22, 30] , relapse age [23] et al.) have been published. on the other hand, some age structure models are established to study pest population control if there is no effective vaccine [15, 25] . Here, we just list some recent works. Iannelli, Martcheva and Li [17] proposed a two-strain epidemic model with vaccination age in which the first strain can infect individuals already infected by the second one. Li, Wang and Ghosh [22] considered a two-dimensional SIS model with vaccination and the rate at which vaccine loses its protective properties with time depends upon vaccination age. Shen and Xiao [30] studied a multi-group SVEIR epidemiological model with the vaccination age and infection age. These results suggest that vaccination is helpful for disease control by decreasing the basic reproduction number, either enhancing the vaccination rate or lengthening the duration of vaccination protection.

It is pointed out that the transmission of an infectious disease depends on both the population behavior and the infectivity of the disease [9] . Mathematically, it is described by the incidence rate of the disease (i.e., the average number of new cases of the disease per unit time). Although bilinear and standard incidence rates are frequently used in classical epidemic models, its nonlinearity was used as the form Sg(I) at disease modelling in [2] , where g(I) is a nonlinear function. Some other forms of nonlinear incidence rates were proposed and studied [29, 35] .

In this paper, we focus on a SEIR-SVS epidemic model with vaccination age and nonlinear incidence of the form Sg(I). Moreover, if we describe the rate at which vaccinated individuals lose immunity by a step function, the age-structure model reduces to a delay differential equation (DDE) model. Our goal is to construct appropriate Lyapunov functionals to show that the socalled basic reproduction number provides the threshold to determine the global dynamics of the age-structured and DDE models.

The paper is organized as follows. In section 2, the age structured and DDE models are formulated. In section 3, for the vaccine-age-structured model, the positivity and boundedness of solutions, the asymptotic smoothness and existence of global attractors of the semi-flows generated by the model, and the existence of equilibria are studied. In sections 4 and 5, by constructing suitable Lyapunov functions, the global asymptotic stability of the age-structured and DDE models are established. In section 6, some numerical simulations on the general age-structured model are carried out. The paper ends with a brief discussion in section 7.

The population is divided into four classes, the susceptible S(t), the exposed E(t), the infectious I(t), the removed R(t), and the vaccinated v(a, t) at time t with vaccine-age a. The vaccineage-structured epidemic model with nonlinear incidence is as follows

with the boundary condition v(0, t) = pS(t), (2.2) and initial conditions

Here, all parameters are assumed to be positive and their biological interpretations are listed in Table 1 The function g(I) satisfies the following assumptions:

Since it is impossible that the age of the vaccinated individuals is infinite, the reasonable

By the continuity of solutions of (2.1), it is necessary to require that pS 0 = v 0 (0). Notice that the variable R(t) does not appear in other equations. Then we may only consider the following reduced system

with the boundary condition v(0, t) = pS(t), and the initial conditions

Along the characteristic line t − a =constant, integrating the fourth equation of (2.5) with the boundary and initial conditions gives Indeed, a vaccinated individual can have the perfect immunity against the infection within a period following vaccination. After the period the vaccine wanes gradually, the vaccinated individual loses the immunity finally, and becomes susceptible. For example, the clinic experiments verified that the neutralizing-antibody response to live SARS-CoV-2 virus was maintained up to 180 days after vaccination and it will be weakened gradually [1] . Based on this fact, the vaccinated class can be divided into two subclasses, which are with perfect and weak immunity, respectively. In order to describe this situation, we choose the losing rate of immunity as the following step function:

where τ is the period of the perfect immunity, γ is the rate at which vaccine wanes. Denote

Then it is the number of vaccinated individuals with weak immunity at time t. From the third equation of (2.5) we have

where v(∞, t) = 0 is used. It follows from (2.6) that v(τ, t) = pS(t − τ )e −µτ for t ≥ τ Then system (2.5) will become as follows

(2.8)

The initial conditions for model (2.8) take the form

In the following we will investigate dynamic behaviors of system (2.5) with conditions (2.2) and (2.3) and system (2.8) with condition (2.9).

This subsection devotes to the preliminaries, including positivity and boundedness.

The function δ(a) satisfies the following assumptions: (F 1 ) δ(a) ∈ L 1 + with essential infimum and supremum, i.e., δ inf and δ sup ;

The standard theory implies that model (2.5) always has a unique nonnegative solution. Furthermore, X is positively invariant and model (2.5) generates a continuous semiflow Φ :

Suppose that x(t) is a solution with the initial condition

Then the semi-flow Φ has the following property.

Proof. Note that

Using the variation of constant formula, one has

Thus, lim sup t→∞ N (t) ≤ A/µ, which implies that Φ is point dissipative. This completes the proof.

As a direct consequence of Proposition 3.1, we have the following results.

The next proposition provides a positive asymptotic lower bound for model (2.5) .

Proof. It follows from the first equation of model (2.5) that

Next, using the second equation of model (2.5) and (3.1), one has

Obviously, it has

Then, combining the third equation of model (2.5) and (3.2) yields that

This implies that

The proof is finished.

There exist two contants T and ϵ such that v(0, t) > ϵ for t ≥ T .

Proof. It follows from v(0, t) = pS(t) and (3.1) that there exists a T such that v(0, t) ≥ pm S ϵ holds. This completes the proof.

In this subsection, the asymptotic smoothness and existence of global attractors of model (2.5) are investigated. The main result is shown by using the following result, which comes from Lemma 3.2.3 in [13] .

has compact closure for each t ≥ T b and any bounded closed set B ⊂ X.

By using Lemma 3.5, we have the following theorem.

Theorem 3.6. The semiflow Φ is asymptotically smooth.

Proof. According to Lemma 3.5, we define two maps

Step 1. To show that condition (i) of Lemma 3.5 holds.

Step 2. To prove that condition (ii) of Lemma 3.5 is valid. According to Proposition 3.2, S, E and I remain in the compact

, which has the compact closure and is independent of x 0 . To do so, we only need to verify that v 2 (a, t) satisfies the following conditions in [31] :

Combining assumption (G 1 ) and Proposition 3.2, one has

which implies that conditions (i), (ii) and (iv) are valid for v 2 (a, t). Now, we check that v 2 (a, t) also satisfies condition (iii). For sufficiently small h ∈ (0, t), one has

(3.7)

Using Proposition 3.2 and Assumption (G 1 ), we know that |S ′ | is bounded by

t has compact closure. Combining Steps 1-2, one can conclude that the semiflow Φ is asymptotically smooth.

Applying the results on the existence of global attractors in Theorem 2.6 [28] , we conclude the following proposition. 

In this subsection, we consider the existence of equilibria. It follows from the equation of v(a, t) of (2.5) that e − ∫ a 0 [µ+δ(ξ)]dξ is the probability for an individual to stay in the vaccinated class for a time units. Denote

then it is easy to see that η < 1, and it represents the probability of leaving the vaccinated class for an individual losing immunity but being alive. Further, it follows from the boundary condition v(0, t) = pS(t) that pη is the per capita rate at which vaccinated individuals return to the susceptible class. Now, we give the main result of this subsection. 3) always has a disease-free equilibrium P 01 (S 01 , 0, 0, v 0 (a)). When R 01 > 1, besides P 01 , it also has a unique endemic equilibrium (2.2) and (2.3) always has a disease-free equilibrium P 01 (S 01 , 0, 0, v 0 (a)), where

.

(3.10)

The endemic equilibrium P *

From the last two equations of (3.11) we have

Substituting them into the first two equations of (3.11) respectively gives

which is equivalent to the following equations

From the first equation of (3.13) we know that I * 1 < εA/[(µ + ε)(µ + α)], and from the second equation of (3.13) we have

.

Substituting it into the first equation of (3.13) gives

(3.14)

is a vertical asymptote of function h 1 (I). For function h 1 (I), we have 

]. Correspondingly, model (2.5) has a unique endemic equilibrium P * 1 (S * 1 , E * 1 , I * 1 , v * (a)) when R 01 > 1.

In order to simplify the process of proving the global stability of the equilibria, we introduce the following lemmas. with conditions w(0, t) = Φ(t), t > 0; w(a, 0) = Ψ(a), a > 0, (3.16) and that functionw(a) satisfies the following ordinary differential equation

where q(a) ∈ C 1 [0, ∞), then the derivative of functionL 1 with respect to t can be expressed as

Proof. It is easy to see that the solution of (3.15) with condition (3.16) is

, a ≥ t > 0. 

Since the functionw(a) satisfies (3.17), we have d dt 

da .

Hence,

Furthermore, using (3.18), we obtain

This completes the proof of Lemma 3.9. 

(ii) For an arbitrary positive number x * the following inequality holds:

Proof. This lemma can easily be understood geometrically. In the following we give an analytical proof.

(i) f ′′ (x) < 0 implies that the function f (x) is convex in (0, ∞), then any tangent line of function f (x) is above the graph of f (x). Since both the graph of f (x) and its tangent line at (0, f (0)) = (0, 0) pass through the origin, we know that f (x) < f ′ (0)x for x > 0.

(ii) Define a function

This implies that Lemma 3.10(ii) is true. The proof of Lemma 3.10 is completed.

With respect to the global stability of equilibria of model (2.5), we have the following theorem.

Theorem 3.11. For system (2.5), the disease-free equilibrium P 01 is globally stable if R 01 ≤ 1, the endemic equilibrium P * 1 is globally stable if R 01 > 1. Proof. We first prove the global stability of the disease-free equilibrium P 01 . Define a Lyapunov functional

where q(a) ∈ C 1 [0, ∞) is to be determined below. Then, applying the equalities A = (µ + p)S 01 − ηpS 01 and v 0 (0) = pS 01 and Lemma 3.9, the derivative of L 11 with respect to time t along the solution of system (2.5) is given by

da.

In order to eliminate the term (3.22) to satisfy d da (q(a)v 0 (a)) = −δ(a)v 0 (a) and q(0) = η.

da.

According to assumption (H) for function g(I), from Lemma 3.10(i) we have that g(I) < g ′ (0)I for I > 0. Hence,

Notice that 1 − x + ln x ≤ 0 for x > 0 and the equality holds if and only if x = 1, then L ′ 11 ≤ 0 as R 01 ≤ 1 and L ′ 11 = 0 if and only if (S(t), E(t), I(t), v(a, t)) = (S 01 , 0, 0, v 0 (a)). Therefore, by the Lyapunov asymptotic stability theorem [12] the disease-free equilibrium P 01 is globally stable if R 01 ≤ 1.

Next, we prove the global stability of the endemic equilibrium P * 1 . Define another Lyapunov functional

where q(a) is a positive and continuous function to be determined later, and the monotonicity of g(I) ensures that the integral ∫ I

du in (0, ∞) has only one extremum which is a global minimum at I * 1 , then L 12 is positive definite with respect to (S * 1 , E * 1 , I * 1 , v * (a)). Applying Lemma 3.10 and the following equalities:

the derivative of L 12 with respect to time t along solutions of (2.5) is given by

da.

If we still choose function q(a) defined in (3.22) , then

da.

Therefore,

Notice that 1 − x + ln x ≤ 0 for x > 0 and the equality holds if and only if x = 1, then, it follows from Lemma 3.10(ii) that L ′ 12 ≤ 0 and L ′ 12 = 0 if and only if (S(t), E(t), I(t), v(a, t)) = (S * 1 , E * 1 , I * 1 , v * (a)). Therefore, by the Lyapunov asymptotic stability theorem [12] the endemic equilibrium P * 1 is globally stable if R 01 > 1. The proof of Theorem 3.11 is completed.

According to the fundamental theory of functional differential equations [12] , it is easy to show the following property. In the following, we consider the existence of the endemic equilibrium of the delay system (2.8), and prove the global stability of the disease-free and endemic equilibria by constructing Lyapunov functionals.

It is easy to see that model (2.8) always has a disease-free equilibrium P 02 (S 02 , 0, V 0 ), where

The endemic equilibrium P * 2 (S * 2 , E * 2 , I * 2 , V * ) with I * 2 > 0 of model (2.8) satisfies the following equilibrium equations

From the last three equations of (4.1) we have

.

Substituting them into the first equation of (4.1) gives

Similar to the analysis for equation (3.14) in Section 3, we know that equation (4.3) has a unique root I * 2 in the interval (0, εA/[(µ + ε)(µ + α)]) as R 02 > 1. Further, it follows from (4.2) that model (2.8) has a unique endemic equilibrium P * 2 (S * 2 , E * 2 , I * 2 , V * ) as R 02 > 1. Therefore, with respect to the existence of equilibria of system (2.8) we have Theorem 4.2. System (2.8) always has the disease-free equilibrium P 02 (S 02 , 0, 0, V 0 ). When R 02 > 1, besides P 02 , it also has a unique endemic equilibrium P * 2 (S * 2 , E * 2 , I * 2 , V * ), where

, and I * 1 is determined by equation (4.3) . With respect to the global stability of equilibria of model (2.8), we have Theorem 4.3. For system (2.8) , the disease-free equilibrium P 02 is globally stable if R 02 ≤ 1, the endemic equilibrium P * 2 is globally stable if R 02 > 1. Proof. We first prove the global stability of the disease-free equilibrium P 02 . Define a Lyapunov functional

Then, applying the equalities A = (µ + p)S 02 − γV 0 and pe −µτ = (µ + γ)V 0 /S 02 , the derivative of L 21 with respect to time t along solutions of system (2.8) is given by

] .

Applying Lemma 3.10(i) and the equality V 0 /S 02 = pe −µτ /(µ + γ) yields

] .

According to Lemma 3.1 in [21] , 2 − S02V (t) . Hence, it follows from the Lyapunov asymptotic stability theorem [12] that the endemic equilibrium P 02 is globally stable if R 02 ≤ 1.

Next, we prove the global stability of the endemic equilibrium P * 2 . Define the other Lyapunov functional by

dθ.

The monotonicity of g(I) ensures that the integral ∫ I

du in (0, ∞) has only one extremum which is a global minimum at I * 2 . Then L 22 is positive definite with respect to (S * 2 , I * 2 , V * ). Applying the equalities

the derivative of L 22 with respect to time t along solutions of (2.8) is given by

Applying the equality γV * /S *

] . 2 , E * 2 , I * 2 , V * ). Therefore, by the Lyapunov asymptotic stability theorem [12] the endemic equilibrium P * 2 is globally stable if R 02 > 1. The proof of Theorem 4.3 is completed.

In this section, we provide some numerical simulations of the general model (2.5) to verify the feasibility of theoretical results. The main parameter values are given in Table 2 . the additional death rate induced by the infectious diseases 0.002 [5] ω the recovery rate from the infectious class 0.025 [37] Model (2.5) is applied to the epidemic of HBV status in China. According to the reference [10] , there is a assumption that the rate of waning vaccine-induced immunity is 0.1 and the immunity protective period is about 10 years ( [10] ). Then δ(a) is taken as δ(a) = 0.1 sin( (a−5)π 10 ). Using parameter values in Table 2 , this subsection concerns on the influence of the adults' vaccination coverage rate on the dynamical behaviors of model (2.5) . Take p = 0.15 and R 01 = 1.25 > 1. According to Theorem 3.11, there exists a unique globally stable endemic equilibrium, which is simulated in Fig. 1 . We have S * 1 = 0.06, E * 1 = 0.94, I * 1 = 0.60, and the proportion of vaccination class is the distribution function of a, particularly, v * (0) = 0.035 and v * (10) = 0.012. Moreover, Fig. 2 shows the distribution of the vaccination population at the endemic equilibrium for all time and a range of age. According to Figs. 1-2, we can see that a lower vaccination rate will lead to the appearance of a stable endemic equilibrium.

If we increase the vaccination rate of adults to p = 0.8, after some computations, we obtain R 01 = 0.25 < 1. It follows from Theorem 3.11 that model (2.5) tends to the disease-free equilibrium which is globally stable (See Fig. 3) . Furthermore, one has S 01 = 0.029, E 01 = 0, I 01 = 0, and the proportion of the vaccinated class is the distribution function of a, particularly, v 0 (0) = 0.023 and v * 0 (10) = 0.008. A comparison between Fig. 1 and Fig. 3 implies that increasing adults' vaccination coverage will be an effective strategy to reduce and control the incidence of some epidemic diseases. 

In this paper, we considered two classes of epidemic models with vaccination by incorporating the vaccine age into a SIR epidemic model with nonlinear incidence. One is model (2.5) with vaccine age, which is constrained by the boundary condition (2.2) and the initial condition (2.3); the other one is model (2.8) with stage structure of vaccination. From the derivation of system (2.8) we know that (2.8) is a special case of (2.5), but they include partial differential equations and delay differential equations, respectively. The thresholds (i.e. R 01 and R 02 ) determining their dynamical behaviors were found. The obtained results show that the two models have the same dynamical behaviors, but there exist some differences on the associated analysis since the types of the two models are different.

On the other hand, since (2.8) is a special case of (2.5), we can also obtain the threshold R 02 from the expression of R 01 . In fact, substituting the step function δ(a) into (3.9) yields η = Then, R 02 can be obtained from R 01 . For epidemic models, the basic reproduction number is defined as the expected number of secondary cases produced by one infective host in an entirely susceptible population [6, 32] . It is easy to see that, in the absence of vaccination, both basic reproduction numbers of models (2.5) and (2.8) are Ag ′ (0)/µ(µ + α) [20] , where A/µ represents the size of the entirely susceptible population. When vaccination is incorporated, from the epidemiological meaning of η we know that a fraction, (1 − η)p/[µ + (1 − η)p], of susceptible individuals is transferred to the vaccinated class and cannot be infected. Then the size of the entirely susceptible population becomes

Therefore, the threshold

is the basic reproduction number of model (2.5) . Further, according to the relation between the two models, R 02 is the basic reproduction number of model (2.8) .

Adjuvanting a subunit COVID-19 vaccine to induce protective immunity

A generalization of the Kermack-Mackendric deterministic model

Loss of vaccine induced immunity to varicella over time

Global analysis of an SIR epidemic model with infection age and saturated incidence

Progress in hepatitis B prevention through universal infant vaccination-China

On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations

The population-level impact of HBV and its vaccination on HIV transmission dynamics

Global stability of an SVIR model with age of vaccination

Vaccination policies and nonlinear force of infection: generalization of an observation by Alexander and Moghadas

Vaccination agains the hepatitis B virus in highly endemic area: waning vaccine-induced immunity and the need for boosterdoses

The transmission dynamics and control of hepatitis B virus in the Gambia

Persistence in infinite-dimensional systems

Lyapunov functions and global stability for age structured HIV infection model

Assessing the efficiency of Wolbachia driven Aedes mosquito suppression by delay differential equations

Mathematical Theory of Age-Structured Population Dynamics

Strain replacement in an epidemic model with super-infection and perfect vaccination

Contributions to the mathematical theory of epidemics-II. The problem of endemicity

Contributions to the mathematical theory of epidemics-III. Further studies of the problem of endemicity

Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission

Global stability of a viral infection model with delays and two types of target cells

Stability and bifurcation of an SIVS epidemic model with treatment and age of vaccination

Global stability of an SEIR epidemic model with age-dependent latency and relapse

Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models

Modeling mosquito population control by a coupled system

Lyapunov functional and global asymptotic stability for an infection-age model

Two-group infection age model including an application to nosocomial infection

Global attractors and steady states for uniformly persistent dynamical systems

Dynamical behavior of an epidemic model with a nonlinear incidence rate

Global stability of a multi-group SVEIR epidemiological model with the vaccination age and infection Age

Dynamical Systems and Population Persistence

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

The stability analysis of an SVEIR model with continuous age-structure in the exposed and infectious classes

Theory of Nonlinear Age-Dependent Population Dynamics

Global analysis of an epidemic model with nonmonotone incidence rate

Global stability of an age-struncture virus dynamics model with Beddington-Deangelis infection function

Modeling the transmission dynamics and control of hepatitis B virus in China

Birth Rate, Death Rate and Natural Growth Rate of Population