Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 45, pp. 1–16. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu MODELING PORCINE PSEUDORABIES WITH AGE STRUCTURE YUHUA LONG, YINING CHEN Abstract. Porcine pseudorabies is an acute and highly contagious viral dis- ease caused by the pseudorabies virus. It inflicts enormous losses to the pig- breeding industry. In this paper, we propose an age-structured mathematical model. We investigate the dynamics of this model characterized by the basic reproduction number <0 = max{<01,<02} by addressing the existence and global stability of equilibria. When <0 < 1, the disease-free equilibrium is unique and globally asymptotically stable. The boundary equilibrium exists and is globally asymptotically stable under the condition <01 < 1 and <02 > 1 or <01 > 1 and <02 < 1 + ε. If both <01 > 1 and <02 > 1 + ε, there is a unique disease-endemic equilibrium which is globally asymptotically stable. 1. Introduction Pseudorabies, known as Aujeszky disease (AD), is an acute and highly contagious viral disease caused by the pseudorabies virus (PRV), and inflicts high economic losses in the swine industry globally [21, 26]. PRV, a double-stranded DNA virus, has a wide range of infection and strong pathogenicity, and belongs to the natural focus disease, which can be found all over the world [14, 30, 32]. Besides pigs, primary hosts and reservoirs of PRV [22], PRV has a broad spectrum of hosts, such as dogs, sheep, cattle, bears, cats, bats, and some avian species [24]. More- over, since an early study describing PRV’s ready infection in human cells proved the PRV’s zoonotic threat theoretically [27], mounting evidence has indicated that PRV has public health significance [15]. Thus, it is necessary to further study for understanding PRV’s potential threat to public health security. According to records, pseudorabies was first discovered in the United States in 1813, which was characterized by a “mad itching disease” of cattle. The sick cattle were extremely itchy and finally died [10]. At that time, the disease was mistakenly regarded as the same disease as rabies, and it was not called Aujeszky’s disease until 1931 [9]. As early as 1902, the Hungarian scholar Aujeszky reported the disease for the first time [1]. Then in 1910, Schniedhofer proved that the disease was caused by the virus. Sabin and Wright confirmed that the virus was herpes virus in 1934. In the next year, Shope found that pigs played an important role in the transmission of the disease. In 1947, Liu first reported pseudorabies in cats [10]. The transmission characteristics of pseudorabies and the infection situation in different periods were 2010 Mathematics Subject Classification. 92D25, 34D23, 92D40. Key words and phrases. Age-structured porcine pseudorabies model; equilibrium; basic reproduction number; global stability. c©2021 Texas State University. Submitted January 22, 2021. Published May 25, 2021. 1 2 Y. LONG, Y. CHEN EJDE-2021/45 studied in [4], and phylogenetic analysis was carried out in [2, 7, 21]. Very recently, considering the vertical transmission, [18] studied the dynamics of pseudorabies by establishing an SEIT mathematical model. PRV infection can elicit nervous system disorders. Its clinical manifestations mainly depend on the virulence and the amount of infection, as well as the age of infected pigs. Among them, the age of infected pigs is the most important factor, and the clinical symptoms of different age groups are also different. The disease of piglets infected with PRV is the most serious. Neonatal piglets infected with PRV will cause a large number of deaths. In clinical diagnosis, the newborn piglets show normal performance on the first day. The disease begins to occur from the second day and reach the peak of death within 3 to 5 days [3, 12]. At the same time, the sick piglets show obvious neurological symptoms, such as drowsiness, singing, vomiting and diarrhea, and the temperature rises above 41◦C. Once the disease occurs, the sick piglets will die within 1 to 2 days. When piglets within 15 days of age are infected with the disease, the disease is very serious and the mortality rate can reach 100%. The incidence rate of PRV in weaned piglets is about 20% ∼ 40%, and the mortality rate is about 10% ∼ 20% [3, 12]. Piglets born more than 60 days after infection show mild symptoms, such as transient fever, mental depression and low mortality, but those piglets grow slowly and have low feed reward. Adult pigs are generally recessive infection, even if there are symptoms occur, they are also very mild and easy to recover and generally recover within 4 ∼ 8 days. At present, there is no specific drug for the disease, only to take measures such as vaccine prevention, timely isolation, elimination of sick pigs, purification of pig population and so on. On the other hand, nowadays, although PRV has been eradicated from domes- ticated pigs in North America and some European countries with only sporadic outbreaks, it continues to be found throughout China [20]. Particularly, with anti- genically different PRV variants have emerged since 2012 [2, 29], it becomes more and more difficult to control porcine pseudorabies with vaccination [20]. Even worse, the number of swine infected with pseudorabies has shown an upward trend, which results in severe restriction of the development of the pig industry and con- siderable economic losses to farmers in recent years [11]. In view of the above observations, it is urgent to develop both the theoretical and clinical method to diagnose and control PRV infection accurately and quickly. Therefore, we propose an age-structured (S1I1S2I2R2) model based on the influence and transmission characteristics of PRV infection on two pig groups of different ages, and study the transmission dynamics of the disease. The outline of this paper is as following: after this introduction, we formulate the mathematical model. In Section 3, we study the existence and dynamic behavior of equilibria by a threshold value, the basic reproduction number, <0. Finally, we draw a brief conclusion in Section 4 to accomplish this paper. 2. Model formulation Considering that PRV infection brings different effects on piglets and adult pigs [3, 12], we divide the swine population into five classes, namely, the susceptible piglets (S1), infected piglets (I1), susceptible adult pigs (S2), infected adult pigs (I2) and recovered adult pigs(R2), such that the total population size is N = EJDE-2021/45 MODELING PORCINE PSEUDORABIES 3 S1 + I1 + S2 + I2 + R2. At first, we assume all parameters are positive and make the following hypotheses: (i) Piglets and adult pigs are kept separately and only the symptomatic in- fected swines are treated. Let the recruit, newborn piglets, is a constant Λ and they are susceptible. Assume a fraction ε of newborn piglets grow up with no symptoms and progress to class S2. (ii) The transmission is assumed to take the form of direct contact between the infectious and the susceptible and the incidence rate is β1S1I1 or β2S2I2 of piglets or adult pigs, respectively. (iii) The symptomatic infected swines are treated. After the treatment, some of piglets grow up and enter class S2 with rate γ. After the treatment, some of adult pigs become asymptomatic and enter the recovery class R2 with rate ν. Some recovered adult pigs with no immunity reenter class S2 with fraction µ. (iv) Let d1 and α denote the natural death rate and the disease-related death rate of piglets, respectively. The adult pig is generally recessive infection, even if the adult pig gets sick, the symptoms are very mild and easy to recover. Therefore, the disease mortality of adult pigs is omitted. Here let d and δ represent the natural death rate and the killing rate of adult pigs, respectively. For convenience, we write d2 = d + δ to be the death rate, including the natural death rate d and the killing rate δ, of adult pigs. Considering the economic effect, mass slaughter of adult pigs leads to d2 � d1. Meanwhile, taking into account of the fact that the smaller the age of the piglets, the higher incidence rate and mortality rate [12], we may assume that γ < ε ≤ α+ γ, d2 ≥ d1 + ε. (2.1) With the above assumptions, we illustrate the model dynamics in Figure 1. Λ- S1 β1S1I1- I1 γI1- S2 β2S2I2- I2 ? d1S1 ? d1I1 ? αI1 ? d2S2 ? d2I2 ? εS1 R2 - νI2 ? d2R2 ? µR2 Figure 1. Progression of infection from the susceptible piglets (S1) through infected piglets (I1), susceptible adult pigs (S2), in- fected adult piglets (I2) and recovered adult pigs (R2) for the model (2.2). 4 Y. LONG, Y. CHEN EJDE-2021/45 Consequently, the model we are concerned with is described by the following differential equations together with nonnegative initial conditions dS1 dt = Λ− β1S1I1 − (d1 + ε)S1 := F1(S1, I1, S2, I2, R2), dI1 dt = β1S1I1 − (d1 + α+ γ)I1 := F2(S1, I1, S2, I2, R2), dS2 dt = εS1 + γI1 − β2S2I2 − d2S2 + µR2 := F3(S1, I1, S2, I2, R2), dI2 dt = β2S2I2 − (d2 + ν)I2 := F4(S1, I1, S2, I2, R2), dR2 dt = νI2 − (d2 + µ)R2 := F5(S1, I1, S2, I2, R2). (2.2) Obviously, for i = 1, 2, 3, 4, 5, Fi : R5 + → R are C1 functions. Moreover, F1(0, 0, 0, 0, 0) = Λ and Fi(0, 0, 0, 0, 0) = 0 for i = 2, 3, 4, 5. Hence, given an initial condition (S1(0), I1(0), S2(0), I2(0), R2(0)) ∈ R5 +, (2.2) admits a unique solution (S1(t), I1(t), S2(t), I2(t), R2(t)) ∈ R5 + through (S1(0), I1(0), S2(0), I2(0), R2(0)) on the existence interval. Let N = N1 + N2 be the total number of pigs, where N1 = S1 + I1 and N2 = S2 +I2 +R2 are the total number of piglets and adult pigs, respectively. Then from system (2.2), we have dN1 dt = Λ− d1N1 − εS1 − (α+ γ)I1, dN2 dt = εS1 + γI1 − d2N2, (2.3) and dN dt = Λ− d1N1 − d2N2 − αI1. (2.4) Taking account of γ < ε ≤ α+ γ and the nonnegativity of the solution, (2.3) leads to dN1 dt ≤ Λ− (d1 + ε)N1, dN2 dt ≤ εN1 − d2N2. Then the above estimation gives N1(t) ≤ Λ d1 + ε − Λ d1 + ε e−(d1+ε)t +N1(0)e−(d1+ε)t, N2(t) ≤ εΛ d2(d1 + ε) − εΛ d2(d1 + ε) e−d2t +N2(0)e−d2t. Consequently, lim sup t→∞ N1(t) ≤ Λ d1 + ε and lim sup t→∞ N2(t) ≤ εΛ d2(d1 + ε) , that is lim sup t→∞ (S1(t)+I1(t)) ≤ Λ d1 + ε , lim sup t→∞ (S2(t)+I2(t)+R2(t)) ≤ εΛ d2(d1 + ε) . (2.5) EJDE-2021/45 MODELING PORCINE PSEUDORABIES 5 Consider d2 � d1, then the nonnegativity of the solutions of system (2.2) and (2.4) lead to dN dt ≤ Λ− d1N. In the same manner as (2.5), it yields that lim sup t→∞ N(t) ≤ Λ d1 . Then lim sup t→∞ (S1(t) + I1(t) + S2(t) + I2(t) +R2(t)) ≤ Λ d1 , which shows that the solution (S1(t), I1(t), S2(t), I2(t), R2(t)) is ultimately bounded and exists on [0,+∞). Denote Ω = { (S1, I1, S2, I2, R2) ∈ R5 +|0 < S1 + I1 + S2 + I2 +R2 ≤ Λ d1 } . (2.6) Similar to [25], one can prove that Ω is a positive invariant set with respect to system (2.2). Therefore, in order to make the model biologically meaningful, here and hereafter, we focus our investigation on the dynamical behavior of system (2.2) on Ω. 3. Dynamics of system (2.2) In this section, we carry out detailed mathematical analysis on dynamics of system (2.2) by considering all possible nonnegative equilibria, their existence and global stability. 3.1. Dynamics of disease-free equilibrium P0. Let Λ− β1S1I1 − (d1 + ε)S1 = 0, β1S1I1 − (d1 + α+ γ)I1 = 0, εS1 + γI1 − β2S2I2 − d2S2 + µR2 = 0, β2S2I2 − (d2 + ν)I2 = 0, νI2 − (d2 + µ)R2 = 0. (3.1) Then (3.1) ensures that system (2.2) always possesses a disease-free equilibrium (DFE) P0 unconditionally and P0 = (S0 1 , I 0 1 , S 0 2 , I 0 2 , R 0 2) = ( Λ d1 + ε , 0, εΛ d2(d1 + ε) , 0, 0 ) . Write <0 be the basic reproduction number of system (2.2) [5, 6], according to [16, 17, 28], we have <0 = max{<01,<02}, where <01 = β1S 0 1 d1 + α+ γ = β1Λ (d1 + ε)(d1 + α+ γ) , <02 = β2S 0 2 d2 + ν = εβ2Λ d2(d1 + ε)(d2 + ν) . Theorem 3.1. Consider system (2.2), the disease-free equilibrium P0 is globally asymptotically stable if <0 < 1 and is unstable if <0 > 1. 6 Y. LONG, Y. CHEN EJDE-2021/45 Proof. Define a scalar function V0 ∈ C1(R5 +,R) as V0 = I1 + I2. Using (2.5), the derivative of V0 along the solutions of system (2.2) is dV0 dt = dI1 dt + dI2 dt =(β1S1 − (d1 + α+ γ))I1 + (β2S2 − (d2 + ν))I2 ≤ ( β1Λ d1 + ε − (d1 + α+ γ) ) I1 + ( εβ2Λ d2(d1 + ε) − (d2 + ν) ) I2 =(d1 + α+ γ)(<01 − 1)I1 + (d2 + ν)(<02 − 1)I2. Thus <0 = max{<01,<02} < 1 guarantees dV0 dt ≤ 0. Moreover, dV0 dt = 0 holds if and only if I1 = I2 = 0, which means that Ω = {P0} is the largest invariant set in {(S1, I1, S2, I2, R2) ∈ Ω|dV0 dt = 0}. Then the global stability of P0 follows immediately from the LaSalle’s invariance principle [23]. Owing to the expression of system (2.2), the Jacobian matrix of system (2.2) at the equilibrium P0 is in the form J(P0) =  −(d1 + ε) −β1S 0 1 0 0 0 0 β1S 0 1 − (d1 + α+ γ) 0 0 0 ε γ −d2 −β2S 0 2 µ 0 0 0 β2S 0 2 − (d2 + ν) 0 0 0 0 ν −(d2 + µ)  , then the corresponding characteristic equation is (λ+ d2)(λ+ d1 + ε)(λ+ d2 + µ)(λ+ (d1 + α+ γ)(<01 − 1)) × (λ+ (d2 + ν)(<02 − 1)) = 0. (3.2) Clearly, (3.2) implies that the eigenvalues of J(P0) are λ1 = −d2 < 0, λ2 = −(d1 + ε) < 0, λ3 = −(d2 + µ) < 0, λ4 = −(d1 + α+ γ)(<01 − 1), λ5 = −(d2 + ν)(<02 − 1). Suppose <0 = max{<01,<02} < 1, which implies λ4 < 0 and λ5 < 0. Hence, all eigenvalues λis (i = 1, 2, 3, 4, 5) are negative. Whereas, if <0 = max{<01,<02} > 1, then we have at least either λ4 > 0 or λ5 > 0. Therefore, J(P0) has at least one positive eigenvalue. As a result, the DFE P0 is unstable. � 3.2. Dynamics of the boundary equilibria P01 and P02. In addition to the DFE P0, (3.1) indicates that system (2.2) admits other equilibria. In this subsec- tion, we devote to studying the existence and stability of boundary equilibria P01 and P02 in the form of (S01 1 , 0, S01 2 , I01 2 , R01 2 ) (only the adult pigs are infected) and (S02 1 , I02 1 , S02 2 , 0, 0) (only the piglets are infected), respectively. Theorem 3.2. If <02 > 1, then system (2.2) has a boundary equilibrium P01 with P01 = ( Λ d1 + ε , 0, d2 + ν β2 , (d2 + µ)(d2 + ν)(<02 − 1) β2(d2 + µ+ ν) , ν(d2 + ν)(<02 − 1) β2(d2 + µ+ ν) ) . Further, if <01 < 1 then P01 is globally asymptotically stable and if <01 > 1 then P01 is unstable. EJDE-2021/45 MODELING PORCINE PSEUDORABIES 7 Proof. Recall (3.1), direct computation shows that system (2.2) just possesses boundary equilibria in the forms of P01 and P02. In the following, we start with seeking the boundary equilibrium P01. Let I01 1 = 0, then Λ− (d1 + ε)S01 1 = 0, εS01 1 − β2S 01 2 I01 2 − d2S 01 2 + µR01 2 = 0, β2S 01 2 I01 2 − (d2 + ν)I01 2 = 0, νI01 2 − (d2 + µ)R01 2 = 0. (3.3) By (3.3), it is straightforward to find that S01 1 = Λ d1 + ε > 0, S01 2 = d2 + ν β2 > 0, I01 2 = (d2 + µ)(d2 + ν)(<02 − 1) β2(d2 + µ+ ν) , R01 2 = νI01 2 d2 + µ , I01 2 > 0 induces R01 2 > 0. Hence, <02 > 1 ensures both I01 2 > 0 and R01 2 > 0. Therefore, system (2.2) has a boundary equilibrium P01 when <02 > 1. Thanks to [13, 31] and by using (3.3), system (2.2) is transformed into the following equivalent system dN1 dt = −d1(N1 −N01 1 )− ε(N1 −N01 1 )− (α+ γ − ε)I1, dI1 dt = β1(N1 −N01 1 − I1)I1 − (d1 + α+ γ)I1 + β1N 01 1 I1, dN2 dt = ε(N1 −N01 1 )− εI1 + γI1 − d2(N2 −N01 2 ), dI2 dt = β2((N2 −N01 2 )− (I2 − I01 2 )− (R2 −R01 2 ))I2, dR2 dt = ν(I2 − I01 2 )− (d2 + µ)(R2 −R01 2 ), (3.4) with N1 = S1 + I1, N01 1 = S01 1 , N2 = S2 + I2 +R2 and N01 2 = S01 2 + I01 2 +R01 2 . To obtain the global asymptotic stability of system (2.2) around P01, motivated by [19], we construct a Lyapunov function as V01 = α β1 I1 + 1 2 (N1 −N01 1 +N2 −N01 2 )2 + d1 + d2 2ε (N2 −N01 2 )2. Then the derivative of V01 along the solutions of System (3.4) is dV01 dt = α β1 dI1 dt + (N1 −N01 1 +N2 −N01 2 ) (dN1 dt + dN2 dt ) + d1 + d2 ε (N2 −N01 2 ) dN2 dt = (N1 −N01 1 +N2 −N01 2 ) ( − d1(N1 −N01 1 )− d2(N2 −N01 2 )− αI1 ) + d1 + d2 ε (N2 −N01 2 ) ( ε(N1 −N01 1 )− εI1 + γI1 − d2(N2 −N01 2 ) ) + α β1 ( β1(N1 −N01 1 − I1)I1 − (d1 + α+ γ)I1 + β1N 01 1 I1 ) = −d1(N1 −N01 1 )2 − d2(N1 −N01 1 )(N2 −N01 2 )− αI1(N1 −N01 1 ) 8 Y. LONG, Y. CHEN EJDE-2021/45 − d1(N1 −N01 1 )(N2 −N01 2 )− d2(N2 −N01 2 )2 − αI1(N2 −N01 2 ) + (d1 + d2)(N1 −N01 1 )(N2 −N01 2 )− (d1 + d2)I1(N2 −N01 2 ) + γ(d1 + d2) ε I1(N2 −N01 2 )− d2(d1 + d2) ε (N2 −N01 2 )2 − αI2 1 + αI1(N1 −N01 1 ) + α β1 (β1N 01 1 − (d1 + α+ γ))I1 = −αI2 1 − d1(N1 −N01 1 )2 − d2(ε+ d1 + d2) ε (N2 −N01 2 )2 − ε(α+ d1 + d2)− γ(d1 + d2) ε I1(N2 −N01 2 )− α β1 (d1 + α+ γ − β1Λ d1 + ε )I1 = −αI2 1 − d1(N1 −N01 1 )2 − d2(ε+ d1 + d2) ε (N2 −N01 2 )2 − ε(α+ d1 + d2)− γ(d1 + d2) ε I1(N2 −N01 2 )− α β1 (d1 + α+ γ)(1−<01)I1 ≤ −αI2 1 − d1(N1 −N01 1 )2 − d2(ε+ d1 + d2) ε (N2 −N01 2 )2 − ε(α+ d1 + d2)− γ(d1 + d2) ε I1(N2 −N01 2 ). (3.5) This last inequality is true because <01 < 1. For convenience, let Î1 = I1, N̂1 = N1 −N01 1 , N̂2 = N2 −N01 2 and rewrite (3.5) as dV01 dt ≤ −αÎ1 2 − d1N̂1 2 − d2(ε+ d1 + d2) ε N̂2 2 − ε(α+ d1 + d2)− γ(d1 + d2) ε Î1N̂2 = −(Î1, N̂1, N̂2)A(Î1, N̂1, N̂2)T , where A = A11 A12 A13 A21 A22 A23 A31 A32 A33  =  α 0 A31 0 d1 0 A13 0 A33  , (3.6) is a real symmetric matrix with A33 = d2(ε+ d1 + d2) ε > 0, A13 = A31 = ε(α+ d1 + d2)− γ(d1 + d2) 2ε . Next we show that A is positive definite. By (3.6), A11 = α > 0, A11A22 −A21A12 = αd1 > 0. (3.7) It is then left to verify that detA > 0. Note that detA =d1(αA33 −A2 13) = αd1d2(ε+ d1 + d2) ε − d1 (ε(α+ d1 + d2)− γ(d1 + d2) 2ε )2 = αd1d2(ε+ d1 + d2) ε − d1ε 2(α+ d1 + d2)2 4ε2 + 2d1εγ(α+ d1 + d2)(d1 + d2) 4ε2 − d1γ 2(d1 + d2)2 4ε2 > d1 4ε2 ( 4εαd2(ε+ d1 + d2) + 2εγ(α+ d1 + d2)(d1 + d2)− 2ε2(α+ d1 + d2)2 ) EJDE-2021/45 MODELING PORCINE PSEUDORABIES 9 = d1 2ε ( 2αd2(ε+ d1 + d2) + γ(α+ d1 + d2)(d1 + d2)− ε(α+ d1 + d2)2 ) . Let ξ = min{ε, α}. Owing to (2.1), there holds ε ≤ min{α+ γ, d2 − d1}, then detA ≥d1(d1 + d2 + ξ) (2αd2 + γ(d1 + d2)− ε(α+ d1 + d2)) 2ε ≥d1(d1 + d2 + ξ)(2αd2 + (ε− α)(d1 + d2)− ε(α+ d1 + d2)) 2ε = αd1(d1 + d2 + ξ)(d2 − d1 − ε) 2ε > 0. (3.8) Then (3.7) and (3.8) yield A is a positive definite matrix. Therefore, dV01 dt ≤ 0 and dV01 dt = 0 if and only if Î1 = 0, N̂1 = 0 and N̂2 = 0, that is, S1 = S01 1 , I1 = 0, S2 = S01 2 , I2 = I01 2 and R2 = R01 2 . Immediately, the LaSalle’s invariance principle [23] ensures system (2.2) is globally asymptotically stable around P01. Next, we show the instability of system (2.2) around P01. Consider the Jacobian matrix J(P01) of system (2.2) at the equilibrium P01, we have J(P01) = −(d1 + ε) −β1S 01 1 0 0 0 0 β1S 01 1 − (d1 + α+ γ) 0 0 0 ε γ −β2I 01 2 − d2 −β2S 01 2 µ 0 0 β2I 01 2 β2S 01 2 − (d2 + ν) 0 0 0 0 ν −(d2 + µ)  . We write m1 =d2 + µ+ β2I 01 2 , m2 =(d2 + µ+ ν)β2I 01 2 , n1 =d1 + ε+ (d1 + α+ γ)(1−<01), n2 =(d1 + ε)(d1 + α+ γ)(1−<01). (3.9) Evidently, combine the positiveness of I01 2 with <01 > 1, (3.9) gives m1 > 0, m2 > 0, n1 < 0, n2 < 0. (3.10) On the other side, the associated characteristic equation of the Jacobian matrix J(P01) of system (2.2) at the equilibrium P01 is (σ + d2)(σ2 +m1σ +m2)(σ2 + n1σ + n2) = 0. (3.11) We denote σis (i = 1, 2, 3, 4, 5) be the roots of (3.11). Then σ1 = −d2, σ2 + σ3 = −m1, σ2σ3 = m2, σ4 + σ5 = −n1, σ4σ5 = n2. (3.12) Joint (3.10) with (3.12), it follows that σ1 < 0, σ2 < 0, σ3 < 0 and either σ4 > 0 or σ5 > 0, which imply that J(P01) has a positive eigenvalue. The proof is complete. � 10 Y. LONG, Y. CHEN EJDE-2021/45 Let ε = ((ε− γ)d1 + εα)(β1Λ− (d1 + ε)(d1 + α+ γ)) γ(d1 + ε)β1Λ + (d1 + ε)(d1 + α+ γ)((ε− γ)d1 + εα) = ((ε− γ)d1 + εα)(d1 + α+ γ)(<01 − 1) γ(d1 + ε)β1Λ + (d1 + ε)(d1 + α+ γ)((ε− γ)d1 + εα) , (3.13) then ε > 0 for <01 > 1. Concerning the existence and the stability of boundary equilibrium in the form of P02, we have the following result. Theorem 3.3. If <01 > 1, then system (2.2) has a boundary equilibrium P02 = (S02 1 , I02 1 , S02 2 , 0, 0) with S02 1 = d1 + α+ γ β1 , I02 1 = (d1 + ε)(<01 − 1) β1 , S02 2 = ε(d1 + α+ γ) + γ(d1 + ε)(<01 − 1) β1d2 . Further, if <02 < 1 + ε, where ε is defined by (3.13), then P02 is globally asymptot- ically stable and if <02 > 1 + ε, then P02 is unstable. Proof. From (3.1), when I02 2 = R02 2 = 0, it is straightforward to find that S02 1 = d1 + α+ γ β1 , I02 1 = (d1 + ε)(<01 − 1) β1 , S02 2 = ε(d1 + α+ γ) + γ(d1 + ε)(<01 − 1) β1d2 . (3.14) Immediately, <01 > 1 ensures that S02 1 , I02 1 , S02 2 , given in (3.14), are all positive. Thus system (2.2) admits a boundary equilibrium P02 for <01 > 1. In the same manner as for (3.4), we turn system (2.2) into its equivalent form dN1 dt = −d1(N1 −N02 1 )− ε((N1 −N02 1 )− (I1 − I02 1 ))− (α+ γ)(I1 − I02 1 ), dI1 dt = β1I1((N1 −N02 1 )− (I1 − I02 1 )), dN2 dt = ε((N1 −N02 1 )− (I1 − I02 1 )) + γ(I1 − I02 1 )− d2(N2 −N02 2 ), dI2 dt = β2I2(N2 −N02 2 − I2 −R2) + β2N 02 2 I2 − (d2 + ν)I2, dR2 dt = νI2 − (d2 + µ)R2. (3.15) Here N1 = S1 + I1, N02 1 = S02 1 + I02 1 , N2 = S2 + I2 +R2 and N02 2 = S02 2 . We define a function V02 : R5 + → R as V02 = α β1 (I1 − I02 1 − I02 1 ln I1 I02 1 ) + 1 2 (N1 −N02 1 +N2 −N02 2 )2 + d1 + d2 2ε (N2 −N02 2 )2. (3.16) EJDE-2021/45 MODELING PORCINE PSEUDORABIES 11 Then the derivative of V02, defined by (3.16), along System (3.15) is dV02 dt = α β1 ( 1− I02 1 I1 )dI1 dt + (N1 −N02 1 +N2 −N02 2 ) (dN1 dt + dN2 dt ) + d1 + d2 ε (N2 −N02 2 ) dN2 dt =− α(I1 − I02 1 )2 − d1(N1 −N02 1 )2 − d2(ε+ d1 + d2) ε (N2 −N02 2 )2 − ε(α+ d1 + d2)− γ(d1 + d2) ε (I1 − I02 1 )(N2 −N02 2 ) =− (Ĩ1, Ñ1, Ñ2)A(Ĩ1, Ñ1, Ñ2)T , where A is a positive definite matrix defined by (3.6) and Ĩ1 = I1 − I02 1 , Ñ1 = N1 −N02 1 , Ñ2 = N2 −N02 2 . Thus dV02 dt ≤ 0 and dV02 dt = 0 is true if and only if Ĩ1 = 0, Ñ1 = 0 and Ñ2 = 0, which implies that S1 = S02 1 , I1 = I02 1 , S2 = S02 2 and I2 = R2 = 0. By the LaSalle’s Invariance Principle, P02 is globally asymptotically stable. Let θis (i = 1, 2, 3, 4, 5) be the eigenvalues of the Jacobian matrix J(P02) of system (2.2) around P02, then θis satisfy∣∣∣∣∣∣∣∣∣ θ + β1I 02 1 + d1 + ε β1S 02 1 0 0 0 −β1I021 θ + d1 + α+ γ − β1S 02 1 0 0 0 −ε −γ θ + d2 β2S 02 2 −µ 0 0 0 θ + d2 + ν − β2S 02 2 0 0 0 0 −ν θ + d2 + µ ∣∣∣∣∣∣∣∣∣ = (θ + d2)(θ + d2 + µ)(θ + d2 + ν − β2S 02 2 ) × ( (θ + β1I 02 1 + d1 + ε)(θ + d1 + α+ γ − β1S 02 1 ) + β1S 02 1 β1I 02 1 ) = (θ + d2)(θ + d2 + µ)(θ + d2 + ν − β2S 02 2 ) × ( θ2 + (d1 + ε)<01θ + (d1 + ε)(d1 + α+ γ)(<01 − 1) ) = 0. It follows that θ1 = −d2 < 0, θ2 = −(d2 + µ) < 0, θ3 = β2S 02 2 − (d2 + ν) (3.17) and θ4, θ5 solve the equation θ2 + (d1 + ε)<01θ + (d1 + ε)(d1 + α+ γ)(<01 − 1) = 0. (3.18) By Vieta’s theorem, (3.18) implies θ4 + θ5 = −(d1 + ε)<01, θ4θ5 = (d1 + ε)(d1 + α+ γ)(<01 − 1). From this and <01 > 1, we have θ4 + θ5 = −(d1 + ε)<01 < 0, θ4θ5 = (d1 + ε)(d1 + α+ γ)(<01 − 1) > 0, which guarantee that both θ4 and θ5 have negative real parts. Therefore, we show θ3 > 0 to obtain the instability of system (2.2) around P02. Let ζ = ε(d1 + α+ γ)<01 γ(d1 + ε)(<01 − 1) + ε(d1 + α+ γ) . 12 Y. LONG, Y. CHEN EJDE-2021/45 Notice that ε > γ and <01 = β1Λ (d1+ε)(d1+α+γ) > 1, then we have ζ = ε(d1 + α+ γ)β1Λ γ(d1 + ε)(β1Λ− (d1 + ε)(d1 + α+ γ)) + ε(d1 + ε)(d1 + α+ γ)2 = ε(d1 + α+ γ)β1Λ γ(d1 + ε)β1Λ + (d1 + ε)(d1 + α+ γ)(ε(d1 + α+ γ)− γ(d1 + ε)) =1 + ε(d1 + α+ γ)β1Λ γ(d1 + ε)β1Λ + (d1 + ε)(d1 + α+ γ)((ε− γ)d1 + εα) − 1 =1 + ε(d1 + α+ γ)β1Λ− γ(d1 + ε)β1Λ− (d1 + ε)(d1 + α+ γ)((ε− γ)d1 + εα) γ(d1 + ε)β1Λ + (d1 + ε)(d1 + α+ γ)((ε− γ)d1 + εα) =1 + ((ε− γ)d1 + εα)(β1Λ− (d1 + ε)(d1 + α+ γ)) γ(d1 + ε)β1Λ + (d1 + ε)(d1 + α+ γ)((ε− γ)d1 + εα) =1 + ε, where ε is defined by (3.13). Then from (3.17), we obtain θ3 =β2S 02 2 − (d2 + ν) = 1 β1d2 (εβ2(d1 + α+ γ) + γβ2(d1 + ε)(<01 − 1)− β1d2(d2 + ν)) = β2 β1d2 ( ε(d1 + α+ γ) + γ(d1 + ε)(<01 − 1)− β1d2(d2 + ν) β2 ) = β2 β1d2<02 ( ε(d1 + α+ γ) + γ(d1 + ε)(<01 − 1)− ε(d1 + α+ γ)<01 <02 ) = β2 β1d2<02 ( (ε(d1 + α+ γ) + γ(d1 + ε)(<01 − 1))<02 − ε(d1 + α+ γ)<01 ) = β2 β1d2<02 ( ε(d1 + α+ γ) + γ(d1 + ε)(<01 − 1) ) (<02 − ζ) = β2 β1d2<02 ( ε(d1 + α+ γ) + γ(d1 + ε)(<01 − 1) ) (<02 − 1− ε). (3.19) Subsequently, <02 > 1 + ε leads to θ3 > 0. The proof is complete. � 3.3. Dynamics of the disease-endemic equilibrium P ∗. In this subsection, we intend to study the existence and stability of the disease-endemic equilibrium P ∗ = (S∗1 , I ∗ 1 , S ∗ 2 , I ∗ 2 , R ∗ 2) with S∗1 > 0, I∗1 > 0, S∗2 > 0, I∗2 > 0 and R∗2 > 0, which implies that both the piglets and adult pigs are infected. We draw a conclusion as follows. Theorem 3.4. Let ε be given by (3.13). If <01 > 1 and <02 > 1 + ε, in addition to the DFE P0 and the boundary equilibria P01 and P02, system (2.2) also has a unique disease-endemic equilibrium P ∗ = (d1 + α+ γ β1 , (d1 + ε)(<01 − 1) β1 , d2 + ν β2 , (d2 + µ)(εS∗1 + γI∗1 − d2S ∗ 2 ) d2(d2 + µ+ ν) , νI∗2 d2 + µ ) . Moreover, P ∗ is globally asymptotically stable. EJDE-2021/45 MODELING PORCINE PSEUDORABIES 13 Proof. In view of our obtained existence results of P0, P01 and P02, here we study P ∗. Notice that S∗1 , I∗1 , S∗2 , I∗2 and R∗2 solve Λ− β1S ∗ 1I ∗ 1 − (d1 + ε)S∗1 = 0, β1S ∗ 1I ∗ 1 − (d1 + α+ γ)I∗1 = 0, εS∗1 + γI∗1 − β2S ∗ 2I ∗ 2 − d2S ∗ 2 + µR∗2 = 0, β2S ∗ 2I ∗ 2 − (d2 + ν)I∗2 = 0, νI∗2 − (d2 + µ)R∗2 = 0. (3.20) Direct calculation yields S∗1 = d1 + α+ γ β1 , I∗1 = (d1 + ε)(<01 − 1) β1 , S∗2 = d2 + ν β2 , I∗2 = (d2 + µ)(εS∗1 + γI∗1 − d2S ∗ 2 ) d2(d2 + µ+ ν) , R∗2 = νI∗2 d2 + µ . It is clear that S∗1 > 0, S∗2 > 0 and I∗1 > 0 for <01 > 1. To show the existence of P ∗, we are in the position to prove I∗2 > 0. Moreover, the expression of I∗2 indicates that I∗2 > 0 is equivalent to εS∗1 +γI∗1 −d2S ∗ 2 > 0. Similar to ((3.19), it follows that εS∗1 + γI∗1 − d2S ∗ 2 = ε(d1 + α+ γ) β1 + γ(d1 + ε)(<01 − 1) β1 − d2(d2 + ν) β2 = 1 β1 ( γ(d1 + ε)(<01 − 1) + ε(d1 + α+ γ)− β1d2(d2 + ν) β2 ) = 1 β1<02 ((ε(d1 + α+ γ) + γ(d1 + ε)(<01 − 1)) (<02 − 1− ε). Therefore, <01 > 1 and <02 > 1 + ε ensure εS∗1 + γI∗1 − d2S ∗ 2 > 0 and system (2.2) has a unique disease-endemic equilibrium P ∗. In the following, we show system (2.2) is globally asymptotically stable around P ∗. Here, the method of our proof is similar to that of Theorem 3.2. For complete- ness, we state the sketch of the remain part of the proof. First, transform system (2.2) into the equivalent system dN1 dt = −d1(N1 −N∗1 )− ε((N1 −N∗1 )− (I1 − I∗1 ))− (α+ γ)(I1 − I∗1 ), dI1 dt = β1I1((N1 −N∗1 )− (I1 − I∗1 )), dN2 dt = ε((N1 −N∗1 )− (I1 − I∗1 )) + γ(I1 − I∗1 )− d2(N2 −N∗2 ), dI2 dt = β2I2((N2 −N∗2 )− (I2 − I∗2 )− (R2 −R∗2)), dR2 dt = ν(I2 − I∗2 )− (d2 + µ)(R2 −R∗2), (3.21) where N1 −N∗1 = S1 − S∗1 + I1 − I∗1 and N2 −N∗2 = S2 − S∗2 + I2 − I∗2 +R2 −R∗2. Next, define a continuously differentiable function V : R5 + → R as V = α β1 (I1 − I∗1 − I∗1 ln I1 I∗1 ) + 1 2 (N1 −N∗1 +N2 −N∗2 )2 + d1 + d2 2ε (N2 −N∗2 )2. 14 Y. LONG, Y. CHEN EJDE-2021/45 Write Ĩ∗1 = I1 − I∗1 , Ñ∗1 = N1 − N∗1 and Ñ∗2 = N2 − N∗2 , then the derivative of V along the solution of System (3.21) is dV dt = −(Ĩ∗1 , Ñ ∗ 1 , Ñ ∗ 2 )A(Ĩ∗1 , Ñ ∗ 1 , Ñ ∗ 2 )T with A is a positive definite matrix given by (3.6), which means that dV dt ≤ 0. Moreover, dVdt = 0 implies the set {(S1, I1, S2, I2, R2) ∈ Ω : dVdt = 0} is the singleton P ∗. This completes the proof. � 4. Summary In this paper, to study the infection transmission of porcine pseudorabies, we have proposed a mathematical model (S1I1S2I2R2) with age structure. We have also derived conditions for the existence, global stability of disease-free equilibrium P0, boundary equilibria P01 and P02 (only either piglets or adult pigs are infected) and disease-endemic equilibrium P ∗ (both piglets and adult pigs are infected). Define a threshold <0 = max{<01,<02}, we have explored if <0 < 1, P0 is globally asymptotically stable and the disease will die out regardless its initial sizes. Conversely, the invasion is always possible if <0 > 1 [6, 8]. If <01 < 1 and <02 > 1 or <01 > 1 and <02 < 1 + ε, system (2.2) admits a globally asymptotically stable boundary equilibrium P01 or P02, which means that either piglets or adult pigs are always disease-endemic, respectively. 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Introduction 2. Model formulation 3. Dynamics of system (??) 3.1. Dynamics of disease-free equilibrium P0 3.2. Dynamics of the boundary equilibria P01 and P02 3.3. Dynamics of the disease-endemic equilibrium P* 4. Summary Acknowledgements References