GLOBAL STABILITY FOR INFECTIOUS DISEASE MODELS THAT INCLUDE IMMIGRATION OF INFECTED INDIVIDUALS AND DELAY IN THE INCIDENCE

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1 Electronic Journal of Differential Equations, Vol (2018, No. 64, pp N: URL: or GLOBAL TABLTY FOR NFECTOU DEAE MODEL THAT NCLUDE MMGRATON OF NFECTED NDVDUAL AND DELAY N THE NCDENCE CHELEA UGGENT, C. CONNELL MCCLUKEY Communicated by Ratnasinham hivaji Abstract. We bein with a detailed study of a delayed model of disease transmission with immiration into both classes. The incidence function allows for a nonlinear dependence on the infected population, includin mass action and saturatin incidence as special cases. Due to the immiration of infectives, there is no disease-free equilibrium and hence no basic reproduction number. We show there is a unique endemic equilibrium and that this equilibrium is lobally asymptotically stable for all parameter values. The results include vector-style delay and latency-style delay. Next, we show that previous lobal stability results for an E model and an V model that include immiration of infectives and non-linear incidence but not delay can be extended to systems with vector-style delay and latency-style delay. 1. ntroduction Many countries throuhout the world have hih numbers of both immirants and short-term visitors. For example, accordin to the 2006 Canadian Census [3], 1, 109, 980 people immirated to Canada, between January 1, 2001 and May 16, 2006 (the day of the census. At the time of the census, there were 6, 186, 950 immirants in Canada, comprisin 19.8% of the population. Additionally, there are millions of short-term visitors to Canada each year. With so many individuals enterin Canada (for example, it is inevitable that some individuals will already be infected with a iven disease at the time of arrival. This makes it desirable to consider models that account for the immiration of infected individuals. An immediate consequence of includin immiration of infected individuals is that the disease-free space is no loner positively invariant. Thus, there is no disease-free equilibrium, and hence no basic reproduction number. n this work, we study disease transmission models that include immiration of infected individuals. The models also include delayed effects. We consider both delay due to vector transmission and delay due to latency. For vector transmitted diseases, one can view the vector as providin a connection between a susceptible individual at time t and an infected individual from an 2010 Mathematics ubject Classification. 34K20, 92D30, 93D30. Key words and phrases. Global stability; Lyapunov function; epidemioloy; immiration. c 2018 Texas tate University. ubmitted eptember 11, Published March 7,

2 2 C. UGGENT, C. C. MCCLUKEY EJDE-2018/64 earlier time, say t τ. This puts a delay in the incidence term, as done by Cooke [4] and Takeuchi et al. [11], and many others since. For mass action models, this vector-style delay enerally results in a term of the form β(t(t τ bein subtracted from the susceptible equation and a correspondin term bein added to the equation for the first infected class. Another way that delay often arises in compartmental models for disease transmission relates to a latency period of duration τ between the time when an individual becomes infected and when they become infectious. For mass action models, it is then common to have a neative term β(t(t to represent the rate at which individuals infected at time t leave the susceptible class and a related positive term β(t τ(t τ to represent the rate at which individuals were infected at time t τ, who are now enterin the infectious class, for example. Another feature that we include in this work is that the force of infection may have a non-linear dependence on the size of the infectious population, so that the incidence function takes the form f(, for some reasonable function f, described in ection 2. This form of incidence includes mass action β and saturatin incidence β 1+m as special cases. We now provide a brief review of earlier work on models that include immiration of infected individuals. n [2], the authors study a non-delayed model. They show that there is a unique equilibrium, which is strictly positive. Usin the Bendixson-Dulac Criterion, they show that the equilibrium is lobally asymptotically stable. They also consider an R model, aain showin that the unique (positive equilibrium is lobally asymptotically stable. This time the proof was based on convertin the system of ordinary differential equations to a scalar interal equation. n each case they first work with mass action incidence β, before extendin their work to the case where β is a function of the total population size. n [8] an ODE model includin standard incidence was studied, findin a unique positive equilibrium. Usin compound matrix techniques, they prove the unique positive equilibrium is lobally asymptotically stable for a portion of the parameter space. n [9], the authors study an ODE model of HV infection with proportional mixin. Usin a Lyapunov function they show that the unique positive equilibrium is lobally asymptotically stable for a portion of the parameter space. The paper [12] presents an ODE model of vector transmission of malaria, accountin for the vector population (mosquitoes explicitly and includin immiration of infected hosts. Usin compound matrix techniques, they prove the unique positive equilibrium is lobally asymptotically stable. n [10] an E model was studied and in [5] an V model (where V stands for vaccinated was studied. For each of these models, which were systems of ordinary differential equations, it was shown that there was a lobally asymptotically stable equilibrium throuh the use of a Lyapunov function. n [1], the authors studied an VR model that includes diffusion within the reion of interest. The unique endemic equilibrium was shown to be lobally asymptotically stable throuh the use of a Lyapunov functional. n [7], an E model with continuous ae-in-class structure for the infected classes is studied. The unique endemic equilibrium was shown to be lobally asymptotically stable throuh the use of a Lyapunov functional.

3 EJDE-2018/64 TABLTY FOR NFECTOU DEAE MODEL 3 The current paper is oranized as follows. n ections 2-5, an model with immiration of infecteds, vector-style delay and non-linear incidence is studied in full detail. n ection 6, it is shown that the same results also apply to a correspondin model with latency-style delay. n ection 7, the lobal dynamics are resolved for two E models, one with vector-style delay and the other with latency-style delay. n ection 8, the lobal dynamics are resolved for two V models, one with vector-style delay and the other with latency-style delay. A discussion of the results is iven in ection An model with immiration and vector-style delay A human population is divided into two classes: susceptible and infectious. The sizes of the classes are denoted by and, respectively. The influxes of new individuals enterin the population into each roup (throuh birth and immiration are denoted by Λ > 0 for the susceptible class and by Λ > 0 for the infectious class. The incidence rate at which susceptibles become infectious is assumed to be linear in but may have a non-linear dependence on the size of the infectious population, takin the form f(, where f is a twice differentiable function satisfyin the followin hypotheses: (H1 f( 0 with equality if and only if = 0. (H2 f ( 0. (H3 f ( 0. These hypotheses were also used in [5, 10]. We assume that the disease is transmitted throuh a vector (such as a mosquito. Followin the work of Cooke [4], we assume that infected vectors become infectious after a fixed time τ, and that the number of infected humans is a ood proxy for the number of infected vectors. This implies that rate of new human infections at time t is (tf((t τ. We note that the form of incidence used here includes 1+m mass action β and saturatin incidence β as special cases. ndividuals leave the susceptible and infectious classes with per capita death rates of µ and µ, respectively. We assume that 0 < µ µ so that the death rate for those infected with the disease is at least as hih as for those who are not infected. The transfer diaram for the model is shown below. Λ µ f((t τ The correspondin system of differential equations is d dt = Λ f((t τ µ (2.1 d dt = Λ + f((t τ µ. The phase space for the system is Y = R 0 C([ τ, 0], R 0, where C([ τ, 0], R 0 is the space of continuous functions from [ τ, 0] to R 0. Λ µ

4 4 C. UGGENT, C. C. MCCLUKEY EJDE-2018/64 Given a function h : [T τ, T ] R 0, we define the associated function h T : [ τ, 0] R 0 by h T (θ = h(t + θ for all θ [ τ, 0]. The initial condition for (2.1 is ( (0, 0 ( = (, φ( Y. Note that we are usin the convention that if the arument for or is omitted, then the variable is to be evaluated at time t; if the variable is to be evaluated at any other time (such as t τ, for example then the arument will be iven explicitly. 3. Equilibria ince d dt =0 = Λ > 0, there is no disease-free equilibrium, and so there is no basic reproduction number. There is, however, an endemic equilbrium. The proof of the followin result is similar to the proof of [10, Proposition 3.1]. Proposition 3.1. There exists a unique equilibrium (, R 2 0. Furthermore,, > 0. Proof. ince d dt =0 = Λ > 0 and d dt =0 = Λ > 0, it follows that there are no equilibria for which either or is zero. Thus, we may restrict our search for equilibria to R 2 >0. olvin d dt + d dt = 0, ives = 1 µ (Λ + Λ µ, and therefore to have positive, we must have < max, where max = Λ +Λ µ. Rearranin d dt = 0 ives H( = 0, where H( = f( + µ µ Λ Λ + Λ µ. (3.1 Recallin from (H1 that f(0 = 0, we note that H(0 = µ µ Λ Λ +Λ > 0. Also, H( tends to neative infinity as increases to max. Thus, there exists at least one zero of H in the interval (0, max. Note that H ( = f ( 2µ2 µ Λ (Λ +Λ µ, which is neative for (0, 3 max. ince H is positive at 0 and concave down on (0, max, it follows that the zero of H in (0, max is unique. The result follows. 4. Local stability Proposition 4.1. The equilibrium (, is locally asymptotically stable. Proof. We bein by determinin the characteristic equation. Let s(t = (t and i(t = (t. Then for sufficiently small s and i, ds dt = d dt = Λ f((t τ µ = Λ ( + sf( + i(t τ µ ( + s Λ ( + s [f( + f ( i(t τ] µ ( + s = [Λ f( µ ] [ f ( i(t τ + sf( + µ s] sf ( i(t τ = [ f ( i(t τ + sf( + µ s] sf ( i(t τ f ( i(t τ sf( µ s, (4.1

5 EJDE-2018/64 TABLTY FOR NFECTOU DEAE MODEL 5 where the first approximation comes from takin a first order Taylor series of f and the second approximation comes from droppin the quadratic term s(tf ( i(t τ. A similar calculation yields di dt f ( i(t τ + sf( µ i. (4.2 We now use the exponential ansatz [ ] [ ] s(t = e λt s(0. i(t i(0 Fillin into (4.1 and (4.2, and cancelin e λt from each side, the equations can be re-written as [ ] [ ] [ ] s(0 0 (f( (M λ 2 2 =, M = + µ f ( e λτ i(0 0 f( f ( e λτ (4.3 µ and 2 2 is the 2 2 identity matrix. There are nontrivial solutions if and only if M λ 2 2 is sinular. Thus, the characteristic equation is where 0 = det (M λ 2 2 = (f( + µ + λ[(µ + λ f ( e λτ ] + f( f ( e λτ = λ 2 + (f( + µ + µ λ + (f( + µ µ (λ + µ f ( e λτ = λ 2 + p 1 λ + p 0 + (q 1 λ + q 0 e λτ, p 1 = f( + µ + µ q 1 = f ( p 0 = (f( + µ µ q 0 = µ f (. (4.4 To assist with the upcomin calculations, we first obtain a useful inequality. Usin (H1 (H3 and followin the proof of [10, Proposition 4.1] it can be shown that f ( f(. Then, usin the fact that d dt is zero at the equilibrium, we can replace f( in this inequality to write f ( µ Λ. t follows that µ > f (. (4.5 We now use a four step approach, part of which comes from the approach described in [13, ection 2], to show that all solutions λ of (4.4 have neative real part. tep A: Consider the matrix M for τ = 0. Note that the (2, 2-entry becomes f ( µ. By (4.5, this is neative and so, for τ = 0, M has the sin pattern [ ]. + Thus, trace(m < 0 and det(m > 0. t follows that the eienvalues of M both have neative real part and so (, is locally asymptotically stable for τ = 0. tep B: n this step, we study the possibility of eienvalues appearin at infinity. n particular, we show for τ 0, that there is an upper bound on the manitude of any eienvalues with positive real part, thereby precludin the possibility of eienvalues appearin at infinity.

6 6 C. UGGENT, C. C. MCCLUKEY EJDE-2018/64 Let K > max{ p 1 + q 1 + 1, p 0 + q 0 }. uppose λ is a solution of (4.4 with Re(λ > 0 and λ > K. Let Z = λ 2 + p 1 λ + p 0 + (q 1 λ + q 0 e λτ. Then, Z λ 2 p 1 λ + p 0 + (q 1 λ + q 0 e λτ λ 2 p 1 λ p 0 q 1 λ e λτ q 0 e λτ λ 2 p 1 λ p 0 q 1 λ q 0 = λ ( λ p 1 q 1 ( p 0 + q 0 > λ (K p 1 q 1 ( p 0 + q 0 > λ ( p 0 + q 0 > K ( p 0 + q 0 > 0. Thus, Z 0 and so λ is not a solution to the characteristic equation. This means that each solution of the characteristic equation either has neative real part or has a manitude of at most K. Combinin this with the result of tep A, it follows that the only possible loss of (local stability that can happen as τ increases from 0, is that eienvalues could cross from the left half-plane to the riht half-plane (but only with manitude less than K. n teps C and D, we rule out that possibility. tep C: For any τ 0, fillin λ = 0 into the expression on the riht-hand side of the characteristic equation (4.4 ives p 0 + q 0 = (f( + µ µ µ f (, which is positive by (4.5. Thus, λ = 0 is never a solution to (4.4. tep D: uppose τ > 0 and suppose λ = ωi (with ω 0 is a solution of (4.4. Replacin λ in (4.4 and separatin the real and imainary parts ives ω 2 p 0 = q 1 ω sin ωτ + q 0 cos ωτ, p 1 ω = q 0 sin ωτ q 1 ω cos ωτ. quarin both equations, addin the results and lettin z = ω 2 > 0, ives z 2 + ( p 2 1 2p 0 q 2 1 z + p 2 0 q 2 0 = 0. (4.6 We now show that the constant and linear terms on the left-hand side of (4.6 are positive. ince z is positive, it will then follow that (4.6 has no valid solutions, and so ωi must not be a characteristic root for the equilibrium. t follows from (4.5 that µ µ > µ f (. This means that p 0 is further from 0 than q 0 is, and so p 2 0 q0 2 > 0. That is, the constant term on the left-hand side of (4.6 is positive. Also, p 2 1 2p 0 q1 2 = ( f( µ + µ 2 ( f ( 2 > µ 2 ( f ( 2 > 0. Thus, the linear coefficient on the left-hand side of (4.6 is positive. Therefore, (4.6 has no positive roots and so ωi cannot be a solution to (4.4. Combinin the results of teps A, B, C, D, it follows that all roots of (4.4 have neative real part. Thus, the equilibrium (, is locally asymptotically stable for all τ Global stability Theorem 5.1. The equilibrium (, is lobally asymptotically stable on the set Y.

7 EJDE-2018/64 TABLTY FOR NFECTOU DEAE MODEL 7 Proof. Let (x = x 1 ln(x, V = ( + (, τ ( f((t σ W = 0 f( µ σ, U = V + f( W. (5.1 Note that d dt =0 = Λ > 0 and d dt =0 = Λ > 0 implies that (t, (t > 0 for all t > 0. Thus, we may assume that V, W and U are well-defined and finite for all t > τ. We bein by calculatin dv dt. Usin Λ = f( + µ and µ = Λ + f(, we have dv ( dt = 1 [Λ f((t τ µ ] + (1 [Λ + f((t τ µ ] = (1 [ f( f((t τ + µ ( ] where + (1 ( 2 = µ [ ( Λ + f( Λ + f((t τ + f( (1 (1 Λ ( 2 + f( = µ ( 2 Λ ( 2 ( 1 ( f((t τ f( + f( C, ] f((t τ f( f((t τ C = 2 + f( f((t τ f( ( f((t τ ( ( ( f((t τ = f( f(. (5.2 (5.3 (This last expression can be checked by usin the definition of to obtain the previous line. Also, dw = d τ ( f((t σ dt dt 0 f( µ σ τ d ( f((t σ = 0 dt f( µ σ τ d ( f((t σ (5.4 = 0 dσ f( µ σ ( f( ( f((t τ = f( f(. To find du dt, we combine (5.2, (5.3 and (5.4, to obtain du dt = µ ( 2 ( 2 Λ + f( µ, (5.5

8 8 C. UGGENT, C. C. MCCLUKEY EJDE-2018/64 where ( f( ( ( ( f((t τ D = f( f(. (5.6 ( By Proposition A.1 from [10], the hypotheses (H1 (H3 ensure that f( f( ( 0 for all > 0. Thus, ( ( f((t τ µ f( 0. Therefore, du dt 0, with equality only if (, = (,. Thus, by Lyapunov s Direct Method, the equilibrium is lobally asymptotically stable. 6. An model with immiration and latency-style delay t is noteworthy that the model studied in ections 2 to 5, which includes delay due to vector transmission, is equivalent to a model that includes delay due to latency. This can be seen by definin Y (t = (t + τ. Then, (2.1 becomes dy (t = Λ Y (tf((t µ Y (t dt (6.1 d(t = Λ + Y (t τf((t τ µ (t, dt which is a similar model, but with a latency-style delay. t follows that our results also apply to (6.1. Theorem 6.1. The equilibrium (, is locally and lobally asymptotically stable under the flow described by ( Two E models with immiration and delay n [10], a model includin susceptible, exposed and infectious classes, with immiration of infected individuals, but no delay, was studied. The transfer diaram for the model is shown below. Λ µ f( Λ E µ E E E n this model, Λ E ives the influx of individuals enterin the system into the exposed class, µ E is the per capita death rate of the exposed class and 1 γ is the averae time spent in the exposed class before movin to the infectious class; other parameters have the same meanin as in the -model introduced in ection 2. The system of differential equations for the model is d dt = Λ f( µ de dt = Λ E + f( (µ E + γe (7.1 d dt = Λ + γe µ, γe Λ µ

9 EJDE-2018/64 TABLTY FOR NFECTOU DEAE MODEL 9 with Λ E, Λ 0 and Λ E + Λ, Λ, µ, µ E, µ, γ > 0. Also, the force of infection f is assumed to satisfy the hypotheses (H1 (H3. n [10], it is shown that there is a unique equilibrium X = (, E, R 3 >0. Usin the Lyapunov function ( V = ( E + E E + f( ( γe, (7.2 it is shown that X is lobally asymptotically stable. n doin so, the authors of [10] obtain the inequality ( 2 (E E 2 D (7.1 V µ Λ E E f( ( 2 E γe Λ [ ( f( ( E f( ( E ] + Ef( + E. (7.3 Notation. For the remainder of this paper, we use notation similar to the previous equation, where D (7.1 V ives the derivative of V with respect to time as the aruments of V(, E, chane accordin to the differential equation (7.1. We now wish to consider delayed versions of (7.1. Consider vector-style delay in the transmission term: d dt = Λ f((t τ µ de dt = Λ E + f((t τ (µ E + γe (7.4 d dt = Λ + γe µ, and latency-style delay in the transmission term d dt = Λ f( µ de dt = Λ E + (t τf((t τ (µ E + γe (7.5 d dt = Λ + γe µ, where τ > 0 in each case. Note that for each of (7.4 and (7.5, the unique equilibrium is X = (, E,, the same as for the ordinary differential equation (7.1. Also note that by makin the substitution Y (t = (t+τ (and then chanin Y to, (7.5 can be shown to be equivalent to (7.4, similar to how (6.1 was shown to be equivalent to (2.1. Thus, we will focus on the stability of just (7.5. To do this, we first state a version of Theorem 5.1 from [6], which ives conditions under which a Lyapunov function V for an ordinary differential equation can be extended to a Lyapunov functional U for a related delay differential equation that includes latency-style delay. n the terminoloy of [6], we are addin delay to the transmission term q(x(t = (tf((t; we also have x j = E, A = 1 and L = E E. Then [6, Theorem 5.1] and its proof ive the followin result. Theorem 7.1. f ( q(x(t D (7.1 V + Aq(X q(x L 0,

10 10 C. UGGENT, C. C. MCCLUKEY EJDE-2018/64 then τ ( X(t σ U = V + Aq(X 0 X dσ is a Lyapunov functional for (7.5 satisfyin [ ( X(t D (7.5 U = D (7.1 V + Aq(X X L ( X(t τ X ] L. Fillin the expressions for q, A and L into ( the theorem conditions, we see that it is necessary to have D (7.1 V + f( f(e less than or equal to 0. Usin f( E (7.3 to replace D (7.1 V, we see that the condition is satisfied. Thus, Theorem 7.1 implies ( 2 (E E 2 D (7.5 U µ Λ E E f( ( 2 E γe Λ [ ( f( ( (t τf((t τe ( E ] + f( + E E. t follows from Lyapunov s Direct Method that the equilibrium X is lobally asymptotically stable under the flow described by (7.5. Due to the equivalence of (7.4 and (7.5, X is also lobally asymptotically stable under the flow described by (7.4. Theorem 7.2. The equilibrium X is lobally asymptotically stable under the flow described by (7.4 and also under the flow described by ( Two vaccination models with immiration and delay n [5], a model includin susceptible, vaccinated, infectious and recovered classes, with immiration of infected individuals, but no delay, was studied. The transfer diaram is: Λ µ (µ+γ µr Λ f( R δ V h( α γ 1V V Λ V µv Λ R n this model, Λ V and Λ R ive the influxes of individuals enterin the system into the vaccinated and recovered classes, µ is the per capita death rate for death that is not related to the disease, γ is the per capita disease-related death rate, α is the per capita vaccination rate, γ 1 is the per capita rate at which individuals in the vaccinated class receive permanent immunity, δ is the per capita recovery rate,

11 EJDE-2018/64 TABLTY FOR NFECTOU DEAE MODEL 11 and h( is the force of infection for vaccinated individuals. Other parameters have the same meanin as in the -model introduced in ection 2. The correspondin differential equations are: d dt = Λ f( (µ + α dv dt = Λ V + α V h( (µ + γ 1 V d dt = Λ + f( + V h( (µ + γ + δ dr dt = Λ R + γ 1 V + δ µr, (8.1 with Λ, Λ V, Λ, Λ R, µ > 0 and α, δ, γ, γ 1 > 0. Also, the functions f and h are assumed to satisfy the hypotheses (H1 (H3 and to satisfy h( f( for all 0. The variable R does not appear in the first three equations and so it is sufficient to only study those three equations. n [5], it is shown that there is a unique equilibrium Z = (, V, R 3 >0. Usin the Lyapunov function ( ( V ( V = + V V +, (8.2 it is shown that Z is lobally asymptotically stable. n doin so, the authors of [5] obtain the inequality ( D (8.1 V (µ + α [ ( f( ( µ ( 2 Λ ( f( ] + f( ( V ( V (µ + γ 1 V V α V ( V h( V h( V h(. (8.3 We now wish to consider delayed versions of (8.1. For the sake of brevity, we omit the equation for dr dt. Consider vector-style delay in the transmission terms: d dt = Λ f((t τ (µ + α dv dt = Λ V + α V h((t τ (µ + γ 1 V d dt = Λ + f((t τ + V h((t τ (µ + γ + δ. and latency-style delay in the transmission terms: d dt = Λ f( (µ + α dv dt = Λ V + α V h( (µ + γ 1 V d dt = Λ + (t τf((t τ + V (t τh((t τ (µ + γ + δ. (8.4 (8.5

12 12 C. UGGENT, C. C. MCCLUKEY EJDE-2018/64 where τ > 0 in each case. We have also assumed that the delay associated with susceptible individuals is the same as the delay associated with vaccinated individuals. Recall that for (7.4 and (7.5, the unique equilibrium is Z = (, E,, the same as for the ordinary differential equation (7.1. By makin the substitutions Y (t = (t + τ and Y V (t = V (t + τ, it can be shown that (8.5 is equivalent to (8.4. Thus, we will focus only on the stability of (8.5. By usin [6, Theorem 5.1] twice (or Theorem 7.1 once, the Lyapunov function V iven in (8.2 that resolves the lobal stability of Z for (8.1 can be extended to the Lyapunov functional U iven by τ ( (t σf((t σ U = V + f( dσ + V h( f( ( V (t σh((t σ V h( 0 τ 0 dσ. By followin the approach used in ection 7, and described in detail in [6], we obtain [ ( f( D (8.5 U =D (8.1 V + f( ( (t τf((t τ ] f( f( [ ( V h( + V f( ( V (t τh((t τ ] V h( V h( ( (µ + α ( µ ( 2 Λ [ ( f( ( (t τf((t τ ] + f( ( V ( V (µ + γ 1 V V α V ( V (t τh((t τ V h( V h( 0 with equality if and only if (, V, = (, V,. By Lyapunov s Direct Method the equilibrium is lobally asymptotically stable under the flow described by (8.5. Due to the equivalence of (8.4 and (8.5, it follows that the equilibrium Z is also lobally asymptotically stable under the flow described by (8.4. Theorem 8.1. The equilibrium Z is lobally asymptotically stable under the flow described by (8.4 and also under the flow described by ( Discussion We have studied, E and V models of disease spread that include immiration of infected individuals and each of delay due to vector transmission delay due to a period of latency. Due to the immiration of infecteds, there is no disease-free equilibrium for any of these models. f one of the systems were somehow in a disease-free state, then infected individuals would enter the population throuh immiration and so the system would no loner be in a disease-free state. inificantly, since there is no disease-free equilibrium, there is no basic reproduction number.

13 EJDE-2018/64 TABLTY FOR NFECTOU DEAE MODEL 13 For the model with vector-style delay, we provide a detailed analysis, showin that for all parameter values there is a unique positive equilibrium and that it is both locally and lobally asymptotically stable. Thus, the level of disease in the population will asymptotically approach the equilibrium level. We then show that this vector-style model is equivalent to an model with latency-style delay so that our results also apply to this new model. Previous works [5, 10] have studied E and V models with immiration of infected individuals, but without delay. Those works used Lyapunov functions to show that the unique positive equilibrium was lobally asymptotically stable. We show that these Lyapunov functions can be extended to Lyapunov functionals, showin that the equilibrium is still lobally asymptotically stable if the system is modified to include either vector-style or latency-style delay. t is apparent that in order to eliminate disease in a connected world, it is necessary either to screen travelers perfectly for infection (so that there is no immiration of infecteds or to treat disease elimination as a lobal problem. Due to the hih level of interconnectedness of today s world, the lobal approach to elimination seems more likely to be successful. Acknowledments. This article was prepared while the first author was a student at Wilfrid Laurier University. C. C. McCluskey is supported by an NERC Discovery Grant. References [1]. Abdelmalek,. Bendoukha; Global asymptotic stability of a diffusive VR epidemic model with immiration of individuals. Elect. J. Diff. Eqns., 2016:1 14, [2] F. Brauer, P. van den Driessche; Models for transmission of disease with immiration of infectives. Math. Biosci., 171: , [3] tatistics Canada; Canadian Census 2006, Population by immirant status and period of immiration. Available from: url = [4] K. L. Cooke; tability analysis for a vector disease model. Rocky Mount. J. Math., 9:31 42, [5]. Henshaw, C. C. McCluskey; Global stability of a vaccination model with immiration. Elect. J. Diff. Eqns., 2015(92:1 10, [6] C. C. McCluskey; Usin Lyapunov functions to construct Lyapunov functionals for delay differential equations. AM Journal on Applied Dynamical ystems, 14 (1 (2015, [7] C. C. McCluskey; Global stability for an E model of infectious disease with ae structure and immiration of infecteds. Math. Biosci. En., 13: , DO /mbe [8] C. C. McCluskey, P. van den Driessche; Global analysis of two tuberculosis models. J. Dynam. Differential Equations, 16(1: , [9] R. Naresh, A. Tripathi, D. harma; Modellin and analysis of the spread of aids epidemic with immiration of hiv infectives. Math. Comput. Modell., 49: , [10] R. P. idel, C. C. McCluskey; Global stability for an E model of infectious disease with immiration. Appl. Math. Comput., 243: , [11] Y. Takeuchi, W. Ma, E. Beretta; Global asymptotic properties of a delay R epidemic model with finite incubation times. Nonlinear Anal., 42: , [12] J. Tumwiine, J. Y. T. Muisha, L.. Luboobi; A host-vector model for malaria with infective immirants. J. Math. Anal. Appl., 361: , [13] R. Xu, Z. Ma; tability of a delayed R epidemic model with a nonlinear incidence rate. Chaos, olitons and Fractals, 41: , Chelsea Uenti Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada address: cuenti@uwo.ca

14 14 C. UGGENT, C. C. MCCLUKEY EJDE-2018/64 C. Connell McCluskey Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, Canada address: