A Reaction-Diffusion Lyme Disease Model with

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SIAM J. APPL. MATH. Vol. 73, No. 6, pp. 2077–2099

c 2013 Society for Industrial and Applied Mathematics

A REACTION-DIFFUSION LYME DISEASE MODEL WITH SEASONALITY∗ YUXIANG ZHANG† AND XIAO-QIANG ZHAO‡ Abstract. This paper is devoted to the study of a reaction-diffusion Lyme disease model with seasonality. In the case of a bounded habitat, we obtain a threshold result on the global stability of either disease-free or endemic periodic solution. In the case of an unbounded habitat, we establish the existence of the disease spreading speed and its coincidence with the minimal wave speed for time-periodic traveling wave solutions. We also estimate parameter values based on some published data and use them to study the Lyme disease transmission in Port Dove, Ontario. Our numerical simulations are consistent with the obtained analytic results. Key words. seasonality, basic reproduction ratio, periodic solutions, spreading speed, traveling waves AMS subject classifications. 35K57, 37C65, 92D30 DOI. 10.1137/120875454

1. Introduction. Lyme disease is a commonly reported tick-borne illness, which was named after Lyme, Connecticut, where the first outbreak in humans in North America was recognized in 1975. The disease is caused by the bacterium Borrelia burgdorferi, which is transmitted to humans through the bite of infected ticks. The ticks live for about two years with three feeding stages: larva, nymph, and adult. Larval and nymphal ticks primarily feed on mice, and adult ticks feed on deer. Larvae that obtain a blood meal drop off their host (mice) and then grow into nymphs. These nymphs seek out their host (mice) for their blood meal. If they succeed, the nymphs pass the spirochete to susceptible mice and mature into adults. Adults feed almost exclusively on deer and mate there. Female adults eventually drop off the deer, lay their eggs nearby, and die. Larvae hatch and acquire the spirochete when they attack an infected mouse for their blood meal. Another tick-to-mouse-to-tick infection cycle happens again. For more information about the infection of Lyme disease, we refer the reader to [3, 4, 13, 15, 16, 17] and references therein. In order to study the effect of a vector’s stage structure on the transmission dynamics and disease spreading velocity, Caraco et al. [4] proposed a reaction and diffusion model for Lyme disease in the northeast United States. The model treats population densities at locations x := (x1 , x2 ) in a continuous two-dimensional space Ω, and parameters for birth, death, infection, and developmental advancement are all positive constants. Recently, Zhao [24] studied the global dynamics of this spatial model for Lyme disease. Note that this model ignores the seasonal pattern in abundance and activities of different stages. As mentioned in [1], seasonal variations in temperature, rainfall, and resource availability are ubiquitous and can exert strong ∗ Received by the editors April 30, 2012; accepted for publication (in revised form) August 30, 2013; published electronically November 12, 2013. This research was supported in part by the NSERC of Canada and the MITACS of Canada. http://www.siam.org/journals/siap/73-6/87545.html † Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada ([email protected]). Current address: Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON, K1N 6N5, Canada ([email protected]). ‡ Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, A1C 5S7, Canada ([email protected]).

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YUXIANG ZHANG AND XIAO-QIANG ZHAO

pressures on population dynamics. For Lyme disease, the ticks develop slowly or become less active in colder temperatures (see [17]), and rainfall is also critically important for the development, survival, and activities of ticks (see [18]). According to the report from the Public Health Agency of Canada on Lyme disease cases in Ontario between 1999 and 2004 [5], most cases occurred in late spring and summer, when the young ticks are most active and people are outdoors more often. To take seasonal influences into account, we modify Caraco et al.’s model to a reaction and diffusion model in a periodic environment. Since tick development and activities are strongly affected by temperatures [15, 16, 17], we assume that the development rates of ticks and their activity rates (biting rates) are time-dependent. Another assumption is the self-regulation mechanism for the tick population, as discussed in [4]. We assume that the self-regulation process is mainly due to the carrying capacity of hosts and some density-dependent death terms. Let M (t, x) and m(t, x) be the densities of susceptible and pathogen-infected mice, L(t, x) be the density of questing larvae, V (t, x) and v(t, x) be the densities of larvae infesting susceptible and pathogen-infected mice, N (t, x) and n(t, x) be the densities of susceptible and infectious questing nymphs, and A(t, x) and a(t, x) be the densities of uninfected and pathogen-infected adult ticks, at time t and location x. In Figure 1.1, we use two schematic diagrams to illustrate the tick-mouse cycle of infection. Accordingly, our spatial model for Lyme disease is governed by the following periodic reaction-diffusion system: M +m ∂M = DM ΔM + rM (M + m) 1 − − μM M − α2 (t)βM n, ∂t KM ∂m = DM Δm + α2 (t)βM n − μM m, ∂t ∂L = r(t)(A + a) − μL L − α1 (t)L(M + m), ∂t ∂V = DM ΔV + α1 (t)M L − V (σ(t) + μV ) − δV (V + v)V, ∂t ∂v = DM Δv + α1 (t)mL − v(σ(t) + μV ) − δV (V + v)v, (1.1) ∂t ∂N = σ(t)[V + (1 − βT )v] − N [γ + α2 (t)(M + m) + μN ], ∂t ∂n = βT σ(t)v − n[γ + α2 (t)(M + m) + μN ], ∂t ∂A = DH ΔA + α2 (t)N [M + (1 − βT )m] − μA A − δA (A + a)A, ∂t ∂a = DH Δa + α2 (t)[(M + m)n + βT mN ] − μA a − δA (A + a)a, ∂t 2

2

∂ ∂ 2 All constant parameters where Δ = ∂x 2 + ∂x2 is the Laplacian operator on R . 1 2 are positive, and r(t), α1 (t), σ(t), and α2 (t) are nonnegative ω-periodic functions. The biological interpretations for the parameters are listed in Table 1.1. We further assume that rM > μM , β ∈ (0, 1), and βT ∈ (0, 1). It is worth mentioning that the term γn(t, x) represents the local risk of Lyme disease to humans under the assumption that nymphs biting humans do not feed long enough to mature into adults, and the diffusion coefficient DH models dispersal of adult ticks while they infest deer. However, deer do not disperse the pathogen and cannot be infected. This also explains why we have ignored the dynamics for humans and deer in model (1.1).



A LYME DISEASE MODEL WITH SEASONALITY

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(a) The schematic diagram for mice.

(b) The schematic diagram for ticks. Fig. 1.1. The illustrative diagram for the tick-mouse cycle of infection.

Our main purpose in this paper is to study the global dynamics of system (1.1) in both bounded and unbounded spatial domains. In section 2, we obtain a threshold result on the global dynamics of (1.1) in a bounded domain Ω. In section 3, we establish the existence of the spreading speed of the disease and its coincidence with the minimal wave speed for periodic traveling waves of system (1.1) when Ω is unbounded. In section 4, we present a case study on the transmission of Lyme disease in Port Dove, Ontario. A short discussion section completes the paper. 2. Threshold dynamics in a bounded domain. In this section, we consider system (1.1) in a bounded domain Ω ⊂ R2 with smooth boundary ∂Ω. We assume that all populations remain confined to the domain Ω for all time, and hence the model system (1.1) is subject to the Neumann boundary conditions ∂m ∂V ∂v ∂A ∂a ∂M = = = = = = 0, ∂ν ∂ν ∂ν ∂ν ∂ν ∂ν


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Table 1.1 The biological interpretations for parameters in model (1.1). rM KM μM DM DH μL μV μN μA δV δA β βT γ r(t) σ(t) α1 (t) α2 (t)

Individual birth rate of mice. Carrying capacity for mice. Mortality rate per mouse. Diffusion coefficients for mice. Diffusion coefficients for deer. Mortality rate per questing tick larva. Mortality rate per feeding tick larva. Mortality rate per questing tick nymph. Mortality rate per adult tick. Self-regulation coefficient for tick larva. Self-regulation coefficient for adult tick. Susceptibility to infection in mice. Susceptibility to infection in ticks. Biting rate per nymph to humans. Individual birth rate of tick at time t. Individual development rate of nymph at time t. Individual biting rate of larva to mice at time t. Individual biting rate of nymph to mice at time t.

∂ where ∂ν represents the differentiation along the outward normal ν to ∂Ω. For the convenience of mathematical analysis, we make a change of variables M = M + m, V = V + v, N = N + n, A = A + a for system (1.1). It then follows that system (1.1) is equivalent to the following: ∂M M = DM ΔM + rM M 1 − − μM M, ∂t KM ∂L = r(t)A − μL L − α1 (t)LM, ∂t ∂V = DM ΔV + α1 (t)ML − V(σ(t) + μV ) − δV V 2 , ∂t ∂N = σ(t)V − N [γ + α2 (t)M + μN ], ∂t ∂A = DH ΔA + α2 (t)N M − μA A − δA A2 , (2.1) ∂t ∂m = DM Δm + α2 (t)β(M − m)n − μM m, ∂t ∂v = DM Δv + α1 (t)mL − v(σ(t) + μV ) − δV Vv, ∂t ∂n = βT σ(t)v − n[γ + α2 (t)M + μN ], ∂t ∂a = DH Δa + α2 (t)[Mn + βT m(N − n)] − μA a − δA Aa. ∂t

Note that the first five equations in (2.1) do not depend on the last four equations. In addition, by the condition rM > μM and a standard convergence result on the logistic type reaction-diffusion equation (see, e.g., Theorem 3.1.5 and the proof of Theorem ¯ R2+ )\{0}, we have 3.1.6 in [23]), it follows that for any M(0, ·) ∈ C(Ω, μM lim M(t, x) = KM 1 − := Q t→∞ rM ¯ Thus, we first analyze the global dynamics of the uniformly for x = (x1 , x2 ) ∈ Ω.



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following limiting system:

(2.2)

∂L = r(t)A − L[μL + α1 (t)Q], ∂t ∂V = DM ΔV + α1 (t)QL − V(σ(t) + μV ) − δV V 2 , ∂t ∂N = σ(t)V − N [γ + α2 (t)Q + μN ], ∂t ∂A = DH ΔA + α2 (t)N Q − μA A − δA A2 . ∂t

¯ R4+ ), and Γ(t, x, y) be the Green function associated ¯ R4 ), X + = C(Ω, Let X = C(Ω, with the Laplacian operator Δ and the Neumann boundary condition, and define [T1 (t, s)φ1 ](x) = e− [T2 (t, s)φ2 ](x) = e− [T3 (t, s)φ3 ](x) = e

t s

t s

(μL +α1 (τ )Q)dτ (σ(τ )+μV )dτ

φ1 (x), Γ(DM (t − s), x, y)φ2 (y)dy,

Ω − s (γ+α2 (τ )Q+μN )dτ t

[T4 (t, s)φ4 ](x) = e−μA (t−s)

Ω

φ3 (x),

Γ(DH (t − s), x, y)φ4 (y)dy.

Then system (2.2) can be written as the following integral equations: t T1 (t, s)r(s)A(s, x)ds, L(t, x) = T1 (t, 0)L(0, x) + 0 t T2 (t, s)(α1 (s)QL(s, x) − δV V 2 (s, x))ds, V(t, x) = T2 (t, 0)V(0, x) + 0 t (2.3) N (t, x) = T3 (t, 0)N (0, x) + T3 (t, s)σ(s)V(s, x)ds, 0 t T4 (t, s)(α2 (s)QN (s, x) − δA A2 (s, x))ds. A(t, x) = T4 (t, 0)A(0, x) + 0

By the theory of abstract semilinear integral equations in [14], it follows that for any φ ∈ X + , system (2.3) admits a unique nonnegative and noncontinuable solution u(t, x, φ) := (L(t, x, φ), V(t, x, φ), N (t, x, φ), A(t, x, φ)) on [0, σφ ) with u(0, ·, φ) = φ. Moreover, it follows from [14, Proposition 1 and Remark 1.4] that system (2.3) admits the comparison principle. Note that the spatially homogeneous system of (2.2) is the following periodic system of ordinary differential equations (ODEs):

(2.4)

dL = r(t)A − L[μL + α1 (t)Q], dt dV = α1 (t)QL − V(σ(t) + μV ) − δV V 2 , dt dN = σ(t)V − N [γ + α2 (t)Q + μN ], dt dA = α2 (t)N Q − μA A − δA A2 , dt


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and every nonnegative solution (L(t), V(t), N (t), A(t)) of (2.4) satisfies d (L(t)+V(t)+N (t)+A(t)) = (r(t)−μA −δA A)A−μL L−μV V−δV V 2 −N (γ+μN ) < 0 dt provided A > (max0≤t≤ω r(t) − μA )/δA . It then follows that solutions of (2.4) are ultimately bounded in R4+ and exist for all t ∈ [0, ∞). Linearizing system (2.4) at (0, 0, 0, 0), we get the following linear cooperative system: dL = r(t)A − L[μL + α1 (t)Q], dt dV = α1 (t)QL − V(σ(t) + μV ), dt dN = σ(t)V − N [γ + α2 (t)Q + μN ], (2.5) dt dA = α2 (t)N Q − μA A. dt Let r1 be the principal Floquet multiplier of system (2.5), that is, the spectral radius of the Poincaré map associated with system (2.5). Then we have the following result. Lemma 2.1. The following statements are valid: (i) If r1 ≤ 1, then (0,0,0,0) is globally asymptotically stable for (2.4) in R4+ . (ii) If r1 > 1, then (2.4) admits a unique positive ω-periodic solution u∗ (t) := (L∗ (t), V ∗ (t), N ∗ (t), A∗ (t)), which is globally asymptotically stable for (2.4) in R4+ \{0}. Proof. Define f = (f1 , f2 , f3 , f4 ) : R4 → R4 by f1 (x1 , x2 , x3 , x4 ) = r(t)x4 − x1 [μL + α1 (t)Q], f2 (x1 , x2 , x3 , x4 ) = α1 (t)Qx1 − x2 (σ(t) + μV ) − δV x22 , f3 (x1 , x2 , x3 , x4 ) = σ(t)x2 − x3 [γ + α2 (t)Q + μN ], f4 (x1 , x2 , x3 , x4 ) = α2 (t)x3 Q − μA x4 − δA x24 . It is easy to verify that fi ≥ 0 for any x = (x1 , x2 , x3 , x4 ) ∈ R4+ with xi = 0, and the Jacobian matrix of f (x1 , x2 , x3 , x4 ) is cooperative for any x ∈ R4+ . Thus, the solution semiflow {Πt }t≥0 determined by (2.4) is monotone in the sense that Πt (x) ≥ Πt (y) provided x ≥ y in R4+ . Next, we show that Πt is strongly monotone for all t ≥ 3ω, that is, Πt (x) Πt (y) whenever t ≥ 3ω and x > y. Let x(t) := (x1 (t), x2 (t), x3 (t), x4 (t)) = Πt (x0 ), y(t) := (y1 (t), y2 (t), y3 (t), y4 (t)) = Πt (y0 ), and z(t) := (z1 (t), z2 (t), z3 (t), z4 (t)) = x(t) − y(t). Then z(t) satisfies the following equations: dz1 = r(t)z4 − z1 [μL + α1 (t)Q], dt dz2 = α1 (t)Qz1 − z2 (σ(t) + μV + δV (x2 (t) + y2 (t))), dt dz3 = σ(t)z2 − z3 [γ + α2 (t)Q + μN ], (2.6) dt dz4 = α2 (t)Qz3 − z4 (μA + δA (x4 (t) + y4 (t))). dt Now it suffices to prove z(t) 0 for all t ≥ 3ω, whenever z(0) = x0 − y0 > 0. Denote dz dt = A(t)z, where A(t) := (aij (t)), 1 ≤ i, j ≤ 4, is the coefficient matrix of the righti hand side of (2.6). Since aij (t) ≥ 0, i = j, we have dz dt ≥ aii (t)zi , 1 ≤ i ≤ 4. Using this




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fact, we can see that zi (t) remains positive for all t ≥ t∗ if it becomes so at t = t∗ . Then it suffices to show that at least one of the components becomes strictly positive and that, once this happens, all other components will eventually become strictly positive. Note that z(0) > 0 implies that at least one of the components of z(0) is strictly positive. Without loss of generality, we suppose z1 (0) > 0. From the above analysis, we know that z1 (t) > 0 for all t ≥ 0. Now we claim that there exists t1 ∈ [0, ω] such that z2 (t1 ) > 0. Otherwise, we have z2 (t) ≡ 0 for all t ∈ [0, ω]. From the z2 equation in (2.6), we further derive that α1 (t)Qz1 ≡ 0 for all t ∈ [0, ω]. Since α1 (t) is periodic and not identically zero, there must be some t¯ ∈ [0, ω] such that z1 (t¯) = 0, which is a contradiction. Similarly, by the z3 equation in (2.6) and the fact that σ(t) is periodic and not identically zero, we can prove there exists t2 ∈ [t1 , t1 +ω] such that z3 (t2 ) > 0. Furthermore, by the z4 equation in (2.6) and the properties of α2 (t), we see that there exists t3 ∈ [t2 , t2 + ω] such that z4 (t3 ) > 0. Since t3 ∈ [0, 3ω], we have proved that Πt is strongly monotone when t ≥ 3ω. Clearly, other cases can be proved in a similar way. Therefore, Π3ω is strongly monotone. It is easy to see that f (x1 , x2 , x3 , x4 ) is strictly subhomogeneous in the sense that f (sx1 , sx2 , sx3 , sx4 ) > sf (x1 , x2 , x3 , x4 ) for all s ∈ (0, 1) and (x1 , x2 , x3 , x4 ) ∈ int(R4+ ). By [23, Theorem 2.3.4] (see also [23, Theorem 3.1.2]), as applied to Π3ω , it follows that statement (i) is valid and that in the case of r1 > 1, there exists a unique 3ω-periodic solution u∗ (t) := (L∗ (t), V ∗ (t), N ∗ (t), A∗ (t)), which is globally asymptotically stable for all solutions of (2.4) with initial values in R4+ \{0}. Clearly, u∗ (0) is a unique fixed point of Π3ω . By the properties of the periodic semiflow, we further get Π3ω (Πω (u∗ (0))) = Πω (Π3ω (u∗ (0))) = Πω (u∗ (0)), which implies that Πω (u∗ (0)) is also a fixed point of Π3ω . By the uniqueness of the fixed point of Π3ω , it follows that Πω (u∗ (0)) = u∗ (0). Thus, u∗ (t) is an ω-periodic solution of (2.4) and statement (ii) is valid. By Lemma 2.1 and the comparison principle, we know that solutions of system (2.2) are ultimately bounded in X + , and hence σφ = ∞ for all φ ∈ X + . Let Φt : X + → X + , t ≥ 0, be the solution semiflow associated with (2.2), that is, Φt (φ) = u(t, ·, φ) for all φ ∈ X + . Define a linear operator L(t)φ = (T1 (t, 0)φ1 , 0, T3 (t, 0)φ3 , 0) and S(t)φ =

0

t

T1 (t, s)r(s)A(s, ·, φ)ds, V(t, ·, φ),

t 0

T3 (t, s)σ(s)V(s, ·, φ)ds, A(t, ·, φ) .

Then Φt (φ) = L(t)φ + S(t)φ. By the same decomposition argument as in the proof of [24, Lemma 3.1], it follows that Φt is an α-contraction operator for any given t > 0. Combining the arguments in Lemma 2.1 and the positivity result for reaction-diffusion equations, we can further show that Φt is strongly positive for all t ≥ 3ω, that is, Φt (φ) 0 for any t ≥ 3ω and initial data φ > 0 in X + . Note that solutions of system (2.4) are also solutions of the reaction-diffusion system (2.2) subject to Neumann boundary conditions. Thus, Lemma 2.1 and the standard comparison principle arguments give rise to the following result. Theorem 2.2. For any given φ ∈ X + , let u(t, ·, φ) be the solution of (2.2) with u(0, ·, φ) = φ. Then the following two statement are valid: (i) If r1 ≤ 1, then (0, 0, 0, 0) is globally asymptotically stable for (2.2) in X + . (ii) If r1 > 1, then u∗ (t) is globally asymptotically stable for (2.2) in X + \{0}.


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Next we consider the global dynamics of the following limiting system:

(2.7)

∂m = DM Δm + α2 (t)β(Q − m)n − μM m, ∂t ∂v = DM Δv + α1 (t)mL∗ (t) − v(σ(t) + μV ) − δV V ∗ (t)v, ∂t ∂n = βT σ(t)v − n[γ + α2 (t)Q + μN ], ∂t ∂a = DH Δa + α2 (t)[Qn + βT m(N ∗ (t) − n)] − μA a − δA A∗ (t)a. ∂t

¯ R+ ). It then follows that for any φ ∈ Y , system ¯ [0, Q] × R2 ) × C(Ω, Let Y = C(Ω, + (2.7) admits a unique solution w(t, x, φ) := (m(t, x, φ), v(t, x, φ), n(t, x, φ), a(t, x, φ)) on [0, ∞) with w(0, ·, φ) = φ, and w(t, x, φ) ∈ Y for all t > 0. Moreover, by the ultimate boundedness of V, N , A, we see that w(t, x, φ) is also ultimately bounded. Note that the spatially homogeneous system associated with (2.7) is the following system: dm = α2 (t)β(Q − m)n − μM m, dt dv = α1 (t)mL∗ (t) − v(σ(t) + μV ) − δV V ∗ (t)v, dt dn = βT σ(t)v − n[γ + α2 (t)Q + μN ], dt da = α2 (t)[Qn + βT m(N ∗ (t) − n)] − μA a − δA A∗ (t)a, dt

(2.8)

and (0, 0, 0, 0) is an ω-periodic solution of (2.8). Linearizing system (2.8) at (0, 0, 0, 0), we get the linear system dm = α2 (t)βQn − μM m, dt dv = α1 (t)mL∗ (t) − v(σ(t) + μV ) − δV V ∗ (t)v, dt dn = βT σ(t)v − n[γ + α2 (t)Q + μN ], dt da = α2 (t)Qn + βT α2 (t)N ∗ (t)m − μA a − δA A∗ (t)a. dt

(2.9)

In order to introduce the basic reproduction ratio for system (2.8), we follow the procedure in [21]. We rewrite system (2.9) as du dt = (F (t) − V (t))u, where ⎛

⎛

μM ⎜ 0 V (t) = ⎜ ⎝ 0 0

0 ⎜ α1 (t)L∗ (t) F (t) = ⎜ ⎝ 0 βT α2 (t)N ∗ (t) 0 σ(t) + μV + δV V ∗ (t) −βT σ(t) 0

0 α2 (t)βQ 0 0 0 0 0 0

0 0 γ + α2 (t)Q + μN −α2 (t)Q

⎞ 0 0 ⎟ ⎟, 0 ⎠ 0

⎞ 0 ⎟ 0 ⎟. ⎠ 0 μA + δA A∗ (t)


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Let Y (t, s), t ≥ s, be the evolution operator of the linear system That is, for each s ∈ R, the matrix Y (t, s) satisfies

du dt

= −V (t)u.

d Y (t, s) = −V (t)Y (t, s) ∀t ≥ s, Y (s, s) = I, dt where I is the 4 × 4 identity matrix. Let Cω be the Banach space of all ω-periodic functions from R to R2 , equipped with the maximum norm. Suppose φ(s) ∈ Cω is the initial distribution of infectious individuals in this periodic environment; then F (s)φ(s) is the rate of new infections produced by the infected individuals who were introduced at time s, and Y (t, s)F (s)φ(s) represents the distribution of those infected individuals who were newly infected at time s and remain in the infected compartments at time t for t ≥ s. Hence, ∞ t Y (t, s)F (s)φ(s)ds = Y (t, t − τ )F (t − τ )φ(t − τ )dτ −∞

0

gives the distribution of accumulative new infection at time t produced by all those infected individuals φ(s) introduced at the previous time. Define a linear operator L : Cω → Cω by ∞ Y (t, t − τ )F (t − τ )φ(t − τ )dτ ∀t ∈ R, φ ∈ Cω . (Lφ)(t) = 0

According to [2, 21], we define the basic reproduction ratio to be R0 := r(L), where r(L) is the spectral radius of L. Let r2 be the principal Floquet multiplier of the linear system (2.9). Then [21, Theorem 2.2] implies that R0 − 1 has the same sign as r2 − 1. Thus, (0, 0, 0, 0) is asymptotically stable if R0 < 1 and unstable if R0 > 1. Since the first three equations in system (2.8) do not depend on the fourth, we consider the following subsystem of system (2.8):

(2.10)

dm = α2 (t)β(Q − m)n − μM m, dt dv = α1 (t)mL∗ (t) − v(σ(t) + μV ) − δV V ∗ (t)v, dt dn = βT σ(t)v − n[γ + α2 (t)Q + μN ]. dt

Let r3 be the principal Floquet multiplier of the following periodic linear system:

(2.11)

dm = α2 (t)βQn − μM m, dt dv = α1 (t)mL∗ (t) − v(σ(t) + μV ) − δV V ∗ (t)v, dt dn = βT σ(t)v − n[γ + α2 (t)Q + μN ]. dt

Comparing systems (2.9) and (2.11), it is easy to see that r3 − 1 has the same sign as r2 − 1. Then we have the following threshold result for system (2.8) in terms of R0 . Lemma 2.3. The following statements are valid: (i) If R0 ≤ 1, then (0,0,0,0) is globally asymptotically stable for (2.8) in [0, Q] × R3+ .



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(ii) If R0 > 1, then (2.8) admits a unique positive ω-periodic solution w∗ (t) := (m∗ (t), v ∗ (t), n∗ (t), a∗ (t)), which is globally asymptotically stable for (2.8) in ([0, Q] × R2+ \{0}) × R+ . Proof. We first show that the following threshold result holds for system (2.10): (a) If r3 ≤ 1, then (0,0,0) is globally asymptotically stable for (2.10) in [0, Q]×R2+. (b) If r3 > 1, then (2.10) has a positive ω-periodic solution (m∗ (t), v ∗ (t), n∗ (t)), which is globally asymptotically stable for (2.10) in ([0, Q] × R2+ )\{0}. ¯0 ∈ Let w(t, ¯ w ¯0 ) be the nonnegative solution of system (2.10) with initial data w [0, Q] × R2+ . Denote X(t) = ∂∂w¯w¯0 (t, w¯0 ). Then X(t) = (xij (t))3×3 satisfies X (t) = A(t)X(t),

X(0) = I,

where A(t) = (aij (t))3×3 is the Jacobian matrix of the right-hand side of system (2.10) evaluated at (m, v, n) = w(t, ¯ w ¯0 ). Since aij (t) ≥ 0, i = j, for all t ≥ 0, we have xij (t) ≥ aii (t)xij (t) for all t ≥ 0, 1 ≤ i, j ≤ 3. It then follows that xij (t) > 0 for all t ≥ t∗ provided xij (t∗ ) > 0 for some t∗ ≥ 0. Since xii (0) = 1, we have xii (t) > 0 for all t ≥ 0, 1 ≤ i ≤ 3. We further prove that xij (t) > 0 for all t ≥ 2ω. Note that xij (t), i = j, satisfy the following equations: x12 (t) = −(α2 (t)βn(t) + μM )x12 (t) + α2 (t)β(Q − m(t))x32 (t), x13 (t) = −(α2 (t)βn(t) + μM )x13 (t) + α2 (t)β(Q − m(t))x33 (t),

x21 (t) = α1 (t)L∗ (t)x11 (t) − (σ(t) + μV + V ∗ (t)δV )x21 (t), x23 (t) = α1 (t)L∗ (t)x13 (t) − (σ(t) + μV + V ∗ (t)δV )x23 (t),

x31 (t) = βT σ(t)x21 (t) + (γ + α2 (t)Q + μN )x31 (t), x32 (t) = βT σ(t)x22 (t) + (γ + α2 (t)Q + μN )x32 (t).

Since xii (t) > 0 for all t ≥ 0, 1 ≤ i ≤ 3, and α1 (t), α2 (t), σ(t) are periodic but not identically zero, it follows from a contradiction argument that there exists t1 ∈ [0, ω] such that x13 (t), x21 (t), x32 (t) > 0 for all t ≥ t1 . Then we can prove that there exists t2 ∈ [t1 , t1 + ω] such that x12 (t), x23 (t), x31 (t) > 0 for all t ≥ t2 . Since t2 ∈ [0, 2ω], we have X(t) 0 for all t ≥ 2ω. Then for any w ¯1 , w ¯2 ∈ R3+ satisfying w ¯2 > w ¯1 , we have 1 ∂w ¯ w(t, ¯ w ¯2 ) − w(t, ¯ w ¯1 ) = (t, w ¯1 + r(w¯2 − w ¯1 ))(w¯2 − w ¯1 )dr 0 ∂ w ¯ 0 0 ¯t (w¯1 ) for all t ≥ 2ω. In particular, provided t ≥ 2ω. This implies that w ¯t (w¯2 ) w we have that w ¯2ω (·) is strongly monotone. By the same argument as in the proof of Lemma 2.1, we see that statements (a) and (b) hold. By the theory of chain transitive sets (see [7] or [23, section 1.2]) and arguments similar to those in the proof of Theorem 2.5, it follows that limt→∞ a(t) = 0 in the case where r3 ≤ 1, and limt→∞ (a(t) − a∗ (t)) = 0 in the case where r3 > 1, where a∗ (t) is the unique positive ω-periodic solution of the limiting equation da = α2 (t)[Qn∗ (t) + βT m∗ (t)(N ∗ (t) − n∗ (t))] − (μA + δA A∗ (t))a. dt Since r3 − 1 has the same sign as R0 − 1, we then complete the proof. The following result shows that R0 is also the threshold value for the global dynamics of system (2.7). Theorem 2.4. For any given φ ∈ Y , let w(t, ·, φ) be the solution of (2.7) with w(0, ·, φ) = φ. Then the following two statement are valid:




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(i) If R0 ≤ 1, then (0, 0, 0, 0) is globally asymptotically stable for (2.7) in Y . ¯ [0, Q)× (ii) If R0 > 1, then w∗ (t) is globally asymptotically stable for (2.7) in (C(Ω, 2 ¯ R+ )\{0}) × C(Ω, R+ ). Proof. Since the first three equations in (2.7) do not depend on the fourth, it suffices to prove that the threshold result is valid for the following subsystem:

(2.12)

∂m = DM Δm + α2 (t)β(Q − m)n − μM m, ∂t ∂v = DM Δv + α1 (t)mL∗ (t) − v(σ(t) + μV ) − δV V ∗ (t)v, ∂t ∂n = βT σ(t)v − n[γ + α2 (t)Q + μN ]. ∂t

ˆ be the unique solution of (2.12) with the initial data φˆ ∈ C(Ω, ¯ [0, Q] × Let w(t, ˆ ·, φ) R2+ )\{0}. By the positivity result for reaction-diffusion equations, it follows that ˆ 0 for all t ≥ 2ω. Note that solutions of system (2.10) are also solutions of w(t, ˆ ·, φ) the reaction-diffusion system (2.12) subject to Neumann boundary conditions. Thus, Lemma 2.3, together with the standard comparison argument, implies that the threshold result is valid for system (2.12). In the rest of this section, we use the theory of chain transitive sets (see [7] or [23, section 1.2]) to establish the following threshold result on the global dynamics for system (2.1). Theorem 2.5. Let r1 > 1. Then the following statements are valid: (i) If R0 ≤ 1, then the disease-free periodic solution (Q, L∗ (t), V ∗ (t), N ∗ (t), A∗ (t), 0, 0, 0, 0) ¯ R+ ) \ {0}) × (X + \ {0}) × is globally attractive for system (2.1) in (C(Ω, 4 ¯ C(Ω, R+ ). (ii) If R0 > 1, then system (2.1) has a unique positive ω-periodic solution (Q, L∗ (t), V ∗ (t), N ∗ (t), A∗ (t), m∗ (t), v ∗ (t), n∗ (t), a∗ (t)), ¯ R+ )\ {0})× (X + \ {0})× which is globally attractive for system (2.1) in (C(Ω, ¯ R+ ). ¯ R3 ) \ {0}) × C(Ω, (C(Ω, + Proof. Let {Ψt }t≥0 be the periodic semiflow associated with system (2.1). That is, Ψt (ψ)(x) := (M(t, x), L(t, x), V(t, x), N (t, x), A(t, x), m(t, x), v(t, x), n(t, x), a(t, x)) ¯ R9+ ). For any given ψ ∈ is the unique solution of (2.1) with initial data ψ ∈ C(Ω, + 4 ¯ ¯ (C(Ω, R+ ) \ {0}) × (X \ {0}) × C(Ω, R+ ), let L be the omega limit set of the discretetime orbit {Ψnω (ψ)}n≥1 . Since every solution is ultimately bounded, we know from [7, Lemma 2.1] (see also [23, Lemma 1.2.1]) that L is an internally chain transitive set for Ψω . Since r1 > 1, Theorem 2.2 implies that lim ((Ψnω (ψ))1 , (Ψnω (ψ))2 , (Ψnω (ψ))3 , (Ψnω (ψ))4 , (Ψnω (ψ))5 )

n→∞

= (Q, L∗ (0), V ∗ (0), N ∗ (0), A∗ (0)).

¯ R4+ ) such that Thus, there exists a subset L1 of C(Ω, L = {(Q, L∗ (0), V ∗ (0), N ∗ (0), A∗ (0))} × L1 .



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For any given φ = (φ1 , φ2 , . . . , φ9 ) ∈ L, there exists a sequence nk → ∞ such that ¯ letting nk → ∞, Ψnωk (ψ) → φ as k → ∞. Since m(nk ω, x) ≤ M(nk ω, x) for all x ∈ Ω, ¯ we obtain 0 ≤ φ6 (x) ≤ φ1 (x) ≡ Q for all x ∈ Ω. It then follows that L1 ⊂ Y . It is easy to see that Ψω |L (Q, L∗ (0), V ∗ (0), N ∗ (0), A∗ (0), φ6 , φ7 , φ8 , φ9 ) = {(Q, L∗ (0), V ∗ (0), N ∗ (0), A∗ (0))} × Υω |L1 (φ6 , φ7 , φ8 , φ9 ), where {Υt }t≥0 is the solution semiflow associated with system (2.7) on Y . Since L is an internally chain transitive set for Ψω , it follows that L1 is an internally chain transitive set for Υω . In the case where R0 ≤ 1, it follows from Theorem 2.4(i) that (0, 0, 0, 0) is globally asymptotically stable. By [7, Theorem 3.1] (see also [23, Theorem 1.2.1]), we have L1 = {(0, 0, 0, 0)}, and hence L = {(Q, L∗ (0), V ∗ (0), N ∗ (0), A∗ (0), 0, 0, 0, 0)}. This implies that statement (i) is valid. In the case where R0 > 1, by Theorem 2.4(ii) and [7, Theorem 3.2] (see also [23, Theorem 1.2.2]), it follows that either L1 = {(0, 0, 0, 0)} or L1 = {(m∗ (0), v ∗ (0), n∗ (0), a∗ (0))}. We further claim that L1 = {(0, 0, 0, 0)}. Suppose, by contradiction, that L1 = {(0, 0, 0, 0)}. Then we have L = {(Q, L∗ (0), V ∗ (0), N ∗ (0), A∗ (0), 0, 0, 0, 0)}. Thus, ¯ and for any > 0, there limt→∞ (m(t, x), v(t, x), n(t, x)) = 0 uniformly for x ∈ Ω, exists T > 0 such that |(M(t, x), L(t, x), V(t, x), N (t, x), A(t, x)) − (Q, L∗ (t), V ∗ (t), N ∗ (t), A∗ (t))| < ¯ Hence, for any t ≥ T , we have for all t ≥ T and x ∈ Ω.

(2.13)

∂m ≥ DM Δm + α2 (t)β(Q − − m)n − μM m, ∂t ∂v ≥ DM Δv + α1 (t)m(L∗ (t) − ) − v(σ(t) + μV ) − δV (V ∗ (t) + )v, ∂t ∂n ≥ βT σ(t)v − n[γ + α2 (t)(Q + ) + μN ]. ∂t

By the assumption on ψ in statement (ii), we further have (m(0, ·), v(0, ·), n(0, ·)) ∈ ¯ R3 )\{0}. Let r be the principal Floquet multiplier of the periodic linear system C(Ω, +

(2.14)

dm = α2 (t)β(Q − )n − μM m, dt dv = α1 (t)m(L∗ (t) − ) − v(σ(t) + μV ) − δV (V ∗ (t) + )v, dt dn = βT σ(t)v − n[γ + α2 (t)(Q + ) + μN ]. dt

Since r3 > 1, we can fix 0 < < min (Q, min0≤t≤ω L∗ (t)) small enough such that r > 1. By a result similar to Theorem 2.4(ii), we see that the Poincaré map of

(2.15)

∂m = DM Δm + α2 (t)β(Q − − m)n − μM m, ∂t ∂v = DM Δv + α1 (t)m(L∗ (t) − ) − v(σ(t) + μV ) − δV (V ∗ (t) + )v, ∂t ∂n = βT σ(t)v − n[γ + α2 (t)(Q + ) + μN ] ∂t




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admits a globally attractive fixed point (m ¯ (0), v¯ (0), n ¯ (0)) 0. In view of (2.13) and (2.15), the comparison principle implies that lim inf n→∞ (m(nω, x), n(nω, x), v(nω, x)) ≥ (m ¯ (0), n ¯ (0), v¯ (0)) 0, which contradicts limt→∞ (m(t, x), v(t, x), n(t, x)) = 0. It then follows that L1 = {(m∗ (0), v ∗ (0), n∗ (0), a∗ (0))}, and hence L = {(Q, L∗ (0), V ∗ (0), V ∗ (0), A∗ (0), m∗ (0), v ∗ (0), n∗ (0), a∗ (0))}. This implies that statement (ii) is valid. 3. Spreading speed and traveling waves. In this section, we consider the spreading speed and traveling waves for system (2.7) in an unbounded spatial habitat Ω. Since all coefficients in (2.7) are spatially homogeneous, it suffices to study the spreading speed in any given direction of R2 . Without loss of generality, we then assume that the spatial domain Ω = R. In view of Theorems 2.2(ii) and 2.4(ii), we assume that r1 > 1 and R0 > 1 throughout this section. Since the first three equations in system (2.7) do not depend on the fourth equation, we first analyze the subsystem (2.12). By the proof of Lemma 2.3, we know that the spatially homogeneous system (2.10) associated with (2.12) admits a globally attractive ω-periodic solution w ˆ∗ (t) = (m∗ (t), v ∗ (t), n∗ (t)) 0. In what follows, we appeal to the theory of spreading speeds and traveling waves developed in [8] for periodic evolution systems to study the spreading speed and monotone traveling waves connecting 0 and w ˆ ∗ (t) for system (2.7). Let C := BC(R, R3 ) be the set of all bounded and continuous functions from R to 3 R equipped with the compact open topology; that is, a sequence ψn converges to ψ in C if and only if ψn (x) converges to ψ(x) in R3 uniformly for x in any compact subset of R. For any ψ 1 = (ψ11 , ψ21 , ψ31 ), ψ 2 = (ψ12 , ψ22 , ψ32 ) ∈ C, we denote ψ 2 ≥ ψ 1 (ψ 2 ψ 1 ) if ψi2 ≥ ψi1 (ψi2 > ψi1 ) for all 1 ≤ i ≤ 3, x ∈ R, and ψ 2 > ψ 1 if ψ 2 ≥ ψ 1 but ψ 2 = ψ 1 . For any vectors a, b in R3 , we can define a ≥ (, >)b similarly. For any β 0 in R3 , we define [0, β] := {ψ ∈ R3 : β ≥ ψ ≥ 0} and Cβ := {ψ ∈ C : β ≥ ψ ≥ 0}. Define the reflection operator R by R[ψ](x) = ψ(−x), and the translation operator Ty by Ty [ψ](x) = ψ(x − y) for any given y ∈ R. Let Q be an operator from Cβ to Cβ . In order to use the results in [8, 10], we need the following assumptions on Q: (A1) Q[R[φ]] = R[Q[φ]], Ty ◦ Q[φ] = Q ◦ Ty [φ] for all φ ∈ Cβ , y ∈ R. (A2) Q : Cβ → Cβ is continuous with respect to the compact open topology. (A3) Q is order preserving in the sense that Q[φ] ≥ Q[ψ] whenever φ ≥ ψ in Cβ . (A4) Q : [0, β] → [0, β] admits exactly two fixed points 0 and β, and limn→∞ Qn [α] = β for any α ∈ [0, β] with 0 α ≤ β. Given a function φ ∈ Cβ and a bounded interval I = [a, b] ⊂ R, we define a function φI ∈ C(I, R3 ) by φI (x) = φ(x). Moreover, for any subset D of Cβ , we define DI = {φI ∈ C(I, R3 ) : φ ∈ D}. In order to obtain the existence of traveling waves, we need the following weak compactness assumption: (A5) For any δ > 0, there exists l = l(δ) ∈ [0, 1) such that for any D ⊂ Cβ and any interval I = [a, b] of the length δ, we have α(Q[D]I ) ≤ lα(DI ), where α is the Kuratowski measure of noncompactness on the Banach space C(I, R3 ). Let {Qt }t≥0 be the solution semiflow associated with system (2.12) on Cwˆ ∗ (0) , that is, ˆ x, φ) Qt (φ)(x) = w(t,

∀φ ∈ Cwˆ ∗ (0) ,

x ∈ R, t ≥ 0.



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It then follows that {Qt }t≥0 is a monotone periodic semiflow and each map Qt is subhomogeneous in the sense that Qt (sφ) ≥ sQt (φ) for all φ ∈ Cwˆ ∗ (0) and s ∈ [0, 1]. Moreover, we have the following observation. Lemma 3.1. The Poincaré map Qω satisfies conditions (A1)–(A5) with β = w ˆ∗ (0). Proof. It is easy to verify that Qω admits conditions (A1)–(A3). By statement (b) in the proof of Lemma 2.3, we see that condition (A4) holds for Qω . Furthermore, by a decomposition argument similar to that in the proof of [24, Lemma 3.1], it follows that (A5) holds for Qω . Note that the results of spreading speeds and traveling waves in [8] are still valid provided that the assumption (A5) in [8] is replaced by the standing assumption (A4) (see [10]). By [8, Theorem 2.1], it then follows that the map Qω : Cwˆ ∗ (0) → Cwˆ ∗ (0) admits a spreading speed c∗ω . In order to estimate c∗ω , we consider the following linear system:

(3.1)

∂m = DM Δm + α2 (t)βQn − μM m, ∂t ∂v = DM Δv + α1 (t)mL∗ (t) − v(σ(t) + μV + δV V ∗ (t)), ∂t ∂n = βT σ(t)v − n[γ + α2 (t)Q + μN ]. ∂t

Let (u1 (t, x), u2 (t, x), u3 (t, x)) = e−μx (¯ u1 (t), u¯2 (t), u ¯3 (t)) be a solution of (3.1). Then (¯ u1 (t), u¯2 (t), u¯3 (t)) satisfies the following ODE system with the initial data u ¯(0) ∈ R3 :

(3.2)

d¯ u1 (t) = μ2 DM u¯1 (t) + α2 (t)βQ¯ u3 (t) − μM u¯2 (t), dt d¯ u2 (t) u1 (t) − (σ(t) + μV + δV V ∗ (t))¯ u2 (t), = μ2 DM u¯2 (t) + α1 (t)L∗ (t)¯ dt d¯ u3 (t) = βT σ(t)¯ u2 (t) − [γ + α2 (t)Q + μN ]¯ u3 (t). dt

Let {Mt }t≥0 be the solution map associated with (3.1). Define Bμt : R3 → R3 as u1 (t, z), u ¯2 (t, z), u ¯3 (t, z)). Bμt (z) := Mt (ze−μx )(0) = (¯ Thus, Bμt (·) is the solution map of the linear system (3.2) on R3 . Let r(μ) be the spectral radius of the Poincaré map Bμω . It is easy to verify that r(μ) is even on R and Bμ2ω = (Bμω )2 is a compact and strongly positive operator (actually, Bμt is strongly positive for all t ≥ 2ω). By [9, Lemma 3.1], it follows that r(μ) > 0, and it is a simple eigenvalue of Bμω with a strongly positive eigenvector w∗ 0. Moreover, it follows from [9, Lemma 3.7] that r(μ) is log convex on R. Using an argument similar to that in the proof of [22, Lemma 2.1], we see that there exists a positive ω-periodic function w(t) such that v(t) = eλ(μ)t w(t) is a solution of (3.2), where λ(μ) = ω1 ln r(μ) and w(0) = w∗ . That is, Bμt (w(0)) = eλ(μ)t w(t). Letting t = ω, we have Bμω (w(0)) = eλ(μ)ω w(0), which implies that eλ(μ)ω is the principal eigenvalue of Bμω with strongly positive eigenvector w(0). Following [9], we define Φ(μ) :=

ln r(μ) 1 λ(μ)ω ln(eλ(μ)ω ) = = μ μ μ

∀μ > 0.

When μ = 0, system (3.2) reduces to the linear system (2.11). Since R0 > 1, we have r(0) = r3 > 1. Thus, limμ→0+ Φ(μ) = ∞. Note that λ(μ) = ω1 ln r(μ) is even and




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convex; then λ(μ) ≥ λ(0) > 0 on R. Since v(t) = eλ(μ)t w(t) is a solution of (3.2), we see that dv2 (t) ≥ [μ2 DM − (σ(t) + μV + δV V ∗ (t))]v2 (t). dt It follows that w2 (t) ≥ μ2 DM − μV − σ(t) − λ(μ) − δV V ∗ (t). w2 (t) Integrating the above inequality from 0 to ω, we get ω ω w2 (t) dt ≥ μ2 DM ω − μV ω − λ(μ)ω − 0= (σ(t) + δV V ∗ (t))dt. w (t) 2 0 0 Then we have λ(μ)ω μV ω Φ(μ) = ≥ μDM ω − − μ μ

ω 0

(σ(t) + δV V ∗ (t))dt →∞ μ

as μ → ∞. Therefore, Φ(μ) attains its positive minimum at some finite value μ∗ . Then we have the following result. Lemma 3.2. Let c∗ω be the spreading speed of map Qω on Cwˆ ∗ (0) . Then c∗ω = inf μ>0 Φ(μ), and hence c∗ω > 0. Proof. It is easy to verify that the map Mt satisfies all conditions (C1)–(C7) in [9] for all t > 0. Comparing systems (2.12) and (3.1), we see that Qt is a lower solution of linear system (3.1) for all t ≥ 0. Then we have Qt (φ) ≤ Mt (φ)

∀φ ∈ Cwˆ ∗ (0) , ∀t ≥ 0.

Fix t = ω. It follows from [9, Theorem 3.1] that c∗ω ≤ inf μ>0 Φ(μ). By the continuity of the solution on the initial data, we know that for any ∈ (0, Q), there exists η > 0 such that the solution w(t, ˆ η¯) of system (2.10) with w(0, ˆ η¯) = η¯ satisfies w(t, ˆ η¯) < ¯ for all t ∈ [0, ω], where ε¯ = ( , , ), η¯ = (η, η, η). Then the comparison principle implies that w(t, ˆ x, φ) ≤ w(t, ˆ η¯) ≤ ¯ ∀x ∈ R, φ ∈ Cη¯, t ∈ [0, ω]. ˆ x, φ) satisfies Thus, for any x ∈ R, t ∈ [0, ω], and φ ∈ Cη¯, w(t,

(3.3)

∂w ˆ1 ≥ DM Δwˆ1 − μM w ˆ1 + α2 (t)β(Q − )w ˆ3 , ∂t ∂w ˆ2 = DM Δwˆ2 + α1 (t)L∗ (t)wˆ1 − (σ(t) + μV + δV V ∗ (t))wˆ2 , ∂t ∂w ˆ3 = βT σ(t)wˆ2 − [γ + α2 (t)Q + μN ]w ˆ3 . ∂t

Let {Mt }t≥0 be the solution semiflow associated with the linear system

(3.4)

∂w ˆ1 = DM Δwˆ1 − μM w ˆ1 + α2 (t)β(Q − )w ˆ3 , ∂t ∂w ˆ2 = DM Δwˆ2 + α1 (t)L∗ (t)wˆ1 − (σ(t) + μV + δV V ∗ (t))wˆ2 , ∂t ∂w ˆ3 = βT σ(t)wˆ2 − [γ + α2 (t)Q + μN ]w ˆ3 . ∂t


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By the comparison principle, we then have Mt (φ) ≤ Qt (φ)

∀φ ∈ Cη¯, t ∈ [0, ω].

Letting t = ω and 0 < < Q small enough, we can perform an analysis on {Mt }t≥0 similar to that for {Mt }t≥0 . It follows from [9, Theorem 3.10] that inf Φ (μ) ≤ c∗ω ≤ inf Φ(μ)

μ>0

μ>0

for all sufficiently small . Letting → 0, we obtain c∗ω = inf μ>0 Φ(μ). Let c∗ := c∗ω /ω. Then the following result shows that c∗ is the spreading speed for system (2.12). Theorem 3.3. Assume R0 > 1. Let w(t, ˆ x, φ) be the solution of system (2.12) with w(0, ˆ ·, φ) = φ ∈ Cwˆ ∗ (0) . Then the following statements hold: (i) For any c > c∗ , if φ ∈ Cwˆ ∗ (0) with 0 ≤ φ w ˆ∗ (0), and φ = 0 outside a bounded interval, then limt→∞,|x|≥ct w(t, ˆ x, φ) = (0, 0, 0). ˆ x, φ) − (ii) For any c ∈ (0, c∗ ), if φ ∈ Cwˆ ∗ (0) with φ ≡ 0, then limt→∞,|x|≤ct (w(t, w ˆ∗ (t)) = 0. Proof. In view of Lemma 3.1, statement (i) is a straightforward consequence of [8, Theorem 2.1] and [10, Theorem 3.4(i)]. For statement (ii), since Qt is subhomogeneous, rσ in [8, Theorem 2.1] can be chosen to be independent of σ 0. Denote rσ = r¯. For any φ ∈ Cwˆ ∗ (0) with φ > 0, from the strong positivity of Qt for t ≥ 2ω, we know that Q2ω (φ) 0. Then there exists a σ 0 in R3 such that Q2ω (φ) σ for x on an interval I of length 2¯ r. Taking w(2ω, ˆ x, φ) as new initial data, we see from [8, Theorem 2.1(ii)] that statement (ii) is valid. Recall that U (t, x−ct) is said to be an ω-periodic traveling wave of (2.12) provided that U (t, z) is ω-periodic in t and w(t, ˆ x) = U (t, x − ct) satisfies (2.12), and we say ˆ∗ (t) and U (t, ∞) = 0 uniformly for U (t, x − ct) connects w ˆ∗ (t) to 0 if U (t, −∞) = w t ∈ [0, ω]. The existence and nonexistence of traveling waves are straightforward consequences of Lemma 3.1; see [8, Theorems 2.2 and 2.3] and [10, Theorems 4.1 and 4.2]. Theorem 3.4. Assume that R0 > 1. Then the following statements are valid: (i) For any c ∈ (0, c∗ ), system (2.12) has no ω-periodic traveling wave U (t, x−ct) connecting w ˆ∗ (t) to 0. (ii) For any c > c∗ , system (2.12) has an ω-periodic traveling wave U (t, x − ct) connecting w ˆ∗ (t) to 0, and U (t, z) is continuous and nonincreasing in z ∈ R. Note that we can regard the fourth equation in system (2.7) as the following nonhomogeneous reaction-diffusion equation: ∂a = DH Δa − (μA + δA A∗ (t))a(t, x) + α2 (t)[Qn(t, x) + βT m(t, x)(N ∗ (t) − n(t, x))]. ∂t By an argument similar to that in the proof of [6, Theorems 3.1 and 3.2], it follows that similar results in Theorems 3.3 and 3.4 are also valid for a(t, x). Thus, c∗ is the spreading speed and the minimal wave speed for monotone periodic traveling waves of system (2.7). 4. A case study. In this section, we do a case study for Lyme disease in Port Dover, Ontario, and present some numerical simulations. According to [15], the duration of development and the questing activity of ticks can be explained largely by temperature effects alone. Thus, we focus on the discussion



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Table 4.1 Values for constant parameters in model (1.1). Parameter rM KM μM DM DH μL μV μN μA δV δA β βT γ

Value 2 × 30.4 3000 0.012 × 30.4 8.84 × 10−4 × 30.4 0.227 × 30.4 0.006 × 30.4 0.003 × 30.4 0.006 × 30.4 0.003 × 30.4 1/(595.35/12) × Q 1/(521.12/12) × 42 1 0.9 0.005 × 30.4

Dimension /Month Dimensionless /Month /km2 /km2 /Month /Month /Month /Month /Month /Month Dimensionless Dimensionless /Month

Reference [3] [13] [17] [13] [13] [17] [17] [17] [17] N/A N/A [17] [4] [4]

Table 4.2 Monthly mean temperature for Port Dover (in o C). Month Temperature Month Temperature

Jan −4.5 Jul 21.2

Feb −1.4 Aug 19.6

Mar 1.9 Sep 16.7

Apr 7.5 Oct 10.4

May 15.4 Nov 3.9

Jun 19.4 Dec −2.1

of the temperature effects on the transmission of Lyme disease. Using the published data in [13, 15, 16, 17] and mean monthly temperature normals at Port Dover from the Canadian meteorological website [19], we can evaluate the temperature-dependent coefficients r(t), α1 (t), σ(t), and α2 (t) and other constant coefficients in our model. In this study, we let the period ω = 12 months. First, we estimate the constant coefficients in our model. Note that in [13, 15, 16, 17], the authors determined some realistically feasible constant coefficients in Lyme disease models based on the valuable data from laboratory study and field observation. We refer the reader to their work and list values of constants coefficients for the Lyme disease models (1.1) in Table 4.1. According to Table 2 in [13], we know that the maximum number of ticks of a given life stage that a mouse and a deer can feed in one year is 595.35 and 521.12, respectively. We suppose that the number of mice and deer is Q and 42, respectively, in the region. Then we estimate δV =

Q , 595.35/12

δA =

42 . 521.12/12

Next, we use the monthly mean temperatures at Port Dover, temperaturedependent questing activity rate for immature ticks, and the relationship between the temperature and the development rate to estimate the periodic coefficients r(t), α1 (t), σ(t), and α2 (t). In this case study, we take January to be the starting point and assume that tick development is zero for all stages when the air temperature is 0o C or below [16]. We list the monthly mean temperature for Port Dover in Table 4.2 according to the temperature statistics in [19]. It follows from Figure 1 in [15] that the preoviposition period of female adults, preeclosion period for egg masses, and premolt period of larvae are given in days,


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respectively, by Y = 1300C −1.42,

Y = 34234C −2.27,

Y = 101181C −2.55,

where C > 0 is the temperature in o C. We assume that five percent of adult ticks are pregnant females, and per-capital egg production by pregnant females is 3000 [17]. Then the temperature-dependent developmental rates for larvae and nymphs per month can be expressed as 30.4 ×

3000 1 1 × and 30.4 × . 20 1300C −1.42 + 34234C −2.27 101181C −2.55

Using the temperature data in Table 4.2 and the curve fitting tool (CFTOOL) in MATLAB, we can fit the time-dependent individual birth rate of tick r(t) and the individual development rate of nymph σ(t) (see Table 1.1) as r(t) = 31.87 − 37.77 cos(πt/6) − 25.75 sin(πt/6) + 5.815 cos(πt/3) + 12.38 sin(πt/3) and σ(t) = 0.2325 − 0.2896 cos(πt/6) − 0.1951 sin(πt/6) + 0.05472 cos(πt/3) + 0.1181 sin(πt/3) − 0.00855 cos(πt/6) − 0.00345 sin(πt/2) + 0.01085 cos(2πt/3) − 0.00433 sin(2πt/3). According to [16, 17], the biting rate of larvae and nymphs to mice are dependent on the mice-finding probability (see Table 1 in [17]) and the activity proportion, where the activity proportion is temperature-dependent (see Figure 3 in [16]). In [17], the daily mice-finding probability of questing larvae and nymphs is expressed as λql = 0.0013m0.515

and λqn = 0.002m0.515,

where m is the total number of mice. The relationship between the temperature and the activity proportion of immature ticks is given in Figure 3 of [16]. Combining the temperature data in Table 4.2, we fit the temperature-dependent activity proportion of immature ticks as θ(t) = 0.08292 − 0.1158 cos(πt/6) − 0.07253 sin(πt/6) + 0.02833 cos(πt/3) + 0.06495 sin(πt/3) + 0.008333 cos(3πt/6) − 0.0125 sin(3πt/6). Thus, in this case study, the monthly biting rate of larvae and nymphs to one mouse (see Table 1.1) can be given by α1 (t) = 30.4 ×

0.0013Q0.515 × θ(t), Q

α2 (t) = 30.4 ×

0.002Q0.515 × θ(t). Q

With the above temperature-dependent coefficients and constants parameters in Table 4.1, we numerically calculate the principal Floquet multiplier of system (2.5) r1 = 3870.6 > 1. Then we use solver ODE45 and the CFTOOL package in MATLAB to find the periodic solution (L∗ (t), V ∗ (t), N ∗ (t), A∗ (t)) for system (2.4). Thanks to [21, Theorem 2.1], we can further numerically compute the basic reproduction ratio R0 for system (2.7). Since all coefficients in our model are spatially homogeneous, without loss of generality, we assume the spatial domain Ω = [−I, I] ⊂ R when Ω is bounded and truncate the infinite domain R to be [−I, I].


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(a) The evolution of v.

(b) The evolution of n.

5

2.5

4

x 10

10

x 10

9 2

8 7 6 n (t)

1.5 *

v*(t)



1

5 4 3

0.5

2 1

0

0

5

10

15

20

t

(c) The numerical solution v∗ (t).

25

0

0

5

10

15

20

25

t

(d) The numerical solution n∗ (t).

Fig. 4.1. The evolution of v and n, and numerical periodic solutions v∗ (t) and n∗ (t) with R0 = 3.625 > 1.

In order to simulate the global dynamics of system (2.1) in a bounded domain, we apply the difference method to the system with the Neumann boundary condition and I = 10, and choose the initial data as πx 4 M (0, x) = L(0, x) = 100 × cos , V (0, x) = × L(0, x), 2I 5 3 1 N (0, x) = × L(0, x), A(0, x) = × L(0, x), 5 2 1 3 m(0, x) = × L(0, x), v(0, x) = × L(0, x), 5 10 1 1 n(0, x) = × L(0, x), a(0, x) = × L(0, x). 4 5 Using the parameter values in Table 4.1 and the periodic coefficients, we numerically calculate the basic reproduction ratio R0 = 3.625 > 1. Figure 4.1 shows the evolution of v(t) and n(t) in system (2.1). If the susceptibility to infection in mice and ticks is, respectively, reduced to β = 0.2 and βT = 0.22 due to some preventive measures, we numerically get R0 = 0.825 < 1. In this situation, Figure 4.2 shows that v(t) and n(t) will eventually approach zero. The simulation results for m(t) and a(t) are also consistent with our analytic result in Theorem 2.5. In the case of unbounded domain, using the given parameters such that R0 = 3.625 > 1, we numerically estimate the spreading speed c∗ω /ω = 0.2644. To simulate





Fig. 4.2. The evolution of v and n with R0 = 0.825 < 1.

(a) The spread of v.

(b) The spread of n.

4

5

4

x 10

5

4.5 4

4

3.5

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0 −40

x 10

4.5

n

v


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−30

−20

−10

0 x

10

20

30

40

0 −40

(c) The density of v at t = nω.

−30

−20

−10

0 x

10

20

30

40

(d) The density of n at t = nω.

Fig. 4.3. The spread of v and n, and the densities of v and n at t = nω with n = 0, 1, 2, 3, 4, 5, respectively.

the spatial spread of the disease, we choose I = 40. Figure 4.3 shows the numerical plots of the solution of system (2.7) with the initial data given by ⎧ ⎨ 0, 10 × (20 − |x|), m(0, x) = ⎩ 100,

|x| ≥ 20, 10 ≤ |x| ≤ 20, |x| ≤ 10,


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Fig. 4.4. The time-periodic traveling waves observed for v and n.

and v(0, x) = 10 × m(0, x),

n(0, x) = 8 × m(0, x),

a(0, x) = 6 × m(0, x).

To observe traveling waves, we choose the initial data as ⎧ −40 ≤ x ≤ −20, ⎨ 1200, 30 × (20 − x), |x| ≤ 20, m(0, x) = ⎩ 0, 20 ≤ x ≤ 40, and v(0, x) = 30 × m(0, x),

n(0, x) =

110 × m(0, x), 3

a(0, x) =

3 × m(0, x). 4

Then the evolution of the solution is as shown in Figure 4.4. 5. Discussion. In this paper, we incorporated seasonal forcing into a reactiondiffusion Lyme disease model. Since seasonal variations are critical for the development of ticks and their activities, we assume that the developmental rate of ticks and their biting rates are time-periodic. We investigated the global dynamics of this model in a bounded and an unbounded habitat. In the case of a bounded habitat, we introduced the basic reproduction ratio R0 for Lyme disease, and further proved that R0 can serve as the threshold parameter for the global stability of either disease-free or positive periodic solution. Biologically, this means that the disease will die out when R0 < 1, and the disease will stabilizes at a positive periodic solution when R0 > 1. In the case of an unbounded habitat, we consider the spatial spread of the disease and the existence of time-periodic traveling waves. We established the existence of the spreading speed of infection and its coincidence with the minimal wave speed for limiting system (2.7). Moreover, we got an implicit formula for the spreading speed in Lemma 3.2, which may be used to compute the spreading speed numerically. For our model, we picked some feasible coefficients and estimated the timeperiodic coefficients using some published data. We numerically calculated the basic reproduction ratio R0 . Since R0 = 3.625 > 1, we can see from Figure 4.1 that the disease stabilizes at a positive periodic state. If the infection susceptibilities β and βT decrease such that the basic reproduction ratio R0 = 0.825 < 1, then Figure 4.2 shows that the disease will die out eventually. In order to consider the spatial propagation of



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the disease in an unbounded domain, we calculated the spreading speed numerically. Figure 4.3 shows that the disease spreads at a certain speed. To control the disease, we may use some strategies to reduce the spreading speed. For example, we may use some chemical methods to reduce the infection susceptibilities or the total number of hosts. Our analytic results and the numerical values of the basic reproduction ratio and the spreading speed may provide some helpful suggestions for disease control. When the spatial domain Ω is bounded, we can also study the global dynamics of model (1.1) under the Robin-type or Dirichlet boundary conditions. In such a case, we can show that solution maps of reaction-diffusion systems (2.2) and (2.12) and their linearizations at zero are α-contractions by a decomposition argument similar to that in [24, Lemma 3.1]. Thus, the abstract threshold type result for monotone and subhomogeneous systems (see [23, Theorem 2.3.4]), together with the generalized Krein–Rutman theorem, can be applied directly to (2.2) and (2.12), respectively. It then follows that the analogues of Theorems 2.2–2.5 still hold true. In these results, however, two numbers r1 and R0 should be replaced by the spectral radii of the Poincaré (period) maps of the linearized systems of (2.2) and (2.12) at zero solution, respectively. We should point out that our model ignores the time delays between tick life stages. It would be more interesting to incorporate time delays into the model. Another challenging problem is to study the spreading speeds and traveling waves in the case where some parameters are spatially dependent. We leave these problems for future investigation. Acknowledgments. We are very grateful to Yijun Lou for his valuable comments and discussions on the Lyme disease modeling and numerical simulations. Our sincere thanks also go to two anonymous referees for their careful reading and helpful comments which led to an improvement of the original manuscript. REFERENCES [1] S. Altizer, A. Dobson, P. Hosseini, P. Hudson, M. Pascual, and P. Rohani, Seasonality and dynamics of infectious disease, Ecology Letters, 9 (2006), pp. 467–484. [2] N. Baca¨ er and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), pp. 421–436. [3] T. Caraco, G. Gardner, W. Maniatty, E. Deelman, and B. K. Szymanski, Lyme disease: Self-regulation and pathogen invasion, J. Theoret. Biol., 193 (1998), pp. 561–575. [4] T. Caraco, S. Glavanakov, G. Chen, J. E. Flaherty, T. K. Ohsumi, and B. K. Szymanski, Stage-structured infection transmission and a spatial epidemic: A model for Lyme disease, The American Naturalist, 160 (2002), pp. 348–359. [5] Descriptive Epidemiology of Lyme Disease in Ontario: 1999–2004, http://www.phacaspc.gc.ca/publicat/ccdr-rmtc/06vol32/dr3221a-eng.php. [6] J. Fang, J. Wei, and X.-Q. Zhao, Spatial dynamics of a nonlocal and time-delayed reactionCdiffusion system, J. Differential Equations, 245 (2008), pp. 2749–2770. [7] M. W. Hirsch, H. L. Smith, and X.-Q. Zhao, Chain transitivity, attractivity, and strong repellors for semidynamical systems, J. Dynam. Differential Equations, 13 (2001), pp. 107–131. [8] X. Liang, Y. Yi, and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems, J. Differential Equations, 231 (2006), pp. 57–77. [9] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflow with applications, Comm. Pure Appl. Math., 60 (2007), pp. 1–40. [10] X. Liang and X.-Q. Zhao, Spreading speeds and traveling waves for abstract monostable evolution systems, J. Funct. Anal., 259 (2010), pp. 857–903. [11] Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math., 70 (2010), pp. 2023–2044. [12] Y. Lou and X.-Q. Zhao, Periodic Ross-Macdonald model with diffusion and advection, Appl. Anal., 89 (2010), pp. 1067–1089.




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[13] N. K. Madhav, J. S. Brownstein, J. I. Tsao, and D. Fish, A dispersal model for the range expansion of blacklegged tick (Acari: Ixodidae), J. Med. Entomol., 41 (2004), pp. 842–852. [14] R. H. Martin and H. L. Smith, Abstract functional differential equations and reactiondiffusion systems, Trans. Amer. Math. Soc., 321 (1990), pp. 1–44. [15] N. H. Ogden, L. R. Lindsay, G. Beauchamp, D. Charron, A. Maarouf, C. J. O’Callaghan, D. Waltner-Toews, and I. K. Barker, Investigation of relationships between temperature and developmental rates of tick Ixodes scapularis (Acari: Ixodidae) in the laboratory and field, J. Med. Entomol., 41 (2004), pp. 622–633. [16] N. H. Ogden, M. Bigras-Poulin, C. J. O’Callaghan, I. K. Barker, L. R. Lindsay, A. Maarouf, K. E. Smoyer-Tomic, D. Waltner-Toews, and D. Charron, A dynamic population model to investigate effects of climate on geographic range and seasonality of tick Ixodes scapularis, Int. J. Parasitol., 35 (2005), pp. 375–389. [17] N. H. Ogden, M. Bigras-Poulin, C. J. O’Callaghan, I. K. Barker, K. Kurtenbach, L. R. Lindsay, and F. Charron, Vector seasonality, host infection dynamics and fitness of pathogens transmitted by the tick Ixodes scapularis, Parasitology, 134 (2007), pp. 209– 227. [18] S. Randolph, Epidemiological uses of a population model for the tick Rhipicephalus appendiculatus, Trop. Med. Int. Health, 4 (1999), pp. A34–A42. [19] Temperature Statistics, http://www.theweathernetwork.com/statistics/CL6131983/caon0554. [20] A. I. Volpert, V. A. Volpert, and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monogr. 140, AMS, Providence, RI, 1994. [21] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dynam. Differential Equations, 20 (2008), pp. 699–717. [22] F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), pp. 496–516. [23] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. [24] X.-Q. Zhao, Global dynamics of a reaction and diffusion model for Lyme disease, J. Math. Biol., 65 (2012), pp. 787–808.
