A NONAUTONOMOUS EPIDEMIC MODEL WITH GENERAL ...

arXiv:1205.5740v2 [math.DS] 14 Jan 2013

A NONAUTONOMOUS EPIDEMIC MODEL WITH GENERAL INCIDENCE RATE AND QUARANTINE ´ CESAR M. SILVA Abstract. We obtain threshold conditions for eradication and permanence of the disease for a nonautonomous SIRQ model with time-dependent incidence rates, given by functions of all compartments in some general family. The threshold conditions are given by some numbers that play the role of the basic reproduction number. Additionally, we obtain simple threshold conditions in the asymptotically autonomous and periodic settings and show that our thresholds coincide with the basic reproduction number in some common autonomous examples.

1. Introduction Quarantine is one of the possible measures for reducing the transmission of diseases and has been used over the centuries to reduce the transmission of human diseases such as leprosy, plague, typhus, cholera, yellow fever and tuberculosis. The concept of quarantine dates back to the fourteenth century and is related to plague. In fact, in 1377, the Rector of the seaport of the old town of Ragusa (modern Dubrovnik) officially issued a thirty-day isolation period: ships coming from infected or suspected to be infected sites were to stay at anchor for thirty days before docking. A similar measure was adopted for land travelers but the period of time was enlarged to forty days. It is from the Italian word for forty quaranta - that comes the term quarantine [9]. Isolation is still nowadays an effective way of controlling infectious diseases. For some milder infectious diseases, quarantined individuals could be individuals that choose to stay home from school or work while for more severe infectious diseases people may be forced into quarantine. To study the effect of quarantine in the spread of a disease, we consider a SIRQ model. Thus, we divide the population into four compartments: the S compartment corresponding to not infected individuals that are susceptible to the disease, the I compartment corresponding to individuals that are infected and not isolated, the Q compartment corresponding to isolated individuals (Q stands for quarantine) and the R compartment including the recovered and immune individuals. We assume in this paper that the quarantined individuals are perfectly isolated from the others so that they do not infect the susceptibles and that the infection confers permanent immunity upon recovery. A major concern when studying epidemic models is to understand the asymptotic behavior of the compartments considered, particularly the infective compartment. Date: May 1, 2014. 2010 Mathematics Subject Classification. 92D30, 37B55, 34D20. Key words and phrases. Epidemic model, quarantine, non-autonomous, stability. Work partially supported by FCT thought CMUBI (project PEst-OE/MAT/UI0212/2011). 1

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´ CESAR M. SILVA

Frequently, in autonomous epidemic models, one of two situations occur: there is a disease-free equilibrium that is the unique equilibrium of the system or the diseasefree equilibrium coexists with an endemic equilibrium. The basic reproduction number is a fundamental tool to obtain the behavior for a given set of parameters: if the basic reproductive number is less than one the disease dies out and if this number exceeds one the disease remains in the population. Several authors studied autonomous epidemic models with a quarantine class. Namely, Feng and Thieme [8] proposed a SIQR model for childhood diseases and showed that isolation can be responsible for the existence of self-sustained oscillations. Their model includes a modified incidence function - the quarantine-adjusted incidence - where the number of per capita contacts between susceptibles and infectives is divided by the number of non isolated individuals. Some dynamical aspects of this SIQR model were studied by Wu and Feng [12]. Namely, these authors studied this model through unfolding analysis of a normal form derived from the model and found Hopf and homoclinic bifurcations associated to this unfolding. A generalized version of Feng and Thieme’s model, allowing infected individuals to recover without passing through quarantine and also disease related deaths, was proposed by Hethcote, Zhien and Shengbing in [5], where this model was compared to SIQR models with different incidence rates. A similar analysis is also undertaken in that paper for SIQS models with different incidence rates. The authors concluded in that work that the SIQR model with quarantine-adjusted incidence seems consistent with the sustained oscillations that are observed in real disease incidences but do not match every aspect of the observed data. Other types of epidemic models with quarantine were studied in the literature. A SEIQR model was considered by Gerberry and Milner in [4] where the authors studied the dynamics and used historical data to discuss the adequacy of the model in the context of childhood diseases. The global dynamic behavior of a different SEIQS epidemic model, with a nonlinear incidence rate of the form βSI, was analyzed by Yi, Zhao and Zhang in [17]. In another direction, Arino, Jordan and van den Driessche [1] studied the effect of quarantine in a general model of a disease that can be transmitted between different species and multiple patches (though in their setting quarantine is present only in the form of travel restriction between patches). A model with twelve different classes was proposed by Safi and Gumel [16] to study the impact of quarantine of latent cases, isolation of symptomatic cases and of an imperfect vaccine in the control of the spread of an infectious disease in the population. The same authors studied a delayed model including a quarantined class [15]. We emphasize that all the above models are autonomous models. However, it is well-known that fluctuations are very common in disease transmission. For instance, the opening and closing of schools tend to make the contact rates vary seasonally in childhood diseases [3] and periodic changes in birth rates of the population are also frequent. It is therefore natural to consider time-dependent parameters in epidemic models and define threshold conditions for extinction and persistence of the disease. Our objective in this paper is to consider a family nonautonomous SIQR models with a large family of contact rates given by general functions that are, in general, time-dependent and far from bilinear, and obtain numbers that play the role of the basic reproductive number in this setting. Related results for nonautonomous SIRVS models were obtained in [11, 13], thought the incidence rates considered in

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those papers are linear in the infectives while here we assume that the incidence rates are, in general, nonlinear functions of all compartments. The structure of the paper is as follows: in section 2 we introduce the model and the main definitions; in section 3 we state our main Theorem; in the next sections we look at some particular cases of our model, namely, in section 4, we particularize for the autonomous model, in section 5, we see that the asymptotically autonomous model has the same thresholds as the limiting autonomous model and in section 6 we obtain thresholds for the periodic model with constant death and recruitment rates; finally, in section 7 we prove some auxiliary results and the main theorem. 2. Generalized nonautonomous SIQR model We consider the following nonautonomous and general incidence rate SIRQ model that generalizes the three autonomous SIRQ models in [5],  ′ S = Λ(t) − ϕ(t, S, R, Q, I) − d(t)S    I ′ = ϕ(t, S, R, Q, I) − [γ(t) + σ(t) + d(t) + α (t)] I 1 , (1) ′  Q = σ(t)I − [d(t) + α (t) + ε(t)]Q 2    ′ R = γ(t)I + ε(t)Q − d(t)R

where S, R, Q and I correspond respectively to the susceptible, recovered, quarantine and infective compartments; Λ(t) is the recruitment rate of susceptible (births and immigration); ϕ(t, S, R, I) is the incidence (into the infective class); d(t) is the per capita natural mortality rate; γ(t) is the recovery rate; σ(t) is the removal rate from the infectives; α1 (t) is the disease related death in the infectives; α2 (t) is the disease related death in the quarantine class; and ε(t) is the removal rate from the quarantine class. Furthermore we assume that Λ, d, γ, σ, α1 , α2 and ε are continuous bounded and nonnegative functions on R+ 0 and that d = inf d(t) > 0 t≥0

and Λ = sup Λ(t) > 0. t≥0

For each δ ∈ R+ 0 define the set ∆δ = {(x1 , x2 , x3 , x4 ) ∈ R4 : 0 ≤ xi ≤ Λ/d + δ}. 5 Assume further that the function ϕ : (R+ 0 ) → R is continuous and nonnegative and that there are K > 0 and M > 0 such that, given 0 ≤ δ ≤ M , we have

ϕ(t, 0, y, w, z) = 0,

(2)

|ϕ(t, x1 , y, w, z) − ϕ(t, x2 , y, w, z)| ≤ K|x1 − x2 |z,

(3)

sup 0 0 such that Rp (λ) > 1 then the infectives I are permanent. 2. If there is a constant λ such that Re,1 (λ) < 1 or if Re,2 < 1 then the infectives I go to extinction and any disease free solution (S(t), 0, R(t), Q(t)) is globally attractive. 4. Autonomous model In this section we are going to consider the autonomous setting. That is, we are going to assume in system (1) that Λ, d, γ, δ, α1 , ε and α2 are constant functions, that Λ, d > 0, and that ϕ is independent of t. We obtain the autonomous system  ′ S = Λ − ϕ0 (S, R, Q, I) − dS    I ′ = ϕ (S, R, Q, I) − [γ + δ + d + α ] I 0 1 (8) ′  Q = δI − [d + α2 + ε]Q    ′ R = γI + εQ − dR 4 where ϕ0 : (R+ 0 ) → R is continuous, nonnegative and satisfies

ϕ0 (0, y, w, z) = 0,

|ϕ0 (x1 , y, w, z) − ϕ0 (x2 , y, w, z)| ≤ K|x1 − x2 |z,

and ϕ0 (x, 0, 0, τ ) ϕ0 (x, y, w, z) ϕ0 (x, 0, 0, τ ) ≥ ≥ inf , τ z 0 0 there is a Tε > 0 such that |x(t) − x1 (t)| < ε for every t ≥ Tε . Let δ ∈]0, M [ and choose εδ > 0 such that, for ε < εδ and t ≥ Tε , we have x(t), x1 (t), x(t) ± ε ∈ ]0, Λ/d + δ[. Leting a = x(t), b = x1 (t) and ε1 , ε2 = 0 we obtain by (6) Z t+λ Z t+λ Z t+λ bI,δ (s, x1 (s)) ds−λKε < bI,δ (s, x(s)) ds < bI,δ (s, x1 (s)) ds+λKε, t

t

t

for every t ≥ Tε . We conclude that, for every 0 < ε < εδ , Z t+λ Z t+λ bI,δ (s, x(s)) ds − lim inf bI,δ (s, x1 (s)) ds < λKε, lim inf t→+∞ t t→+∞ t

and thus

lim inf t→+∞

Z

t+λ

bI,δ (s, x(s)) ds = lim inf t→+∞

t

Z

t+λ

bI,δ (s, x1 (s)) ds.

t

Letting δ → 0 we conclude that rp (λ) (and thus Rp (λ)) is independent of the chosen solution. Using (7), the same reasoning shows that re,1 (λ) and re,2 (and thus Re,1 (λ) and Re,2 ) are also independent of the particular solution. This proves the lemma. 7.2. Proof of Theorem 1. Assume that Rp (λ) > 1 (and thus rp (λ) > 0) and let (S(t), I(t), R(t), Q(t)) be any solution of (1) with S(T0 ) > 0, I(T0 ) > 0, R(T0 ) > 0 and Q(T0 ) > 0 for some T0 ≥ 0. Let x∗ (t) be some fixed solution of (5) with x∗ (0) > 0. Then, by ii) in Proposition 2 we have x∗ (T0 ) > 0. Since rp (λ) > 0, using (6) we conclude that there are constants δ1 ∈]0, M [, N > 0 and T1 ≥ 0 such that Z t+λ (15) bI,δ1 (s, x∗ (s) − θ) ds > N t

for all t ≥ T1 and θ > 0 sufficiently small, say for 0 < θ ≤ ε¯. By iii) and iv) in Proposition 1 we may assume that for t ≥ T0 we have (S(t), I(t), R(t), Q(t)) ∈ ∆δ1 . Define   ε0 d N , ε¯ and ε1 = . (16) ε0 = min 4Kλ 2K(Λ/d + δ1 ) We will show that lim sup I(t) ≥ ε1 . (17) t→+∞

Assume by contradiction that (17) is not true. Then there exists T2 ≥ 0 satisfying I(t) < ε1 for all t ≥ T2 . Consider the auxiliary equation x′ = Λ(t) − d(t)x − K(Λ/d + δ1 )ε1 .

(18)

Let x¯(t) be the solution of (5) with x ¯(T0 ) = S(T0 ) and let x(t) be the solution of (18) with x(T0 ) = S(T0 ). By iv) in Proposition 2 and (16), for all t ≥ T0 , we have ε0 K (19) |x(t) − x ¯(t)| ≤ (Λ/d + δ1 )ε1 = . d 2 According to iii) in Proposition 2, x∗ (t) is globally uniformly attractive on R+ 0. Thus there exists T3 > 0 such that, for all t ≥ T3 , we have ε0 (20) |¯ x(t) − x∗ (t)| ≤ . 2

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From our assumptions about the function ϕ, we have ϕ(t, S(t), R(t), Q(t), I(t)) ≤ KS(t)I(t) < K(Λ/d + δ1 )ε1 , for all t ≥ max{T0 , T2 }, and thus, by the first equation in (1), S ′ > Λ(t) − d(t)S − K(Λ/d + δ1 )ε1 .

(21)

for all t ≥ max{T0 , T2 }. Comparing (18) and (21) we conclude that S(t) > x(t) for all t ≥ max{T0 , T2 }. Write T4 = max{T0 , T1 , T2 , T3 }. By (19) and (20) we have, for all t ≥ T4 , ε0 ≥ x∗ (t) − ε0 . (22) S(t) > x(t) ≥ x¯(t) − 2 By (22) and (6) in Lemma 1 we get, for all t ≥ T3 , bI (t, S(t)) > bI (t, x∗ (t) − ε0 ) − 2Kε0 . According to (16) we have ε0 ≤ ε¯ and by (15) we get   Z t 1 ∗ bI,δ1 (τ, x (τ ) − ε0 ) dτ > (t − T3 ) − 1 N. λ T3

(23)

(24)

By the second equation in (1) we obtain   ϕ(t, S(t), R(t), Q(t), I(t)) − (γ(t) + σ(t) + d(t) + α1 (t)) I(t) I ′ (t) = I(t) and thus, integrating from T3 to t and using (4), (23), (24) and (16), we have I(t) = I(T3 ) e ≥ I(T3 ) e > I(T3 ) e

Rt

ϕ(s,S(s),R(s),Q(s),I(s))/I(s)−(γ(s)+δ(s)+d(s)+α1 (s)) ds

Rt

bI,δ1 (τ,S(s)) ds

Rt

bI,δ1 (τ,x∗ (s)−ε0 ) ds−2Kε0 (t−T3 )

T3

T3

T3 N

> I(T3 ) e( λ −2Kε0 )(t−T3 )−N N

≥ I(T3 ) e 2λ (t−T3 )−N , and we conclude that I(t) → +∞. This contradicts the assumption that I(t) < ε1 for all t ≥ T2 . From this we conclude that (17) holds. Next we will prove that for some constant ℓ > 0 we have in fact lim inf I(t) > ℓ t→+∞

(25)

for every solution (S(t), I(t), R(t), Q(t)) with S(T0 ) > 0, I(T0 ) > 0, R(T0 ) > 0 and Q(T0 ) > 0. By (15), there is a λ1 ≥ λ such that for all ξ ≥ λ1 , t ≥ T1 and 0 < θ < ε¯, we have Z t+ξ (26) bI,δ1 (s, x∗ (s) − θ) ds > N. t

We proceed by contradiction. Assume that (25) does not hold. Then there exists a sequence of initial values (xn )n∈N , with xn = (Sn , Rn , Qn , In ) with Sn > 0, Rn > 0, Qn > 0 and In > 0 such that ε1 lim inf I(t, xn ) < 2 , t→+∞ n

´ CESAR M. SILVA

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where I(t, xn ) denotes the solution of (1) with initial conditions S(T1 ) = Sn , I(T1 ) = In , R(T1 ) = Rn and Q(T1 ) = Qn . By (17), given n ∈ N, there are two sequences (tn,k )k∈N and (sn,k )k∈N with T1 < sn,1 < tn,1 < sn,2 < tn,2 < · · · < sn,k < tn,k < · · · and lim sn,k = +∞, such that k→+∞

I(sk,n , xn ) =

ε1 , n

I(tk,n , xn ) =

ε1 n2

(27)

and

ε1 ε1 < I(t, xn ) < , 2 n n By the second equation in (1) we have

for all t ∈]sn,k , tn,k [.

(28)

I ′ (t, xn ) = I(t, xn )[(ϕ(t, S(t, xn ), R(t, xn ), Q(t, xn ), I(t, xn ))/I(t, xn )− − (γ(t) + δ(t) + d(t) + α1 (t))] ≥ −(γ(t) + σ(t) + d(t) + α1 (t))I(t, xn ) ≥ −γ0 I(t, xn ), where γ0 = sups≥0 {γ(s) + δ(s) + d(s) + α1 (s)}. Therefore we obtain Z tk,n ′ I (τ, xn ) dτ ≥ −γ0 (tk,n − sk,n ) sk,n I(τ, xn ) and thus I(tk,n , xn ) ≥ I(sk,n , xn ) e−γ0 (tk,n −sk,n ) . By (27) we get n1 ≥ e−γ0 (tk,n −sk,n ) and therefore we have log n tk,n − sk,n ≥ → +∞ (29) γ0 as n → +∞. According to (28), for all t ∈]sk,n , tk,n [, we have I(t, xn ) < ε1 /n < ε1 and we conclude that ϕ(t, S, R, Q, I) ≤ K(Λ/d + δ1 )ε1 for all t ∈]sk,n , tk,n [. Thus by (1) we obtain S ′ ≥ Λ(t) − d(t)S − K(Λ/d + δ1 )ε1 for all t ∈]sk,n , tk,n [. By comparison we have S(t, xn ) ≥ x(t) for all t ∈]sk,n , tk,n [, where x is the solution of (18) with x(sk,n ) = S(sk,n , xn ). Using (19) we obtain, for all t ∈ ]sk,n , tk,n [, ε0 (30) |x(t) − x ¯(t)| ≤ , 2 ∗ where x¯(t) is a solution of (5) with x¯(sk,n ) = S(sk,n , xn ). Since x (t) is globally uniformly attractive, there exists T ∗ > 0, independent of n and k, such that ε0 (31) |¯ x(t) − x∗ (t)| ≤ , 2 for all t ≥ sk,n + T ∗ . By (29) we can choose B > 0 such that tn,k − sn,k > λ1 + T ∗ for all n ≥ B. Given n ≥ B, by (26), (30) and (31) and by the second equation in (1) we get R

tk,n ε1 b (τ,S(τ,xn )) dτ ∗ sk,n +T ∗ I,δ1 = I(t , x ) = I(s + T , x ) e k,n n k,n n 2 n R ε1 tk,n ∗ bI,δ1 (τ,x∗ (τ )−ε0 ) dτ ε1 ≥ 2 e sk,n +T > 2. n n

13

This leads to a contradiction and establishes that lim inf I(t) > ℓ > 0. Thus, 1. in t→+∞

Theorem 1 is established. Assume now that Re,1 (λ) < 0 for some λ > 0 or that Re,2 < 0. Then we can choose δ1 ∈ [0, M ], N > 0 and T0 > 0 such that Z t+λ (32) bS,δ1 (s, x∗ (s) + θ) ds < −N t

or

Z 1 t (33) bS,δ1 (s, x∗ (s) + θ) ds < −N t 0 for all t ≥ T0 and θ > 0 sufficiently small. By iii) in Proposition 1 we can assume that (S(t), I(t), R(t), Q(t)) ∈ ∆δ1 for t ≥ T0 . Given a solution of (1) with S(T0 ) > 0, R(T0 ) > 0, Q(T0 ) > 0 and I(T0 ) > 0 we have S ′ (t) ≤ Λ(t) − d(t)S for all t ≥ T0 . For any solution x(t) of (5) with x(T0 ) = S(T0 ) we have by comparison S(t) ≤ x(t) for all t ≥ T0 . Let ε2 > 0 satisfy   1 N min ,1 . (34) ε2 = 4K λ

Since x∗ (t) is globally uniformly attractive, there exists T1 > T0 such that, for all t > T1 , we have |x(t) − x∗ (t)| < ε2 . Therefore, for all t ≥ T1 , we have S(t) ≤ x(t) ≤ x∗ (t) + ε2 .

(35)

Thus, by the second equation in (1) and by (4) we conclude that ϕ(t, S(t), R(t), Q(t), I(t)) I ′ (t) = − (γ(t) + σ(t) + d(t) + α1 (t)) I(t) I(t) ϕ(t, S(t), 0, 0, τ ) ≤ sup − (γ(t) + σ(t) + d(t) + α1 (t)) τ 0 0 or that Re,2 < 0, we let (S(t), R(t), Q(t), I(t)) be a solution of (1) with non-negative initial conditions and (S0 (t), 0, R0 (t), Q0 (t)) be a disease-free solution of (1) with non-negative initial conditions. By iii) in Proposition 1 we can assume that for some δ ∈]0, M [, we have (S(t), I(t), R(t), Q(t)), (S0 (t), 0, R0 (t), Q0 (t)) ∈ ∆δ1 for t ≥ T0 . Thus S0 (t), R0 (t), Q0 (t), S(t), R(t), Q(t) < Λ/d + δ1 , for t ≥ T0 . By Theorem 2 we have lim I(t) = 0. Therefore given ε > 0 there t→+∞

exists Tε ≥ T0 such that I(t) < ε, S(t), R(t), Q(t) < Λ/d + δ1 for t ≥ Tε . Thus Λ(t) − d(t)S ≥ S ′ (t) ≥ Λ(t) − d(t)S − K(Λ/d + δ1 )ε, for all t ≥ Tε . By comparison we get x1 (t) ≥ S(t) ≥ x2 (t)

(36)

where x1 (t) is the solution of (5) and x2 (t) is the solution of (18) with x1 (Tε ) = x2 (Tε ) = S(Tε ). By iv) in Proposition 2 we obtain, for all t ≥ T1 , K(Λ/d + δ1 )ε , (37) d Since S0 (t) is a solution of (5), it is globally uniformly attractive, by iii) in Proposition 2. Thus, there is T2 ≥ Tε such that |x2 (t) − x1 (t)| ≤

|x1 (t) − S0 (t)| ≤ ε

(38)

for all t ≥ T2 . By (36), (37) and (38) we conclude that   K(Λ/d + δ1 ) K(Λ/d + δ1 )ε ≥ S0 (t) − + 1 ε, S(t) ≥ x2 (t) ≥ x1 (t) − d d

(39)

for all t ≥ T2 , and also, by (36) and (38),

S(t) ≤ S0 (t) + ε

(40)

for all t ≥ T2 . By (39) and (40) we obtain, since ε > 0 can be made arbitrarily small by taking t ≥ Tε , we have lim |S(t) − S0 (t)| = 0.

(41)

t→+∞

Let T3 ≥ Tε be such that |I(t) − I0 (t)| < ε for every t ≥ T3 and write u(t) = Q(t) − Q0 (t). By the third equation in (1), we get −ζu(t) ≤ u′ (t) ≤ εσ1 − ζu(t), where ζ = sups≥0 (d(s) + α2 (s) + ε(s)) and σ1 = supt≥0 σ(t). We obtain u(T3 ) e−ζ(t−T3 ) ≤ u(t) ≤ u(T3 ) e−ζ(t−T3 )−

εσ1 ζ

.

and thus lim |Q(t) − Q0 (t)| = lim u(t) = 0.

t→+∞

(42)

t→+∞

Let T4 ≥ T1 be such that |Q(t) − Q0 (t)| < ε for every t ≥ T4 and write w(t) = R(t) − R0 (t). By the fourth equation in (1) and since I0 (t) = 0 and for all t ≥ 0, we get −εq − d1 w(t) ≤ w′ (t) ≤ εp − dw(t), where p = supt≥0 (γ(t) + ε(t)), q = supt≥0 ε(t) and d1 = supt≥0 d(t). We obtain w(T4 ) e

−d1 (t−T4 )− dεq

1

εp

≤ w(t) ≤ w(T4 ) e−d(t−T4 )− d .

15

and thus lim |R(t) − R0 (t)| = lim w(t) = 0.

t→+∞

t→+∞

(43)

By (41), (42), (43) and since all the infectives go to extinction, we conclude that any disease-free solution (S0 (t), 0, R0 (t), Q0 (t)) with nonnegative initial conditions is globally attractive. We obtain 2. in Theorem 1 and this concludes our proof. References [1] J. Arino, R. Jordan, P. van den Driessche, Quarantine in a multi-species epidemic model with spatial dynamics, Math. Biosc. 206, 4660 (2007) [2] C. Castillo-Chavez, H.R. Thieme, Asymptotically autonomous epidemic models, in: O. Arino, D. E. Axelrod, M. Kimmel, M. Langlais (Eds.), Mathematical Population Dynamics: Analisys and Heterogenity, Wuerz, Winnipeg, Canada, 1995, p. 33. [3] K. Dietz, The incidence of infectious diseases under the influence of seasonal fluctuations, Lecture Notes in Biomathematics, vol. 11, pp. 1-5. Berlin-Heidelberg-New York, Springer, 1976. [4] , D. J. Gerberry and F. A. Milner, An SEIQR model for childhood diseases, J. Math. Biol., 59, 4 (2009) [5] H. Hethcote, M. Zhien, L. Shengbing, Effects of quarantine in six endemic models for infectuous diseases, Math. Biosc., 180, 141-160 (2002) [6] H. Hethcote, P. van den Driessche, Some epidemiological model with nonlinear incidence, J. Math. Biol., 29, 271-287 (1991) [7] K. Mischaikow, H. Smith and H. R. Thieme, Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347, 1669-1685 (1995) [8] Z. Feng and H. R. Thieme, Recurrent Outbreaks of Childhood Diseases Revisited: The Impact of Isolation, Math. Biosc., 93-130 (1995) [9] G. F. Gensini, M. H. Yacoub, A. A. Conti, The concept of quarantine in history: from plague to SARS, Journal of infection, 49, 257-261 (2004) [10] L. Markus, Asymptotically autonomous differential systems, Contributions to the theory of Nonlinear Oscillations 111 (S. Lefschetz, ed.), Ann. Math. Stud., 36, Princeton University Press, Princeton, NJ, 17-29 (1956) [11] E. Pereira, C. M. Silva and J. A. L. Silva, A Generalized Non-Autonomous SIRVS Model, Math. Meth. Appl. Sci., to appear [12] L. Wu, Z. Feng, Homoclinic Bifurcation in a SIQR Model for Childhood Diseases, J. Differential Equations, 168, 150-167 (2000) [13] T. Zhang, Z. Teng and S. Gao, Threshold conditions for a nonautonomous epidemic model with vaccination, Applicable Analysis, 87, 181-199 (2008) [14] W. M. Liu, H. W. Hethcote, S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25, 359380 (1987) [15] M. A. Safi, A. B. Gumel, The effect of incidence functions on the dynamics of a quarantine/isolation model with time delay, Nonlinear Anal. - Theory Methods and Applications, 12, 215-235 (2011) [16] M. A. Safi, A. B. Gumel, Mathematical analysis of a disease transmission model with quarantine, isolation and an imperfect vaccine, Computers & Mathematics with Applications, 61, 30443070 (2011) [17] N. Yi, Z. Zhao and Q, Zhang, Bifurcations of an SEIQR epidemic model, International Journal of Informations and Systems Sciences, 5, 296-310 (2009) ´ tica, Universidade da Beira Interior, 6201-001 C. Silva, Departamento de Matema ˜ , Portugal Covilha E-mail address: [email protected]

arXiv:1205.5740v2 [math.DS] 14 Jan 2013

A NONAUTONOMOUS EPIDEMIC MODEL WITH GENERAL INCIDENCE AND QUARANTINE ´ CESAR M. SILVA Abstract. We obtain conditions for eradication and permanence of the infectives for a nonautonomous SIQR model with time-dependent parameters, that are not assumed to be periodic. The incidence is given by functions of all compartments and the threshold conditions are given by some numbers that play the role of the basic reproduction number. We obtain simple threshold conditions in the autonomous, asymptotically autonomous and periodic settings and show that our thresholds coincide with the ones already established. Additionally, we obtain threshold conditions for the general nonautonomous model with mass-action, standard and quarantine-adjusted incidence.

1. Introduction Quarantine is one of the possible measures for reducing the transmission of diseases and has been used over the centuries to reduce the transmission of human diseases such as leprosy, plague, typhus, cholera, yellow fever and tuberculosis. The concept of quarantine dates back to the fourteenth century and is related to plague. In fact, in 1377, the Rector of the seaport of the old town of Ragusa (modern Dubrovnik) officially issued a thirty-day isolation period: ships coming from infected or suspected to be infected sites were to stay at anchor for thirty days before docking. A similar measure was adopted for land travelers but the period of time was enlarged to forty days. It is from the Italian word for forty quaranta - that comes the term quarantine [9]. Isolation is still nowadays an effective way of controlling infectious diseases. For some milder infectious diseases, quarantined individuals could be individuals that choose to stay home from school or work while for more severe infectious diseases people may be forced into quarantine. To study the effect of quarantine in the spread of a disease, we consider a SIQR model. Thus, we divide the population into four compartments: the S compartment corresponding to not infected individuals that are susceptible to the disease, the I compartment corresponding to individuals that are infected and not isolated, the Q compartment corresponding to isolated individuals (Q stands for quarantine) and the R compartment including the recovered and immune individuals. We assume in this paper that the quarantined individuals are perfectly isolated from the others so that they do not infect the susceptibles and that the infection confers permanent immunity upon recovery. Date: May 1, 2014. 2010 Mathematics Subject Classification. 92D30, 37B55, 34D20. Key words and phrases. Epidemic model, quarantine, non-autonomous, stability. Work partially supported by FCT thought CMUBI (project PEst-OE/MAT/UI0212/2011). 1

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´ CESAR M. SILVA

A major concern when studying epidemic models is to understand the asymptotic behavior of the compartments considered, particularly the infective compartment. Frequently, in autonomous epidemic models, one of two situations occur: there is a disease-free equilibrium that is the unique equilibrium of the system or the diseasefree equilibrium coexists with an endemic equilibrium. The basic reproduction number is a fundamental tool to obtain the behavior for a given set of parameters: if the basic reproductive number is less than one the disease-free equilibrium is locally asymptotically stable and if this number exceeds one the disease remains in the population. Several authors studied autonomous epidemic models with a quarantine class. Namely, Feng and Thieme [8] proposed a SIQR model for childhood diseases and showed that isolation can be responsible for the existence of self-sustained oscillations. Their model includes a modified incidence function - the quarantine-adjusted incidence - where the number of per capita contacts between susceptibles and infectives is divided by the number of non isolated individuals. Some dynamical aspects of this SIQR model were studied by Wu and Feng [19]. Namely, these authors studied this model through unfolding analysis of a normal form derived from the model and found Hopf and homoclinic bifurcations associated to this unfolding. A generalized version of Feng and Thieme’s model, allowing infected individuals to recover without passing through quarantine and also disease related deaths, was proposed by Hethcote, Zhien and Shengbing in [5], where this model was compared to SIQR models with different incidences. A similar analysis is also undertaken in that paper for SIQS models with different incidences. The authors concluded in that work that the SIQR model with quarantine-adjusted incidence seems consistent with the sustained oscillations that are observed in real disease incidences but do not match every aspect of the observed data. Other types of epidemic models with quarantine were studied in the literature. A SEIQR model was considered by Gerberry and Milner in [4] where the authors studied the dynamics and used historical data to discuss the adequacy of the model in the context of childhood diseases. The global dynamic behavior of a different SEIQS epidemic model, with a nonlinear incidence of the form βSI, was analyzed by Yi, Zhao and Zhang in [16]. In another direction, Arino, Jordan and van den Driessche [1] studied the effect of quarantine in a general model of a disease that can be transmitted between different species and multiple patches (though in their setting quarantine is present only in the form of travel restriction between patches). A model with twelve different classes was proposed by Safi and Gumel [15] to study the impact of quarantine of latent cases, isolation of symptomatic cases and of an imperfect vaccine in the control of the spread of an infectious disease in the population. The same authors studied a delayed model including a quarantined class [14]. We emphasize that all the above models are autonomous models. However, it is well-known that fluctuations are very common in disease transmission. For instance, the opening and closing of schools tend to make the contact rates vary seasonally in childhood diseases [3] and periodic changes in birth rates of the population are also frequent. It is therefore natural to consider time-dependent parameters in epidemic models and define threshold conditions for extinction and persistence of the disease. Our objective in this paper is to consider a family nonautonomous SIQR models with a large family of contact rates given by general functions that are, in general,

3

time-dependent and far from bilinear, and obtain numbers that play the role of the basic reproductive number in this setting. Related results for nonautonomous SIRVS models were obtained in [12, 20], thought the incidences considered in those papers are linear in the infectives while here we assume that the incidences are, in general, nonlinear functions of all compartments. Note that in [18, 13] some methods were developed to obtain thresholds for general periodic models. These methods apply to our setting when we assume the coefficients to be periodic. We emphasise that this is not necessarily our case, since our coefficients need not be periodic. Additionally, our thresholds are given by explicit formulas (see section 3). The structure of the paper is as follows: in section 2 we introduce the model and the main definitions; in section 3 we state our main Theorem; in the next sections we look at some particular cases of our model, namely, in section 4 we particularize for the autonomous model, in section 5 we see that the asymptotically autonomous model has the same thresholds as the limiting autonomous model, in section 6 we obtain thresholds for the periodic model with constant death and recruitment rates and, in section 7 we consider the general nonautonomous model with mass-action, standard and quarantine-adjusted incidences; in section 8 we prove some auxiliary results and the main theorem; finally, in section 9 we present some simulations that illustrate our result. 2. Generalized nonautonomous SIQR model We consider the following nonautonomous and general incidence SIQR model that generalizes the three autonomous SIQR models in [5],  ′ S = Λ(t) − ϕ(t, S, R, Q, I) − d(t)S    I ′ = ϕ(t, S, R, Q, I) − [γ(t) + σ(t) + d(t) + α (t)] I 1 , (1) Q′ = σ(t)I − [d(t) + α2 (t) + ε(t)]Q    ′ R = γ(t)I + ε(t)Q − d(t)R

where S, R, Q and I correspond respectively to the susceptible, recovered, quarantine and infective compartments; Λ(t) is the recruitment of susceptible (births and immigration); ϕ(t, S, R, Q, I) is the incidence (into the infective class); d(t) is the per capita natural mortality rate; γ(t) is the recovery rate; σ(t) is the removal rate from the infectives; α1 (t) is the disease related death in the infectives; α2 (t) is the disease related death in the quarantine class; and ε(t) is the removal rate from the quarantine class. We will assume that ϕ, Λ, d, γ, σ, α1 , α2 and ε are continuous bounded and nonnegative functions on R+ 0 and that there are ωd , ωΛ > 0 such that d− ωd > 0 where we are using the notation Z t+ωh h(s) ds h− = lim inf ωh t→+∞

t

and Λ− ωΛ > 0,

and h+ ωh = lim sup t→+∞

(2) Z

t+ωh

h(s) ds,

t

that we will keep on using throughout the paper. For bounded h we will also use the notation hS = sup h(t) and hI = inf (t). t>0

t>0

Next we state some simple facts about problem (1).

´ CESAR M. SILVA

4

Proposition 1. We have the following: i) all solutions (S(t), I(t), Q(t), R(t)) of (1) with S(t0 ) > 0, I(t0 ) > 0, Q(t0 ) > 0 and R(t0 )) > 0 verify S(t) > 0, I(t) > 0, Q(t) > 0 and R(t)) > 0 for all t > t0 ; ii) all solutions (S(t), I(t), Q(t), R(t)) of (1) with S(t0 ) > 0, I(t0 ) > 0, Q(t0 ) > 0 and R(t0 )) > 0 verify S(t) > 0, I(t) > 0, Q(t) > 0 and R(t)) > 0 for all t > t0 ; iii) If (S(t), I(t), Q(t), R(t)) is a solution of (1) with nonnegative initial conditions then there is a constant K > 0 such that lim sup S(t) + I(t) + Q(t) + R(t) 6 K. t→+∞

iv) There are constants C, T > 0 such that, if I(t) < β for all t > T then Q(t) < Cβ

and

R(t) < Cβ

(3)

for all t sufficiently large. Proof. Properties i), ii) are a simple consequence of the nonnegativeness and the boundedness of the parameter functions. Using (2), property iii) follows easily. Assuming that I(t) < β for all t > Te, by the third equation in (1), we get Q′ (t) < βσ(t) − [d(t) + α2 (t) + ε(t)]Q(t)

−

d and thus, by (2), for some T > Te sufficiently large and C0 = 2ωωdd , we have Z t R Rt t Q(t) = e− T d(s)+α2 (s)+ε(s) ds Q(T ) + e− u d(s)+α2 (s)+ε(s) ds βσ(u) du T Z t 6 e−C0 (t−T −ωd ) Q(T ) + βσS e−C0 (t−u−ωd ) du

6 e−C0 (t−T −ωd ) K +

T C 0 ωd

βσS e C0

  1 − e−C0 (t−T )

and thus, taking C1 = σS eC0 ωd /(2C0 ), we have Q(t) 6 C1 β for t sufficiently large. A similar argument applied to the third equation in (1) shows that there is a constant C2 > 0 (independent of β) such that R(t) 6 C2 β for t sufficiently large. Choosing C = max{C1 , C2 } we obtain property iv).  Consider the auxiliary equation x′ = Λ(t) − d(t)x,

(4)

Proposition 2. We have the following: i) Given t0 > 0, all solutions x(t) of equation (4) with initial condition x(t0 ) > 0 are nonnegative for all t > 0; ii) Given t0 > 0, all solutions x(t) of equation (4) with initial condition x(t0 ) > 0 are positive for all t > 0; iii) Given t0 > 0 and a solution x(t) of equation (4) with initial condition x(t0 ) > 0, there are M > 0 and t1 > 0 such that x(t) > M for all t > t1 ; iv) Each fixed solution x(t) of (4) with initial condition x(t0 ) > 0 is bounded and globally uniformly attractive on [0, +∞[; v) There is a constant D > 0 such that, if x(t) is a solution of (4) and x ˜(t) is a solution of the system x′ = Λ(t) − d(t)x + f (t)

(5)

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with x ˜(t0 ) = x(t0 ) then x(t) − x(t)| 6 D sup |f (t)|. sup |˜ t>t0

t>t0

Proof. For t0 > 0, the solution of (4) with initial condition x(t0 ) = x0 > 0 is given by Z t R R t − t d(s) ds e− u d(s) ds Λ(u) du. (6) x(t) = e t0 x0 + t0

Since t 7→ Λ(t) is nonnegative, equation (6) immediately implies i) and ii). Since t 7→ d(t) is bounded, by and (6) and (2) we obtain Z t Rt Rt − t−ω d(s) ds Λ e− u d(s) ds Λ(u) du x(t) = e x(t − ωΛ ) + > e−dS ωΛ > e−dS ωΛ

Z

t−ωΛ

t

Λ(u) du

t−ωΛ Λ− ωΛ /2,

for all t sufficiently large. Thus we have iii). By (2), since t 7→ Λ(t) is bounded, we get for all t sufficiently large Z t R R t − t d(s) ds e− u d(s) ds Λ(u) du x0 + x(t) = e t0 < x0 + ΛS

Z

t0

t

t0

 −  d t−u −1 du exp − ωd 2

2ωd − < x0 + − ed /2 ΛS , d

and we conclude that each solution x(t) of (4) with initial condition x(t0 ) > 0 is bounded. Additionally, if x, x1 are two solutions of (4) with initial condition x0 = x(t0 ) > 0 and x1,0 = x1 (t0 ) > 0 we have |x(t) − x1 (t)| 6 e

−

Rt

d(s) ds

t0

|x0 − x1,0 |,

and we get iv). Finally, if x(t) is a solution of (4) and x ˜(t) is a solution of (5) with x ˜(t0 ) = x(t0 ) then, setting u(t) = x ˜(t) − x(t), we obtain the problem u′ = −d(t)u + f (t), with u(t0 ) = 0. The above problem has solution Z t R t e− u d(s) ds f (u) du u(t) = t0

6 sup |f (t)| t>t0

Z

t

e−

Rt u

d(s) ds

du

t0

6 D sup |f (t)| t>t0

where D =

2ωd d−

exp



 −

d 2

. Thus, we obtain v).



´ CESAR M. SILVA

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For each θ, δ ∈ R+ 0 with δ > θ define the set ∆θ,δ = {(x1 , x2 , x3 , x4 ) ∈ R4 : θ 6 x1 6 δ ∧ 0 6 xi 6 δ, i = 2, 3, 4}. We note that every solution (S(t), I(t), Q(t), R(t)) of our system stays in the region ∆0,K (where K is givel by iii) in Proposition 1) for every t ∈ R+ 0 sufficiently large. We need some additional assumptions about our system. Assume that: H1) the functions x 7→ ϕ(t, x, y, w, z) are non decreasing and ϕ(t, 0, y, w, z) = 0; H2) given θ > 0 there is Kθ > 0 such that |ϕ(t, x1 , y, w, z) − ϕ(t, x2 , y, w, z)| 6 Kθ |x1 − x2 |z, for each (t, x1 , y, w, z), (t, x2 , y, w, z) ∈ R+ 0 × ∆θ,K ; H3) There is N > 0 such that, for every δ ∈ ]0, K], y, w, z ∈ ]0, δ] and t ∈ R+ 0 we have ϕ(t, x, y, w, z) ϕ(t, x, 0, 0, τ ) 6 < N. inf 0 0 (note that we require that m1 and m2 are independent of the given solution with positive initial condition in I); ii) the infectives I go to extinction in system (1) if lim I(t) = 0 for all solutions t→∞

of (1). 3. Statment of the results For each solution x(t) of (4) with x(0) > 0, define the function bδ (t, x(t)) =

ϕ(t, x(t), 0, 0, δ) − (γ(t) + σ(t) + d(t) + α1 (t)) δ

and the numbers rp (λ) = lim inf

t→ +∞

re (λ) = lim sup t→ +∞

Z

t+λ

lim inf bδ (s, x(s)) ds, δ→0

t

Z

Rp (λ) = erp (λ)

t

t+λ

lim sup bδ (s, x(s)) ds, δ→0

and Re (λ) = ere (λ) .

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Contrarily to what one could expect, the next technical lemma that will be proved in Section 8 shows that the numbers above do not depend on the particular solution x(t) of (4) with x(0) > 0. The numbers Rp (λ), Re (λ) play the role of the basic reproduction number (defined for autonomous systems and also called quarantine number) in the non-autonomous setting. We have the following result whose proof will be given in section 8. Lemma 1. We have the following: 1. Let ε > 0 be sufficiently small and 0 < θ 6 K. If a, b ∈ ]θ, K[

and

a − b < ε,

then |bδ (t, a) − bδ (t, b)| < εKθ .

(7)

2. The numbers Rp (λ) and Re (λ) are independent of the particular solution x(t) with x(0) > 0 of (4). Note that, by (7) we have lim inf bδ (t, a) − lim inf bδ (t, b) < εKθ δ→0

and

δ→0

lim sup bδ (t, a) − lim sup bδ (t, b) < εKθ . δ→0 δ→0

(8)

(9)

We now state our main theorem on the permanence and extinction of the infectives in system (1). A proof of this result will be given in section 8. Theorem 1. We have the following for the system (1). 1. If there is a constant λ > 0 such that Rp (λ) > 1 then the infectives I are permanent. 2. If there is a constant λ > 0 such that Re (λ) < 1 then the infectives I go to extinction and any disease free solution (S(t), 0, R(t), Q(t)) is globally attractive. 4. Autonomous model In this section we are going to consider the autonomous setting. Namely, we are going to assume in system (1) that Λ, d, γ, σ, α1 , ε and α2 are constant functions, that Λ, d > 0, and that ϕ is independent of t. We obtain the autonomous system  ′ S = Λ − ϕ0 (S, R, Q, I) − dS    ′  I = ϕ0 (S, R, Q, I) − [γ + σ + d + α1 ] I (10) ′  Q = σI − [d + α2 + ε]Q    ′ R = γI + εQ − dR

4 where ϕ0 : (R+ 0 ) → R is continuous, nonnegative and satisfies H1), H2) and H3). In this setting we have that the auxiliary equation (4) admits the constant solution x(t) = Λ/d. Thus we obtain   ϕ0 (Λ/d, 0, 0, δ) − (γ + σ + d + α1 ) λ, Rp (λ) = erp (λ) rp (λ) = lim inf δ→0 δ

´ CESAR M. SILVA

8

and

  ϕ0 (Λ/d, 0, 0, δ) − (γ + σ + d + α1 ) λ, re (λ) = lim sup δ δ→0

Re,1 (λ) = ere (λ) .

It is now easy to establish a result that is a version of Theorem 1 in the particular case of autonomous systems. Define RpAUT = lim inf δ→0

ϕ0 (Λ/d, 0, 0, δ) δ(γ + σ + d + α1 )

and ReAUT = lim sup δ→0

ϕ0 (Λ/d, 0, 0, δ) . δ(γ + σ + d + α1 )

RpAUT

For any λ > 0, we have Rp (λ) > 1 if and only if > 1 and Re (λ) < 1 if and only if ReAUT < 1 and this implies the following result. Theorem 2. We have the following for the autonomous system (10). 1. If RpAUT > 1 then the infectives I are permanent; 2. If ReAUT < 1 then the infectives I go to extinction and any disease free solution is globally attractive. Note that, if the incidence has the particular form ϕ0 (S, R, Q, I) = ψ(S, R, Q)g(I)I,

(11)

where g is a continuous, bounded and nonnegative function, then RpAUT =

ψ(Λ/d, 0, 0) lim inf g(δ) δ→0

γ + σ + d + α1

ψ(Λ/d, 0, 0) lim sup g(δ) and ReAUT =

δ→0

γ + σ + d + α1

and we conclude that the infectives are permanent (resp. go to extinction) if lim inf g(δ) > δ→0

Naturally, if

γ + σ + d + α1 ψ(Λ/d, 0, 0)

(resp. lim sup g(δ) < δ→0

γ + σ + d + α1 ). ψ(Λ/d, 0, 0)

  γ + σ + d + α1 ∈ lim inf g(δ), lim sup g(δ) δ→0 ψ(Λ/d, 0, 0) δ→0 we get no information about the asymptotic behavior of the infectives. Assuming now that g is constant, we have RpAUT = ReAUT and we can recover the results obtained in [5]. In fact, the autonomous SIQR model with mass-action incidence (ϕ0 (S, R, Q, I) = βSI), the autonomous SIQR model with standard incidence (ϕ0 (S, R, Q, I) = βSI/(S + I + Q)) and the autonomous SIQR model with quarantine-adjusted incidence (ϕ0 (S, R, Q, I) = βSI/(S + I + Q + R)) are all in the conditions we have assumed and our threshold values for those model are RpAUT = ReAUT = βΛ/[d(γ + σ + d + α1 )] for mass-action incidence and RpAUT = ReAUT = β/(γ + σ + d + α1 ) for standard and quarantine-adjusted incidences. For these models, the numbers RpAUT = ReAUT coincide with the quarantine reproduction number obtained in [5]. Note that, to put the referred autonomous models considered in [5] in our context, we need to have the standard incidence and the quarantine-adjusted incidence continuous in ∆δ,0 . This constitutes no problem since the functions ϕ0 associated to these incidences can be extended to a continuous function in ∆δ,0 , setting ϕ0 (0, R, 0, 0) = 0 for the standard incidence and ϕ0 (0, 0, 0, 0) = 0 for the quarantine-adjusted incidence. I p−1 Set ψ(S, Q, R) = S and g(I) = 1+αI q with p, q > 0, α > 0, in (11). This family of contact rates was considered for instance in [10, 6]. Our threshold conditions show

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that if p < 1 then the disease is permanent (independently of the other parameters) and if p = 1, setting ψ(Λ/d, 0, 0) R= , γ + σ + d + α1 the disease is permanent if R > 1 and goes to extinction if R < 1. 5. Asymptotically autonomous model In this section we are going to consider the asymptotically autonomous SIQR model. That is, additionally to the assumptions on Theorem 1, we are going to assume for system (1) that there is a continuous function ϕ0 such that lim ϕ(t, S, R, Q, I) = ϕ0 (S, R, Q, I)

t→∞

for each (S, R, Q, I) ∈ R4 and that the time-dependent parameters are asymptotically constant: β(t) → β, Λ(t) → Λ, d(t) → d, γ(t) → γ, σ(t) → σ, α1 (t) → α1 , α2 (t) → α2 and ε(t) → ε as t → +∞. Denoting by F (t, S, R, Q, I) the right hand side of (1) and by F0 (S, R, Q, I) the right hand side of the limiting system, that is of (10), we also need to assume that lim F (t, S, R, Q, I) = F0 (S, R, Q, I),

t→+∞

4 with uniform convergence on every compact set of (R+ 0 ) , and that (S, R, Q, I) 7→ F (t, S, R, Q, I) and (S, R, Q, I) 7→ F0 (S, R, Q, I) are locally Lipschitz functions. There is a general setting that will allow us to study this case. Let f : R×Rn → R and f0 : Rn → R be continuous and locally Lipschitz in Rn . Assume also that the nonautonomous system x′ = f (t, x) (12)

is asymptotically autonomous with limit equation x′ = f0 (x),

(13)

that is, assume that f (t, x) → f0 (x) as t → +∞ with uniform convergence in every compact set of Rn . The following theorem is a particular case of a result established in [11] (for related results and applications see for example [2, 7]). Theorem 3. Let Φ(t, t0 , x0 ) and ϕ(t, t0 , y0 ) be solutions of (12) and (13) respectively. Suppose that e ∈ Rn is a locally stable equilibrium point of (13) with attractive region   W (e) =

y ∈ Rn : lim ϕ(t, t0 , y) = e t→+∞

and that WΦ ∩ W (e) 6= ∅, where WΦ denotes the omega limit of Φ(t, t0 , x0 ). Then lim Φ(t, t0 , x0 ) = e.

t→+∞

Since (R+ )4 is the attractive region for any solution of system (10) with initial condition in (R+ )4 and the omega limit of every orbit of the asymptotically autonomous system with I(t0 ) > 0 is contained in (R+ )4 , we can use Theorem 2 and Theorem 3 to obtain the following result. Theorem 4. We have the following for the asymptotically autonomous systems above. 1. If RpAUT > 1 then the infectives I are permanent;

´ CESAR M. SILVA

10

2. If ReAUT < 1 then the infectives I go to extinction and any disease free solution is globally attractive. 6. Periodic model with constant natural death and recruitment In this section we are going to consider a periodic SIQR model. Additionally to our assumptions on the function ϕ and the parameter functions in Theorem 1, we are going to assume in system (1) that Λ(t) = Λ and d(t) = d are constant functions, that there is a T > 0 such that ϕ(t, S, R, Q, I) = ϕ(t + T, S, R, Q, I) and that the remaining time-dependent parameter functions are periodic functions with period T . We have in this case the constant solution x(t) = Λ/d and therefore Z t+T ϕ(s, Λ/d, 0, 0, δ) − (γ(s) + σ(s) + d + α1 (s)) ds rp (λ) = lim inf lim inf t→+∞ t δ→0 δ Z T ϕ(s, Λ/d, 0, 0, δ) = lim inf ds − (¯ γ+σ ¯+d+α ¯ 1 )T δ→0 δ 0 and similarly

Z

T

ϕ(s, Λ/d, 0, 0, δ) ds − (¯ γ+σ ¯ +d+α ¯1 )T, δ δ→0 0 RT where f¯ denotes the average of f in the interval [0,T]: f¯ = T1 0 f (s) ds. Define Z ϕ(s, Λ/d, 0, 0, δ) 1 T P ER ds lim inf Rp (λ) = T 0 δ→0 δ(¯ γ+σ ¯+d+α ¯1) re (λ) =

lim sup

and

ReP ER (λ)

1 = T

Z

0

T

lim sup δ→0

ϕ(s, Λ/d, 0, 0, τ ) ds. τ (¯ γ +σ ¯+d+α ¯1)

Theorem 5. We have the following for the periodic system with constant recruitment and death rates. 1. If RpP ER > 1 then the infectives I are permanent; 2. If ReP ER < 1 then the infectives I go to extinction and any disease free solution is globally attractive. If we assume in the periodic model above that the incidence has the particular form ϕ(t, S, R, Q, I) = β(t)SI with β(t + T ) = β(t), then we obtain ¯ βΛ (14) RdP ER (λ) = ReP ER (λ) = d(¯ γ +σ ¯ +d+α ¯1 ) and if we assume in that model that ϕ(t, S, R, Q, I) = β(t)SI/(S + R + Q + I) or that ϕ(t, S, R, Q, I) = β(t)SI/(S + R + I) with β(t + T ) = β(t) we get β¯ . (15) RdP ER (λ) = ReP ER (λ) = γ¯ + σ ¯+d+α ¯1 If the parameter functions are all constant, by (14) and (15) we obtain again the thresholds for the autonomous system with mass-action, standard and quarantineadjusted incidence. The thresholds (14) and (15) can also be obtained using the methods developed in [18, 13] for general periodic epidemic models, that constitute periodic versions of the general autonomous model considered in [17].

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7. General model with mass-action, standard and quarantine-adjusted incidence In this section we are going to consider a nonautonomous SIQR model with mass-action, standard and quarantine-adjusted incidence. Additionally to our assumptions on the function ϕ and the parameter functions in Theorem 1, we are going to consider the particular cases of mass-action, standard and quarantineadjusted incidence. We will also assume that natural death and recruitment are constant. We have in this case the constant solution x(t) = Λ/d. For mass-action incidence, ϕ(t, S, I, Q, R) = β(t)SI, we have Z t+λ Λ rp (λ) = lim inf β(s) − (γ(s) + σ(s) + d + α1 (s)) ds t→+∞ t d   Λ − − − − 6λ β − [γλ + σλ + d + (α1 )λ ] d λ and similarly re (λ) > λ Define



 Λ + + + + β − [γλ + σλ + d + (α1 )λ ] . d λ

RpSIM (λ) =

d[γλ−

+

+

σλ+

and RpSIM (λ) =

d[γλ+

Λβλ− + d + (α1 )− λ]

σλ−

Λβλ+ . + d + (α1 )+ λ]

Theorem 6. We have the following for the general non-autonomous system with mass-action incidence and constant recruitment and death rates. 1. If there is λ > 0 such that RpSIM (λ) > 1 then the infectives I are permanent; 2. If there is λ > 0 such that ReSIM (λ) < 1 then the infectives I go to extinction and any disease free solution is globally attractive. For standard and quarantine-adjusted incidence, respectively ϕ(t, S, I, Q, R) = we have

Z

β(t)SI S+I +R+Q

and

ϕ(t, S, I, Q, R) =

t+λ

β(t)SI , S+I +Q

Λ/d − (γ(s) + σ(s) + d + α1 (s)) ds t→+∞ t δ→0 Λ/d + δ
6 λ βλ− − [γλ− + σλ− + d + (α1 )− λ]

rp (λ) = lim inf

and similarly Define

β(t) lim inf


re (λ) > λ βλ+ − [γλ+ + σλ+ + d + (α1 )+ λ] . RpS/QA (λ) =

γλ−

+

σλ−

βλ− + d + (α1 )− λ

+

σλ+

βλ+ . + d + (α1 )+ λ

and RpS/QA (λ) =

γλ+

´ CESAR M. SILVA

12

Theorem 7. We have the following for the general non-autonomous system with mass-action incidence and constant recruitment and death rates. S/QA 1. If there is λ > 0 such that Rp (λ) > 1 then the infectives I are permanent; S/QA 2. If there is λ > 0 such that Re (λ) < 1 then the infectives I go to extinction and any disease free solution is globally attractive. 8. Proof of Theorem 1 8.1. Proof of Lemma 1. Assume that ε > 0, that 0 < θ 6 K, that a, b ∈]θ, δ[ and that a − b < ε. We have, by H2), for every 0 6 τ 6 δ, |ϕ(t, a, 0, 0, δ) − ϕ(t, b, 0, 0, δ)| 6 Kθ |a − b|δ < Kθ δε. Therefore, ϕ(t, a, 0, 0, δ) ϕ(t, b, 0, 0, δ) ϕ(t, a, 0, 0, δ) − Kθ ε < < + Kθ ε, (16) δ δ δ and, adding and subtracting γ(t) + σ(t) + d(t) + α1 (t), we get (7). We will now show that in fact rp (λ), re (λ) are independent of the particular solution x(t) of (4) with x(0) > 0. In fact, by iv) in Proposition 2, for every ε > 0 and every solution x1 (t) of (4) with x1 (0) > 0 there is a Tε > 0 such that |x(t) − x1 (t)| < ε for every t > Tε . Choose εK > 0 such that, for ε < εK and t > Tε , we have x(t), x1 (t), x(t) ± ε ∈ ]θ, K[, for some θ > 0 that depends on x and x1 . Leting a = x(t), b = x1 (t) we obtain by (8) Z t+λ Z t+λ lim inf bδ (s, x1 (s)) ds − λKθ ε < lim inf bδ (s, x(s)) ds δ→0

t

δ→0

t


Tε . We conclude that, for every 0 < ε < εK , Z t+λ Z t+λ lim inf bδ (s, x(s)) ds − lim inf lim inf bδ (s, x1 (s)) ds < λKθ ε, lim inf t→+∞ t t→+∞ t δ→0 δ→0

and thus

lim inf t→+∞

Z

t

t+λ

lim inf bδ (s, x(s)) ds = lim inf t→+∞

δ→0

Z

t

t+λ

lim inf bδ (s, x1 (s)) ds. δ→0

Letting δ → 0 we conclude that rp (λ) (and thus Rp (λ)) is independent of the chosen solution. Using (9), the same reasoning shows that re (λ) (and thus Re (λ)) are also independent of the particular solution. This proves the lemma. 8.2. Proof of Theorem 1. Assume that Rp (λ) > 1 (and thus rp (λ) > 0) and let (S(t), I(t), R(t), Q(t)) be any solution of (1) with S(T0 ) > 0, I(T0 ) > 0, R(T0 ) > 0 and Q(T0 ) > 0 for some T0 > 0. By iii) in Proposition 1 we may assume that for t > T0 we have (S(t), I(t), R(t), Q(t)) ∈ ∆0,K . Since rp (λ) > 0, using (8) we conclude that there are constants δ1 ∈]0, K[ and Θ > 0 such that Z t+λ

inf

t

0 0 sufficiently large, say t > T1 , and ε > 0 sufficiently small, say for 0 < ε 6 ε¯, where x∗ is any fixed solution of (4) with x∗ (0) > 0. Note that, by ii) in

13

Proposition 2 we have x∗ (T0 ) > 0 and by by iii) in Proposition 2 we may assume (eventually using a bigger T1 ) that x∗ (t) > θ for some θ > 0 and t > T1 . Define     δ1 ε 0 Θ , (18) , ε¯ and ε1 = min δ1 , , ε0 = min 4Kθ λ C 2DN where Kθ is given by H2), N is given by H3), C is given by iv) in Proposition 1 and D is given by v) in Proposition 2. We will show that lim sup I(t) > ε1 . (19) t→+∞

Assume by contradiction that (19) is not true. Then there exists T2 > 0 satisfying I(t) < ε1 for all t > T2 . Consider the auxiliary equation x′ = Λ(t) − d(t)x − N ε1 ,

(20)

where N is given by H3). Let x¯(t) be the solution of (4) with x ¯(T0 ) = S(T0 ) and let x(t) be the solution of (20) with x(T0 ) = S(T0 ). By (18) and v) in Proposition 2 we obtain ε0 (21) |x(t) − x ¯(t)| 6 DN ε1 6 , 2 for all t > T0 . According to iv) in Proposition 2, x∗ (t) is globally uniformly attractive on R+ 0. Therefore, there exists T3 > 0 such that, for all t > T3 , we have ε0 (22) |¯ x(t) − x∗ (t)| 6 . 2 By H3) we have 0 6 ϕ(t, S(t), R(t), Q(t), I(t)) 6 N I(t) < N ε1 , for all t > max{T0 , T2 }, and thus, by the first equation in (1), Λ(t) − d(t)S > S ′ > Λ(t) − d(t)S − N ε1 .

(23)

for all t > max{T0 , T2 }. Comparing (20) and (23) we conclude that S(t) > x(t) for all t > max{T0 , T2 }. Write T4 = max{T0 , T1 , T2 , T3 }. By (21) and (22) we have, for all t > T4 , ε0 ε0 x∗ (t) + ε0 > x∗ (t) + >x ¯(t) > S(t) > x(t) > x ¯(t) − > x∗ (t) − ε0 , 2 2 and thus |S(t) − x∗ (t) + ε0 | 6 |S(t) − x∗ (t)| + ε0 < 2ε0 , (24) for all t > T4 . By (24) and (8) in Lemma 1 we get, for all t > T4 , bδ (t, S(t)) > bδ (t, x∗ (t) − ε0 ) − 2Kθ ε0 .

(25)

For t > 0 sufficiently large, say t > T5 , we have by (3) and (18), Q(t) < Cε1 < δ1 and R(t) < Cε1 < δ1 and thus I(t), Q(t), R(t) ∈ ]0, δ1 [. We may assume that T5 > T4 . According to (18) we have ε0 6 ε¯ and by (17) we get   Z t 1 (t − T5 ) − 1 Θ. (26) inf bδ (τ, x∗ (s) − ε0 ) ds > λ T5 0 I(T5 ) e > I(T5 ) e

Rt

T5

ϕ(s,S(s),R(s),Q(s),I(s))/I(s)−(γ(s)+δ(s)+d(s)+α1 (s)) ds

Rt

inf bδ (s,S(s)) ds T5 0 0 we have in fact lim inf I(t) > ℓ

(27)

t→+∞

for every solution (S(t), I(t), R(t), Q(t)) with S(T0 ) > 0, I(T0 ) > 0, R(T0 ) > 0 and Q(T0 ) > 0. By (17), for all ξ > λ, t > T1 and 0 < θ < ε¯, we have Z t+ξ (28) inf bδ (s, x∗ (s) − θ) ds > N. t

0 0, Rn > 0, Qn > 0 and In > 0 such that ε1 lim inf I(t, xn ) < 2 , t→+∞ n where I(t, xn ) denotes the solution of (1) with initial conditions S(T1 ) = Sn , I(T1 ) = In , R(T1 ) = Rn and Q(T1 ) = Qn . By (19), given n ∈ N, there are two sequences (tn,k )k∈N and (sn,k )k∈N with T1 < sn,1 < tn,1 < sn,2 < tn,2 < · · · < sn,k < tn,k < · · · and lim sn,k = +∞, such that k→+∞

I(sk,n , xn ) =

ε1 , n

I(tk,n , xn ) =

ε1 n2

(29)

and

ε1 ε1 < I(t, xn ) < , n2 n By the second equation in (1) we have

for all t ∈]sn,k , tn,k [.

I ′ (t, xn ) = I(t, xn )[(ϕ(t, S(t, xn ), R(t, xn ), Q(t, xn ), I(t, xn ))/I(t, xn )− − (γ(t) + δ(t) + d(t) + α1 (t))] > −(γ(t) + σ(t) + d(t) + α1 (t))I(t, xn ) > −(γS + σS + dS + (α1 )S )I(t, xn ),

(30)

15

Therefore we obtain Z tk,n ′ I (τ, xn ) dτ > −(γS + σS + dS + (α1 )S )(tk,n − sk,n ) sk,n I(τ, xn ) and thus I(tk,n , xn ) > I(sk,n , xn ) e−(γS +σS +dS +(α1 )S )(tk,n −sk,n ) . By (29) we get 1 −(γS +σS +dS +(α1 )S )(tk,n −sk,n ) and therefore we have n >e tk,n − sk,n >

log n → +∞ γS + σS + dS + (α1 )S

(31)

as n → +∞. According to (30), for all t ∈]sk,n , tk,n [, we have I(t, xn ) < ε1 /n < ε1 and we conclude that ϕ(t, S, R, Q, I) 6 N ε1 for all t ∈]sk,n , tk,n [. Thus by (1) we obtain S ′ > Λ(t) − d(t)S − N ε1 for all t ∈]sk,n , tk,n [. By comparison we have S(t, xn ) > x(t) for all t ∈]sk,n , tk,n [, where x is the solution of (20) with x(sk,n ) = S(sk,n , xn ). Using (21) we obtain, for all t ∈ ]sk,n , tk,n [, ε0 |x(t) − x ¯(t)| 6 , (32) 2 where x¯(t) is a solution of (4) with x¯(sk,n ) = S(sk,n , xn ). Since x∗ (t) is globally uniformly attractive, there exists T ∗ > 0, independent of n and k, such that ε0 (33) |¯ x(t) − x∗ (t)| 6 , 2 for all t > sk,n + T ∗ . By (31) we can choose B > 0 such that tn,k − sn,k > λ + T ∗ for all n > B. Given n > B, by (28), (32) and (33) and by the second equation in (1) we get Rt

k,n ε1 ∗ lim inf bδ (τ,S(τ,xn )) dτ = I(tk,n , xn ) > I(sk,n + T ∗ , xn ) e sk,n +T δ→0 2 n R tk,n ∗ ε1 ε1 ∗ lim inf bI,δ1 (τ,x (τ )−ε0 ) dτ > 2. > 2 e sk,n +T δ→0 n n

This leads to a contradiction and establishes that lim inf I(t) > ℓ > 0. Thus, 1. in t→+∞

Theorem 1 is established. Assume now that Re (λ) < 1 (and thus re (λ) < 0) for some λ > 0. Then we can choose Θ > 0 and T0 > 0 such that Z t+λ bδ (s, x∗ (s) + ε) ds < −Θ, (34) t

for all t > T0 and ε > 0 sufficiently small, say ε < ε¯. By iii) in Proposition 1 we can assume that (S(t), I(t), R(t), Q(t)) ∈ ∆0,K for t > T0 . We will show that lim sup I(t) = 0. (35) t→+∞

By (1) and H3) we have S ′ (t) 6 Λ(t) − d(t)S

´ CESAR M. SILVA

16

for all t > T0 . For any solution x(t) of (4) with x(T0 ) = S(T0 ) we have by comparison S(t) 6 x(t) for all t > T0 . Since x∗ (t) is globally uniformly attractive, there exists ε1 < ε¯ and T1 > T0 such that, for all t > T1 , we have |x(t) − x∗ (t)| < ε1 . Therefore, for all t > T1 , we have S(t) 6 x(t) 6 x∗ (t) + ε1 .

(36)

Thus, by the second equation in (1) and by H4) we conclude that I ′ (t) ϕ(t, S(t), R(t), Q(t), I(t)) = − (γ(t) + σ(t) + d(t) + α1 (t)) I(t) I(t) ϕ(t, S(t), 0, 0, δ) − (γ(t) + σ(t) + d(t) + α1 (t)) 6 lim sup δ δ→0 6 lim sup bδ (t, S(t)) δ→0

and finally log

I(t) = I(T1 )

Z

t

T1

I ′ (τ ) dτ 6 I(τ )

Therefore, by H1), (36) and (9) we have I(t) = I(T1 ) e

Rt

T1

lim sup bδ (s,S(s)) ds δ→0

Z

t

lim sup bδ (s, S(s)) ds.

T1

δ→0

6 I(T1 ) e

Rt

T1

lim sup bδ (s,x∗ (s)+ε1 ) ds δ→0

.

By (34) we conclude that I(t) 6 I(T1 ) e−(

t−T1 λ

+1)Θ

.

Therefore lim I(t) = 0 and this proves that the infectives go to extinction. t→+∞

Next, still assuming that Re (λ) < 0 for some λ > 0, we let (S(t), R(t), Q(t), I(t)) be a solution of (1) with non-negative initial conditions and (S0 (t), 0, R0 (t), Q0 (t)) be a disease-free solution of (1) with non-negative initial conditions. By iii) in Proposition 1 we can assume that (S(t), I(t), R(t), Q(t)), (S0 (t), 0, R0 (t), Q0 (t)) ∈ ∆0,K for t > T0 . Thus S0 (t), R0 (t), Q0 (t), S(t), R(t), Q(t) < K, for t > T0 . By Theorem 2 we have lim I(t) = 0. Therefore given ε > 0 there t→+∞

exists Tε > T0 such that I(t) < ε, S(t), R(t), Q(t) < K for t > Tε . Thus Λ(t) − d(t)S > S ′ (t) > Λ(t) − d(t)S − εN, for all t > Tε . By comparison we get x1 (t) > S(t) > x2 (t)

(37)

where x1 (t) is the solution of (4) and x2 (t) is the solution of (20) with ε1 = ε and x1 (Tε ) = x2 (Tε ) = S(Tε ). By v) in Proposition 2 we obtain, for all t > T1 , |x2 (t) − x1 (t)| 6 εN D,

(38)

Since S0 (t) is a solution of (4), it is globally uniformly attractive, by iv) in Proposition 2. Thus, there is T2 > Tε such that |x1 (t) − S0 (t)| 6 ε

(39)

17

for all t > T2 . By (37), (38) and (39) we conclude that S(t) > x2 (t) > x1 (t) − εN D > S0 (t) − (1 + N D)ε,

(40)

for all t > T2 , and also, by (37) and (39), S(t) 6 S0 (t) + ε

(41)

for all t > T2 . Since ε > 0 can be made arbitrarily small by taking t > Tε , by (40) and (41) we have lim |S(t) − S0 (t)| = 0.

(42)

t→+∞

Let T3 > Tε be such that |I(t) − I0 (t)| < ε for every t > T3 and write u(t) = Q(t) − Q0 (t). By the third equation in (1), we get −(dS + (α2 )S + εS )u(t) 6 u′ (t) 6 εσS − (dI + (α2 )I + εI )u(t). We obtain u(t) > u(T3 ) e−(dS +(α2 )S +εS )(t−T3 ) and u(t) 6 u(T3 ) e−(dI +(α2 )I +εI )(t−T3 ) +

εσS (1 − e−(dI +(α2 )I +εI )(t−T3 ) ) . dI + (α2 )I + εI

and thus lim |Q(t) − Q0 (t)| = lim u(t) 6

t→+∞

t→+∞

εσS . dI + (α2 )I + εI

(43)

Since ε > 0 can be made arbitrarily small we get lim |Q(t) − Q0 (t)| = 0.

t→+∞

Let T4 > T1 be such that |Q(t) − Q0 (t)| < ε for every t > T4 and write w(t) = R(t) − R0 (t). By the fourth equation in (1) and since I0 (t) = 0 and for all t > 0, we get −dS w(t) 6 w′ (t) 6 ε(γS + εS ) − dI w(t). We obtain w(T4 ) e−dS (t−T4 ) 6 w(t) 6 w(T4 ) e−dI (t−T4 ) +

ε(γS + εS )(1 − e−dI (t−T4 ) ) . dI

and thus lim |R(t) − R0 (t)| = lim w(t) 6

t→+∞

t→+∞

ε(γS + εS ) . dI

(44)

Since ε > 0 can be made arbitrarily small we get lim |R(t) − R0 (t)| = 0.

t→+∞

By (42), (43), (44) and since all the infectives go to extinction, we conclude that any disease-free solution (S0 (t), 0, R0 (t), Q0 (t)) with nonnegative initial conditions is globally attractive. We obtain 2. in Theorem 1 and this concludes our proof.

´ CESAR M. SILVA

18

9. Simulation We carried out some experiments to illustrate our results. We considered some nonautonomous and nonperiodic models with mass-action and quarantine adjusted incidences. Firstly consider a nonautonomous model with mass-action and parameters Λ = 0.001, d = 0.035, γ(t) = 0.4, σ(t) = 0.01, a1 (t) = a2 (t) = ε(t) = 0.2 and β(t) = α(1 − 0.7 sin(0.3t))(2 − e−t ). For any λ > 0, we obtain RpSIM (λ) = lim inf 0.0886α(λ − 4.667 sin(.3t + .15λ) sin(.15λ)) t→∞

and ReSIM (λ) = lim sup 0.0886α(λ − 4.667 sin(.3t + .15λ) sin(.15λ)) t→∞

(where the numbers RpSIM (λ) and ReSIM (λ) are given in section 7). For instance, setting α = 9 and λ = 21, we have RpSIM (λ) = 1.10599 > 1 and we can conclude that the infectives are permanent and, setting α = 8 and λ = 21, we have ReSIM (λ) = 0.98310 < 1 and we can conclude that the infectives go to extinction. Figures 1 and 2 illustrate the referred situations. 0.0030

7. ´ 10-7

0.0025

6. ´ 10-7 5. ´ 10-7

0.0020 4. ´ 10-7 0.0015 3. ´ 10-7 0.0010

2. ´ 10-7

0.0005

1. ´ 10-7

0

100

200

300

400

Figure 1. RpSIM = 1.10599

500

0

100

200

300

400

500

Figure 2. ReSIM = 0.98310

Now consider a nonautonomous model with quarantine adjusted incidence and the same parameters: Λ = 0.001, d = 0.035, γ(t) = 0.4, σ(t) = 0.01, a1 (t) = a2 (t) = ε(t) = 0.2 and β(t) = α(1 − 0.7 sin(0.3t))(2 − e−t ). For any λ > 0, we obtain in this case RpS/QA (λ) = lim inf 3.10078α(λ − 4.667 sin(.3t + .15λ) sin(.15λ)) t→∞

and ReS/QA (λ) = lim sup 3.10078α(λ − 4.667 sin(.3t + .15λ) sin(.15λ)) t→∞

S/Q S/Q (where the numbers Rp (λ) and Rp (λ) are given in section 7). Setting α = 0.25 S/QA (λ) = 1.07527 > 1 and we can conclude that the and λ = 0.2, we have Rp infectives are permanent and, setting α = .23 and λ = 0.2, we have ReSIM (λ) =

0.989247 < 1 and we can conclude that the infectives go to extinction. Figures 3 and 4 illustrate the referred situations.

19 1. ´ 10-8 0.0010 8. ´ 10-9 0.0008

6. ´ 10-9

0.0006

0.0004

4. ´ 10-9

0.0002

2. ´ 10-9

0

200

400

600

800

Figure 3. RpS/QA = 1.07527

0

200

400

600

800

1000

Figure 4. ReS/QA = 0.989247

References [1] J. Arino, R. Jordan, P. van den Driessche, Quarantine in a multi-species epidemic model with spatial dynamics, Math. Biosc. 206, 4660 (2007) [2] C. Castillo-Chavez, H.R. Thieme, Asymptotically autonomous epidemic models, in: O. Arino, D. E. Axelrod, M. Kimmel, M. Langlais (Eds.), Mathematical Population Dynamics: Analisys and Heterogenity, Wuerz, Winnipeg, Canada, 1995, p. 33. [3] K. Dietz, The incidence of infectious diseases under the influence of seasonal fluctuations, Lecture Notes in Biomathematics, vol. 11, pp. 1-5. Berlin-Heidelberg-New York, Springer, 1976. [4] , D. J. Gerberry and F. A. Milner, An SEIQR model for childhood diseases, J. Math. Biol., 59, 4 (2009) [5] H. Hethcote, M. Zhien, L. Shengbing, Effects of quarantine in six endemic models for infectuous diseases, Math. Biosc., 180, 141-160 (2002) [6] H. Hethcote, P. van den Driessche, Some epidemiological model with nonlinear incidence, J. Math. Biol., 29, 271-287 (1991) [7] K. Mischaikow, H. Smith and H. R. Thieme, Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions, Trans. Amer. Math. Soc., 347, 1669-1685 (1995) [8] Z. Feng and H. R. Thieme, Recurrent Outbreaks of Childhood Diseases Revisited: The Impact of Isolation, Math. Biosc., 93-130 (1995) [9] G. F. Gensini, M. H. Yacoub, A. A. Conti, The concept of quarantine in history: from plague to SARS, Journal of infection, 49, 257-261 (2004) [10] W. M. Liu, H. W. Hethcote, S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25, 359380 (1987) [11] L. Markus, Asymptotically autonomous differential systems, Contributions to the theory of Nonlinear Oscillations 111 (S. Lefschetz, ed.), Ann. Math. Stud., 36, Princeton University Press, Princeton, NJ, 17-29 (1956) [12] E. Pereira, C. M. Silva and J. A. L. Silva, A Generalized Non-Autonomous SIRVS Model, Math. Meth. Appl. Sci., to appear [13] C. Rebelo, A. Margheri, N. Baca¨ er, Persistence in seasonally forced epidemiological models, J. Math. Biol. 64, 933949 (2012) [14] M. A. Safi, A. B. Gumel, The effect of incidence functions on the dynamics of a quarantine/isolation model with time delay, Nonlinear Anal. - Theory Methods and Applications, 12, 215-235 (2011) [15] M. A. Safi, A. B. Gumel, Mathematical analysis of a disease transmission model with quarantine, isolation and an imperfect vaccine, Computers & Mathematics with Applications, 61, 30443070 (2011) [16] N. Yi, Z. Zhao and Q, Zhang, Bifurcations of an SEIQR epidemic model, International Journal of Informations and Systems Sciences, 5, 296-310 (2009) [17] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci, 180, 2948 (2002) [18] W. Wang, X.-Q. Zhao, Threshold Dynamics for Compartmental Epidemic Models in Periodic Environments, J. Dyn. Diff. Equat. 20, 699717 (2008)

20

´ CESAR M. SILVA

[19] L. Wu, Z. Feng, Homoclinic Bifurcation in a SIQR Model for Childhood Diseases, J. Differential Equations, 168, 150-167 (2000) [20] T. Zhang, Z. Teng and S. Gao, Threshold conditions for a nonautonomous epidemic model with vaccination, Applicable Analysis, 87, 181-199 (2008) ´ tica, Universidade da Beira Interior, 6201-001 C. Silva, Departamento de Matema ˜ , Portugal Covilha E-mail address: [email protected]