A nonautonomous epidemic model with general incidence rate and quarantine

0.47 MB·English

by  César M. Silva

A NONAUTONOMOUS EPIDEMIC MODEL WITH GENERAL
INCIDENCE RATE AND QUARANTINE
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1 CE´SARM.SILVA
0
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Abstract. We obtain threshold conditions for eradication and permanence

n of the disease for a nonautonomous SIRQ model with time-dependent inci-
a dence rates, given by functions of all compartments in some general family.
J Thethresholdconditionsaregivenbysomenumbersthatplaytheroleofthe

4 basic reproduction number. Additionally, we obtain simple threshold condi-
1 tions in the asymptotically autonomous and periodic settings and show that
our thresholdscoincidewiththe basicreproductionnumber insomecommon

] autonomous examples.
S
D
.
h
1. Introduction
t
a
m Quarantine is one of the possible measures for reducing the transmission of dis-
eases and has been used over the centuries to reduce the transmission of human
[
diseases such as leprosy, plague, typhus, cholera, yellow fever and tuberculosis.

2 The concept of quarantine dates back to the fourteenth century and is related
v
to plague. In fact, in 1377, the Rector of the seaport of the old town of Ragusa
0
(modern Dubrovnik) officially issued a thirty-day isolation period: ships coming
4
7 from infected or suspected to be infected sites were to stay at anchor for thirty
5 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 -
5
0 quaranta - that comes the term quarantine [9].
2 Isolationis still nowadaysan effective way of controllinginfectious diseases. For
1 some milder infectious diseases, quarantined individuals could be individuals that
:
v choose to stay home from school or work while for more severe infectious diseases
i people may be forced into quarantine.
X
To study the effect of quarantine in the spread of a disease, we consider a SIRQ
r
a model. Thus,wedividethepopulationintofourcompartments: theScompartment
corresponding to not infected individuals that are susceptible to the disease, the I
compartmentcorrespondingtoindividualsthatareinfectedandnotisolated,theQ
compartment corresponding to isolated individuals (Q stands for quarantine) and
the R compartment including the recovered and immune individuals. We assume
inthispaperthatthequarantinedindividualsareperfectlyisolatedfromtheothers
sothattheydonotinfectthesusceptiblesandthattheinfectionconferspermanent
immunity upon recovery.
Amajorconcernwhenstudyingepidemicmodelsistounderstandtheasymptotic
behavior of the compartments considered, particularly the infective compartment.
Date:January15,2013.
2010 Mathematics Subject Classification. 92D30,37B55,34D20.
Key words and phrases. Epidemicmodel,quarantine,non-autonomous, stability.
WorkpartiallysupportedbyFCTthought CMUBI(projectPEst-OE/MAT/UI0212/2011).
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2 CE´SARM.SILVA
Frequently,inautonomousepidemic models,one oftwosituationsoccur: there is a
disease-freeequilibriumthatisthe uniqueequilibriumofthe systemorthe disease-
free 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 oscilla-
tions. Theirmodelincludesamodifiedincidencefunction-thequarantine-adjusted
incidence-wherethenumberofpercapitacontactsbetweensusceptiblesandinfec-
tivesisdividedbythe numberofnonisolatedindividuals. Somedynamicalaspects
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
proposedbyHethcote,ZhienandShengbingin[5],wherethismodelwascompared
to SIQR models with different incidence rates. A similar analysis is also under-
taken 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
studiedthedynamicsandusedhistoricaldatatodiscusstheadequacyofthemodel
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 ana-
lyzed 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). AmodelwithtwelvedifferentclasseswasproposedbySafiandGumel[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
thepopulation. Thesameauthorsstudiedadelayedmodelincludingaquarantined
class [15].
We emphasize that allthe above models are autonomousmodels. However,it is
well-knownthatfluctuationsareverycommonindiseasetransmission. Forinstance,
theopeningandclosingofschoolstendtomakethecontactratesvaryseasonallyin
childhooddiseases[3]andperiodicchangesinbirthratesofthe populationarealso
frequent. Itisthereforenaturaltoconsidertime-dependentparametersinepidemic
modelsanddefinethresholdconditionsforextinctionandpersistenceofthedisease.
Ourobjectiveinthis paperistoconsiderafamily nonautonomousSIQRmodels
with a large family of contact rates given by generalfunctions 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
3
those papers are linear in the infectives while here we assume that the incidence
rates are, in general, nonlinear functions of all compartments.
Thestructureofthepaperisasfollows: insection2weintroducethemodeland
the main definitions; in section 3 we state our main Theorem; in the next sections
welookatsomeparticularcasesofourmodel,namely,insection4,weparticularize
fortheautonomousmodel,insection5,weseethattheasymptoticallyautonomous
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)+α2(t)+ε(t)]Q
R′ =γ(t)I+ε(t)Q−d(t)R
whereS, R, QandI correspondrespectivelytothe susceptible,recovered,quaran-
tine andinfective compartments;Λ(t) is the recruitmentrate of susceptible (births
andimmigration); ϕ(t,S,R,I) is the incidence (into the infective class); d(t) is the
per capita naturalmortality rate; γ(t) is the recoveryrate; σ(t) is the removalrate
from the infectives; α (t) is the disease related death in the infectives; α (t) is the
1 2
disease related death in the quarantine class; and ε(t) is the removalrate from the
quarantineclass. FurthermoreweassumethatΛ,d,γ,σ,α ,α andεarecontinuous
1 2
bounded and nonnegative functions on R+ and that
0
d= inf d(t)>0 and Λ=supΛ(t)>0.
t≥0 t≥0
For each δ ∈R+ define the set
0
∆ ={(x ,x ,x ,x )∈R4: 0≤x ≤Λ/d+δ}.
δ 1 2 3 4 i
Assume further that the function ϕ : (R+)5 → R is continuous and nonnegative
0
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,x ,y,w,z)−ϕ(t,x ,y,w,z)|≤K|x −x |z, (3)
1 2 1 2
ϕ(t,x,0,0,τ) ϕ(t,x,y,w,z) ϕ(t,x,0,0,τ)
sup ≥ ≥ inf , (4)
0<τ<λ/d+δ τ z 0<τ<λ/d+δ τ
for all (t,0,y,w,z), (t,x,y,w,z), (t,x ,y,w,z),(t,x ,y,w,z) ∈ R+ × ∆ . Note
1 1 0 δ
that,by (2)and(3),the aboveinfimums andsupremumsarefinite. Notealsothat,
for contact rates that are not dependent on y and w, (4) is always verified.
Now, we need to make some definitions. We say that
i) the infectives I are permanent in the solution (S(t),I(t),R(t),V(t)) of (1) if
there exist constants 0<m <m such that
1 2
m <liminfI(t)≤limsupI(t)<m ;
1 2
t→∞ t→∞
4 CE´SARM.SILVA
ii) the infectives I go to extinction in the solution (S(t),I(t),R(t),V(t)) of (1) if
lim I(t)=0;
t→∞
iii) the infectives I are permanent in system (1) if there exist constants 0<m <
1
m such that
2
m <liminfI(t)≤limsupI(t)<m ,
1 2
t→∞ t→∞
foreverysolution(S(t),I(t),R(t),V(t))of(1)withS (t),I (t),R (t),V (t)>0
0 0 0 0
(note that we require that m and m are independent of the given solution
1 2
with positive initial condition in I);
iv) the infectives I go to extinction in system (1) if lim I(t) = 0 for all solution
t→∞
of (1).
3. Statment of the results
Consider the auxiliary equation
x′ =Λ(t)−d(t)x, (5)
and, for each solution x(t) of (5) with x(0)>0, the functions
ϕ(t,x(t),0,0,τ)
b (t,x(t))= inf −(γ(t)+σ(t)+d(t)+α (t))
I,δ 1
0<τ<Λ+δ τ
d
and
ϕ(t,x(t),0,0,τ)
b (t,x(t))= sup −(γ(t)+σ(t)+d(t)+α (t)).
S,δ 1
τ
0<τ<Λ+δ
d
Finally, define the numbers
t+λ
r (λ)= limliminf b (s,x(s))ds,
p I,δ
δ→0 t→∞ Zt
t+λ
r (λ)= limlimsup b (s,x(s))ds,
e,1 S,δ
δ→0 t→∞ Zt
1 t
r = limlimsup b (s,x(s))ds.
e,2 S,δ
δ→0 t→∞ t Z0
and
R (λ)=erp(λ), R (λ)=ere,1(λ) and R =ere,2.
p e,1 e,2
Contrarilyto what one couldexpect, the next technical lemma that will be proved
inSection7showsthatthenumbersabovedonotdependontheparticularsolution
x(t) of (5) with x(0) > 0. The numbers R (λ), R (λ) andR play the role of
p e,1 e,2
the basic reproduction number (defined for autonomous systems and also called
quarantine number) in the non-autonomous setting.
We have the following result.
Lemma 1. We have the following:
1. Let ε,ε ,ε >0 and 0≤δ ≤M. If
1 2
a,b,a±ε ,b±ε ∈ ]0,Λ/d+δ[ and a−b<ε,
1 2
then
|b (t,a±ε )−b (t,b±ε )|<(ε+ε +ε )K (6)
I,δ 1 I,δ 2 1 2
and
|b (t,a±ε )−b (t,b±ε )|<(ε+ε +ε )K. (7)
S,δ 1 S,δ 2 1 2
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2. The numbers R (λ), R (λ) and R are independent of the particular so-
p e,1 e,2
lution x(t) with x(0)>0 of (5).
We now state our main theorem on the permanence and extinction of the infec-
tives in system (1).
Theorem 1. We have the following for the system (1).
1. If there is a constant λ > 0 such that R (λ) > 1 then the infectives I are
p
permanent.
2. IfthereisaconstantλsuchthatR (λ)<1orifR <1thentheinfectives
e,1 e,2
I gotoextinctionandanydiseasefreesolution(S(t),0,R(t),Q(t))isglobally
attractive.
4. Autonomous model
In this section we aregoing to consider the autonomous setting. That is, we are
going to assume in system (1) that Λ,d,γ,δ,α ,ε and α are constant functions,
1 2
that Λ,d>0, and that ϕ is independent of t. We obtain the autonomous system
S′ =Λ−ϕ (S,R,Q,I)−dS
0
I′ =ϕ (S,R,Q,I)−[γ+δ+d+α ]I
 0 1 (8)
Q′ =δI −[d+α2+ε]Q
R′ =γI+εQ−dR
where ϕ0 :(R+0)4 →R is continuous, nonnegative and satisfies
ϕ (0,y,w,z)=0, |ϕ (x ,y,w,z)−ϕ (x ,y,w,z)|≤K|x −x |z,
0 0 1 0 2 1 2
and
ϕ (x,0,0,τ) ϕ (x,y,w,z) ϕ (x,0,0,τ)
0 0 0
sup ≥ ≥ inf ,
0<τ<λ/d+δ τ z 0<τ<λ/d+δ τ
for all (0,y,w,z),(x ,y,w,z),(x ,y,w,z) ∈ ∆ . In this setting we have that the
1 2 δ
auxiliary equation (5) admits the constant solution x(t)=Λ/d. Thus we obtain
ϕ (Λ/d,0,0,τ)
r (λ)= inf 0 −(γ+δ+d+α ) λ, R (λ)=erp(λ),
p 1 p
"0<τ<Λ τ #
d
ϕ (Λ/d,0,0,τ)
r (λ)= sup 0 −(γ+δ+d+α ) λ, R (λ)=ere,1(λ)
e,1 1 e,1
τ
"0<τ<Λ #
d
and
ϕ (Λ/d,0,0,τ)
r (λ)= sup 0 −(γ+δ+d+α ), R (λ)=ere,2(λ).
e,2 1 e,2
τ
0<τ<Λ
d
It is now easy to establish a result that is a versionof Theorem 1 in the particular
case of autonomous systems. Define
ϕ (Λ/d,0,0,τ) ϕ (Λ/d,0,0,τ)
RAUT = inf 0 and RAUT = sup 0 .
p 0<τ<Λd (γ+δ+d+α1)τ e 0<τ<Λd (γ+δ+d+α1)τ
For any λ>0, we have R (λ)>1 if and only if RAUT >1 and R (λ)<1 if and
p p e,1
only if RAUT <1 and this implies the following result.
e
Theorem 2. We have the following for the autonomous system (8).
6 CE´SARM.SILVA
1. If RAUT >1 then the infectives I are permanent;
p
2. If RAUT < 1 then the infectives I go to extinction and any disease free
e
solution is globally attractive.
Note that, if the incidence rate has the particular form
ϕ (S,R,Q,I)=ψ(S,R,Q)g(I)I, (9)
0
where g is a continuous, bounded and nonnegative function, then
ψ(Λ/d,0,0)Λ inf g(τ) ψ(Λ/d,0,0)Λ sup g(τ)
0<τ<Λ 0<τ<Λ
RAUT = d and RAUT = d
p d(γ+δ+d+α ) e d(γ+δ+d+α )
1 1
and we conclude that the infectives are permanent (resp. go to extinction) if
d(γ+δ+d+α ) d(γ+δ+d+α )
1 1
inf g(τ)> (resp. sup g(τ)< ).
0<τ<Λd Λψ(Λ/d,0,0) 0<τ<Λd Λψ(Λ/d,0,0)
Naturally, if
d(γ+δ+d+α )
1
∈g([0,Λ/d])
Λψ(Λ/d,0,0)
we get no information about the asymptotic behavior of the infectives.
Assuming now that g is constant, we have RAUT = RAUT and we can re-
p e
cover the results obtained in [5]. In fact, the autonomous SIQR model with
simple incidence (ϕ (S,R,Q,I) = βSI), the autonomous SIQR model with stan-
0
dard incidence (ϕ (S,R,Q,I) = βSI/(S + I + Q)) and the autonomous SIQR
0
model with quarantine-adjusted incidence (ϕ (S,R,Q,I)=βSI/(S+I+Q+R))
0
are all in the conditions we have assumed and our threshold values for those
model are RAUT = RAUT = βΛ/[d(γ + δ + d + α )] for simple incidence and
p e 1
RAUT = RAUT = β/(γ +δ +d+α ) for standard and quarantine-adjusted inci-
p e 1
dence. Forthese models,the numbersRAUT =RAUT coincide withthe quarantine
p e
reproduction number obtained in [5]. Note that, to put the referred autonomous
modelsconsideredin[5]inourcontext,weneedtohavethestandardincidenceand
the quarantine-adjusted incidence continuous in ∆ . This constitutes no problem
δ
since the functions ϕ associated to these incidence rates can be extended to a
0
continuousfunction in∆ , setting ϕ (0,R,0,0)=0 forthe standardincidence and
δ 0
ϕ (0,0,0,0)=0 for the quarantine-adjusted incidence.
0
In some situations, we may be able to obtain information about extinction but
not about permanence or vice versa. To illustrate this fact we consider the family
ofcontactratesobtainedbysettingϕ(S,Q,R)=S andg(I)= Ip withp,q >0,
1+αIq
α ≥ 0, in (9). This family of contact rates was considered for instance in [14, 6].
Ourthresholdconditionsabovegivenorelevantinformationaboutthepermanence
of the disease but it is easy to see that if
(Λ/d)p−2− 1+α(Λ/d)q−4 (γ+δ+d+α )<0
1
then the disease goes to extin(cid:2)ction. (cid:3)
5. Asymptotically autonomous model
In this section we are going to consider the asymptotically autonomous SIRQ
model. That is, additionally to the assumptions on Theorem 1, we are going to
7
assume for system (1) that there is a continuous function ϕ such that
0
lim ϕ(t,S,R,Q,I)=ϕ (S,R,Q,I)
0
t→∞
for each (S,R,Q,I) ∈ R4 and that the time-dependent parameters are asymptot-
ically constant: β(t) → β, Λ(t) → Λ, d(t) → d, γ(t) → γ, σ(t) → δ, α (t) → α ,
1 1
α (t) → α and ε(t) → ε as t → +∞. Denoting by F(t,S,R,Q,I) the right hand
2 2
side of (1) and by F (S,R,Q,I) the right hand side of the limiting system, that is
0
of (8), we also need to assume that
lim F(t,S,R,Q,I)=F (S,R,Q,I),
0
t→+∞
with uniform convergence on every compact set of (R+)4 and we will also assume
0
that (S,R,Q,I) 7→ F(t,S,R,Q,I) and (S,R,Q,I) 7→ F (S,R,Q,I) are locally
0
Lipschitz functions.
There is a general setting that will allow us to study this case.
Let f : R×Rn → R and f : Rn → R be continuous and locally Lipschitz in
0
Rn. Assume also that the nonautonomous system
x′ =f(t,x) (10)
is asymptotically autonomous with limit equation
x′ =f (x), (11)
0
that is, assume that f(t,x)→g(x) as t→+∞ with uniform convergence in every
compactsetofRn. Thefollowingtheoremisaparticularcaseofaresultestablished
in [10] (for related results and applications see for example [2, 7]).
Theorem 3. Let Φ(t,t ,x ) and ϕ(t,t ,y ) be solutions of (10) and (11) respec-
0 0 0 0
tively. Suppose that e∈Rn is a locally stable equilibrium point of (11) with attrac-
tive region
W(e)= y ∈Rn : lim ϕ(t,t ,y)=e
0
t→+∞
(cid:26) (cid:27)
and that W ∩W(e) 6= ∅, where W denotes the omega limit of Φ(t,t ,x ). Then
Φ Φ 0 0
lim Φ(t,t ,x )=e.
0 0
t→+∞
Since (R+)4 is the attractive region for any solution of system (8) with initial
condition in (R+)4 and the omega limit of every orbit of the asymptotically au-
tonomous system with I(t ) > 0 is contained in (R+)4, we can use Theorem 3 to
0
obtain the following result.
Theorem 4. We have the following for the asymptotically autonomous systems
above.
1. If RAUT >1 then the infectives I are permanent;
p
2. If RAUT < 1 then the infectives I go to extinction and any disease free
e
solution is globally attractive.
6. Periodic model with constant natural death and recruitment
In this section we are going to consider a periodic SIRQ 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, 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
8 CE´SARM.SILVA
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
t+T ϕ(s,Λ/d,0,0,τ)
r (λ)= limliminf sup −(γ(s)+δ(s)+d+α (s))ds
p 1
δ→0t→+∞Zt 0<τ<Λd+δ τ
T ϕ(s,Λ/d,0,0,τ)
= sup ds−(γ¯+δ¯+d+α¯ )T
1
τ
Z0 0<τ<Λd
and similarly
T ϕ(s,Λ/d,0,0,τ)
r (λ)= inf ds−(γ¯+δ¯+d+α¯ )T,
e,1 1
Z0 0<τ<Λd τ
where f¯denotes the average of f in the interval [0,T]: f¯= 1 T f(s)ds. Define
T 0
1 T ϕ(s,Λ/d,0,0,τ) R
RPER(λ)= sup ds
p T τ(γ¯+δ¯+d+α¯ )
Z0 0<τ<Λd 1
and
1 T ϕ(s,Λ/d,0,0,τ)
RPER(λ)= inf ds.
e T Z0 0<τ<Λd τ(γ¯+δ¯+d+α¯1)
Theorem 5. We have the following for the periodic system with constant recruit-
ment and death rates.
1. If RPER >1 then the infectives I are permanent;
p
2. If RPER < 1 then the infectives I go to extinction and any disease free
e
solution is globally attractive.
If we assume in the periodic modelabove that the incidence rate has the partic-
ular form ϕ(t,S,R,Q,I)=β(t)SI with β(t+T)=β(t), then we obtain
β¯Λ
RPER(λ)=RPER(λ)= . (12)
d e d(γ¯+δ¯+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¯
RPER(λ)=RPER(λ)= , (13)
d e γ¯+δ¯+d+α¯
1
and, assuming that the parameter functions are all constant, by (12) and (13) we
obtain again the thresholds for the autonomous system with simple, standard and
quarantine-adjusted incidence.
7. Proof of Theorem 1
We will divide the proofin severalsteps. Namely, first we establishsome simple
results about the solutions of (1), then we analyze the non-homogeneous linear
system (5) and finally we prove the theorem.
Proposition 1. We have the following:
i) all solutions (S(t),I(t),Q(t),R(t)) of (1) with nonnegative initial conditions
are nonnegative for all t≥0;
ii) all solutions (S(t),I(t),Q(t),R(t)) of (1) with positive initial conditions are
positive for all t≥0;
9
iii) If (S(t),I(t),Q(t),R(t)) is a solution of (1)with nonnegativeinitial conditions
then
limsupS(t)+I(t)+Q(t)+R(t)≤Λ/d.
t→+∞
Proof. Properties i) and ii) are easy to prove. In fact, since t7→Λ(t) and t7→µ(t)
are bounded, setting U(t) = S(t)+I(t)+R(t)+V(t) and adding the equations
in (1) we obtain for nonnegative initial conditions,
U′ =Λ(t)−d(t)U −α (t)I −α (t)Q≤Λ−dU,
1 2
where Λ = sup Λ(t) and d = inf d(t). Moreover, since the general solu-
t∈R+ t∈R+
0 0
tion of equation V′ = Λ−dV is V(t) = Ce−dt+Λ/d, we obtain by comparison
property iii). (cid:3)
Proposition 1 shows that, for every ε > 0, every solution (S(t),I(t),Q(t),R(t))
of our system stays in the region
B ={(x,y,w,z)∈(R+)4: x+y+w+z ≤Λ/d+δ}⊂∆
δ δ
for every t∈R+ sufficiently large.
0
For t ≥0, the solution of (5) with initial condition x(t )=x >0 is given by
0 0 0
t
x(t)=e−Rtt0d(s)dsx0+ e−Rutd(s)dsΛ(u)du
Zt0
and this allows us to establish easily the following proposition.
Proposition 2. We have the following:
i) Given t ≥0, all solutions x(t) of equation (5) with initial condition x(t )≥0
0 0
are nonnegative for all t≥0;
ii) Given t ≥0, all solutions x(t) of equation (5) with initial condition x(t )>0
0 0
are positive for all t≥0;
iii) Each fixed solution x(t) of (5) with initial condition x(t )≥0 is bounded and
0
globally uniformly attractive on [0,+∞[;
iv) if x(t) is a solution of (5) and x˜(t) is a solution of the system
x′ =Λ(t)−d(t)x+f(t)
with x˜(t )=x(t ) then
0 0
sup|x˜(t)−x(t)|≤1/d sup|f(t)|.
t≥t0 t≥t0
7.1. Proof of Lemma 1. Assume that ε,ε ,ε > 0, 0 ≤ δ ≤ M, a, b, a±ε ,
1 2 1
b±ε ∈]0,Λ/d+δ[ and that a−b<ε. We have, by (3),
2
|ϕ(t,a±ε ,0,0,τ)−ϕ(t,b±ε ,0,0,τ)|≤K|a−b±ε ±ε |τ <Kτ(ε+ε +ε ).
1 2 1 2 1 2
Therefore,
ϕ(t,a±ε ,0,0,τ) ϕ(t,b±ε ,0,0,τ)
1 2
−K(ε+ε +ε )<
1 2
τ τ
(14)
ϕ(t,a±ε ,0,0,τ)
1
< +K(ε+ε +ε ),
1 2
τ
andthus,takingtheinfimumandthe supremuminτ ∈]0,Λ/d+δ[inequation(14)
and adding and subtracting γ(t)+σ(t)+d(t)+α (t), we get (6) and (7).
1
We will now show that in fact r (λ), r (λ) and r (λ) are independent of the
p e,1 e,2
particular solution x(t) of (5) with x(0) > 0. In fact, by iii) in Proposition 2, for
10 CE´SARM.SILVA
every ε > 0 and every solution x (t) of (5) with x (0) > 0 there is a T > 0 such
1 1 ε
that |x(t)−x (t)|<ε for every t≥T . Let δ ∈]0,M[and choose ε >0 such that,
1 ε δ
for ε < ε and t ≥ T , we have x(t),x (t),x(t)±ε ∈]0,Λ/d+δ[. Leting a = x(t),
δ ε 1
b=x (t) and ε ,ε =0 we obtain by (6)
1 1 2
t+λ t+λ t+λ
b (s,x (s))ds−λKε< b (s,x(s))ds< b (s,x (s))ds+λKε,
I,δ 1 I,δ I,δ 1
Zt Zt Zt
for every t≥T . We conclude that, for every 0<ε<ε ,
ε δ
t+λ t+λ
liminf b (s,x(s))ds−liminf b (s,x (s))ds <λKε,
I,δ I,δ 1
(cid:12)(cid:12)t→+∞ Zt t→+∞Zt (cid:12)(cid:12)
(cid:12) (cid:12)
and thu(cid:12)s (cid:12)
(cid:12) (cid:12)
t+λ t+λ
liminf b (s,x(s))ds =liminf b (s,x (s))ds.
I,δ I,δ 1
t→+∞ t→+∞
Zt Zt
Letting δ → 0 we conclude that r (λ) (and thus R (λ)) is independent of the
p p
chosen solution. Using (7), the same reasoning shows that r (λ) and r (and
e,1 e,2
thus R (λ) andR ) are also independent of the particular solution. This proves
e,1 e,2
the lemma.
7.2. Proof of Theorem 1. Assume that R (λ)>1 (and thus r (λ)>0) and let
p p
(S(t),I(t),R(t),Q(t)) be any solution of (1) with S(T )>0, I(T )>0, R(T )>0
0 0 0
and Q(T )>0 for some T ≥0.
0 0
Let x∗(t) be some fixed solution of (5) with x∗(0)>0. Then, by ii) in Proposi-
tion 2 we have x∗(T ) > 0. Since r (λ) > 0, using (6) we conclude that there are
0 p
constants δ ∈]0,M[, N >0 and T ≥0 such that
1 1
t+λ
b (s,x∗(s)−θ)ds>N (15)
I,δ1
Zt
for all t ≥ T and θ > 0 sufficiently small, say for 0 < θ ≤ ε¯. By iii) and iv) in
1
Proposition1wemayassumethatfort≥T wehave(S(t),I(t),R(t),Q(t))∈∆ .
0 δ1
Define
N ε d
0
ε =min ,ε¯ and ε = . (16)
0 1
4Kλ 2K(Λ/d+δ )
(cid:26) (cid:27) 1
We will show that
limsupI(t)≥ε . (17)
1
t→+∞
Assume by contradiction that (17) is not true. Then there exists T ≥0 satisfying
2
I(t)<ε for all t≥T . Consider the auxiliary equation
1 2
x′ =Λ(t)−d(t)x−K(Λ/d+δ )ε . (18)
1 1
Let x¯(t) be the solution of (5) with x¯(T ) = S(T ) and let x(t) be the solution
0 0
of (18) with x(T ) = S(T ). By iv) in Proposition 2 and (16), for all t ≥ T , we
0 0 0
have
K ε
0
|x(t)−x¯(t)|≤ (Λ/d+δ )ε = . (19)
1 1
d 2
According to iii) in Proposition 2, x∗(t) is globally uniformly attractive on R+.
0
Thus there exists T >0 such that, for all t≥T , we have
3 3
ε
|x¯(t)−x∗(t)|≤ 0. (20)
2