Criterion of positivity for semilinear problems with applications in biology

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by  Michel Duprez

Noname manuscript No.
(will be inserted by the editor)
Criterion of positivity for semilinear problems with
applications in biology
Michel Duprez · Antoine Perasso
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Received: date/Accepted: date
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n Abstract The goal of this article is to provide an useful criterion of posi-
a tivity and well-posedness for a wide range of infinite dimensional semilinear
J
abstract Cauchy problems. This criterion is based on some weak assumptions
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on the non-linear part of the semilinear problem and on the existence of a
1
stronglycontinuoussemigroupgeneratedbythedifferentialoperator.Toillus-

] trate a large variety of applications, we exhibit the feasibility of this criterion
P
throughthreeexamplesinmathematicalbiology:epidemiology,predator-prey
A
interactions and oncology.
.
h
Keywords Positivity · Well-Posedness · Dynamic Systems · Semilinear
t
a Problems · Population Dynamics
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Mathematics Subject Classification (2000) 35A01 · 35B09 · 35Q92 ·
[
92D25

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v
4
1 1 Introduction
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5
Inawide rangeofmathematicalmodelling ofnaturalphenomena,the quanti-
0
ties that are described throughthe mathematical system haveto satisfy some
.
1
0
M.Duprez
7 InstitutdeMathmatiques deMarseilleUMRCNRS7373
1 Universit´eAix-Marseille
: 39,rueF.JoliotCurie,13453MarseilleCedex13,France
v
Tel.:[phone]
i
X E-mail:[email protected]
Correspondingauthor
r
a A.Perasso
Chrono-environnementUMRCNRS6249
Universit´eBourgogneFranche-Comt´e
16routedeGray,25000Besan¸con, France
2 MichelDuprez,AntoinePerasso
positivity properties to ensure physical reality. For instance, when consider-
ing the evolution of matter quantities, such as in biology (or also physics [1],
chemistry [15],...), the positivity of the solutions of the underlying dynamical
system is a crucial prerequisite to achieve the well-posedness of the problem
and to guarantee its physical relevance.
A significant proportion of dynamical systems that describe the evolution
over time of matter quantities are non-linear, but it oftenly appears that the
non-linear effects can be seen as perturbations of linear dynamics, leading to
such a differential formulation:
lineardynamics perturbations
y′(t)= Ay(t) + f(y(t),t) ,t≥0, (1)

y(0)=y , z }| { z }| {
0
wherey(t)denotesthemodeledmatterquantityattimet,thatmathematically
liesinaBanachlattice.Whenimposinganon-negativeinitialconditiony ,the
0
questionofpositivityisthencrucialtostudy.Inthecaseofafinitedimensional
operatorA,this questionhasbeenextensivelystudied(see [14]andreferences
therein for general results). However, to our knowledge, we don’t know any
general criterion of positivity in the case where A is a differential operator,
i.e. when the first equality in (1) rewrites as partial differential equations
(PDEs),whilesuchdifferentialoperatorsareextensivelyusedinmathematical
biology,or also in many other applied mathematical sciences.For instance, in
the specific case of biology, let us mention the use of structured population
dynamicsmodels,wheretheoperatorisoftransporttype,ortheuseofdiffusive
processes,wheremodelsincorporateaLaplacianoperator(see[13]forareview
of positivity results in reaction-diffusion systems).
Thegoalofthisarticleistoprovideanusefulcriterionofwell-posednessand
positivity for the semilinear problem (1) for wide ranges of linear differential
operators A and non-linear functions f, and then to illustrate the feasibility
of this criterion through three examples of models arising from mathematical
biology: epidemiology, predation and oncology.
This article is structured as follows: Section 2 is dedicated to the intro-
duction of three concrete biological models, described by semilinear PDEs,
for which the positivity of solutions must necessarilly be satisfied. Then we
tackle in Section 3 the formulation and the proof of the criterionof positivity
and well-posedness. This criterion is based on the formulation of an abstract
semilinear Cauchy Problem, studied using a semigroup approach. Finally, in
Section4,weapplythecriteriontothe biologicalmodelsofSection2toprove
the well-posedness and the positivity of their solution.
2 Three biological examples
In this section, we introduce three examples of semilinear evolutionary prob-
lemsinmathematicalbiologyforwhichthepositivityandwell-posednesshave
Criterionofpositivityforsemilinearproblemswithapplicationsinbiology 3
tobeprovedforbiologicalpurpose.Thematterquantitiesthataremodelledin
those three examples, i.e. populations, predator/prey or cell densities, evolve
with respectto the time t≥0.The epidemiologicalandpredator-preymodels
deal with transport process, with a non-constant velocity in the epidemiolog-
ical case and a non-local boundary condition in the predator-prey case, while
the model in oncology deals with diffusive PDEs.
One can note that through those specific examples, a large spectrum of bio-
logical models are involved: PDE structured population models (see [6] and
references therein) and reaction-diffusion models.
Epidemiology The first example on which we focus deals with epidemiology.
When modeling the transmission of disease between individuals, a common
way is to split the population densities into two sub-classes that are the sus-
ceptible class (S) and the infected class (I). From such a splitting results
theclassicalepidemiologicalmodelofSItype[5].Furthermore,lotsofdiseases
(influenza,HIV,prionpathologies...)haveavaryingintensityduringtheirevo-
lution that may be important to take into account in the modeling process.
This phenomena was recently described in [10,11], where the disease inten-
sity was incorporatedinto the infected class,leading to the formulationof the
following infection load-structured epidemiological model of transport type:
S′(t)=γ−(µ +α)S(t)−S(t)T(βI)(t), t≥0,
0
∂ I(t,i)=−∂ (νiI(t,i))−µ(i)I(t,i)+φ(i)S(t)T(βI)(t), t≥0, i∈J,
 t i
νκI(t,κ)=αS(t), t≥0,
S(0)=S , I(0,·)=I (·),
0 0
where the infection load is i ∈J =(κ,+∞)⊂ R , T is the integral opera(t2o)r
+
defined for some integrable fonction h on J by
T :h→ h(i)di
ZJ
and the epidemiological parameters satisfy the following assumptions:
- β, µ , ν, α>0 and γ ≥0;
0
- φ ∈ C∞(J) is a non-negative function such that lim φ(i) = 0 and
i→+∞
φ(i)di=1, µ∈L∞(J) is such that µ(i)≥µ for almost every i∈J.
J 0
ForRa biological relevance, it is clear that for each positive initial condition
(S ,I (·)),thedensitiesS(t)andI(t,·)inProblem(2)havetoremainpositive
0 0
whenever they exist.
Predator-prey interactions When considering predator-prey interactions, the
age of the prey is a key factor of selection for the predator. It is therefore
natural to add a structuration of the prey densities according to their age. In
4 MichelDuprez,AntoinePerasso
doing so,the classicalLotka-Volterramodel, thatwasinitially anODE model
[8], turns into the following PDE model, that is developed in [12]:
∂ x(t,a)+∂ x(t,a)=−µ(a)x(t,a)−y(t)γ(a)x(t,a), t≥0, a≥0,
t a
y′(t)=αy(t) ∞γ(a)x(t,a)da−δy(t), t≥0, a≥0,
 0 (3)
x(t,0)= 0∞βR(a)x(t,a)da, t≥0
x(0,·)=x (·), y(0)=y ,
R0 0
where x and y denote the density of preys and predators, respectively. The
assumptions on the parameters are the following:
- α ∈]0,1[, δ > 0 are constant parameters that respectively denote the as-
similation coefficient of ingested preys and the basic mortality rate of the
predators;
- µ,γ,β ∈L∞(R )areage-dependentfunctionsthatrepresent,respectively,
+ +
thebasicmortalityrateofthepreys,thepredationrateandthebirthrate.
Toensureacertainrealism,wewantthatthedensitiesofpreysxandpredators
y remain positive given a positive initial data (x ,y ).
0 0
Oncology Thethirdapplicationisamodelthatdescribesthegrowthofabrain
tumour published in [3]. The model aims at studying a treatment method of
tumor cells through a problem of controllability. The tumor and normal cells
areincompetitionfortheresourcesandaresubjecttoadrugtreatmentwhose
role is to decrease the cell densities. Even if some normal cells are destroyed,
the key point here is that the drug affects more the tumor ones.
To make explicit the model, let us consider Ω a bounded domain of RN,
N ∈ N∗, with boundary ∂Ω of class C2 and for a fixed T > 0, let Q =
T
Ω×(0,T)andΣ =∂Ω×(0,T).Theevolutionproblemisthenwrittenusing
T
the following three semilinear heat equations, where the variables (t,x) are
delibarately avoided for a better reading:
∂ y =d ∆y +a g (y )y −(α y +κ y )y ,
t 1 1 1 1 1 1 1 1,2 2 1,3 3 1
∂ y =d ∆y +a g (y )y −(α y +κ y )y ,
 t 2 2 2 2 2 2 2 2,1 1 2,3 3 2
∂∂tnyy3i(=t)d=3∆∇yy3i(−t)a·3−→ny3=+0u,, t≥0,i∈{1,...,3}, (4)
y(x,0)=y (x), x∈Ω,
where y := (y ,y ,y )∗, −→0n denotes the external normalized normal to the
1 2 3
boundary ∂Ω. Here y (t,x) stands for the density of tumor cells, y (t,x) the
1 2
density of normal tissue and y (t,x) the drug concentration at any vector
3
position x and time t. In the latter problem, the growth rates of cells are
defined by the functions g according to the following logistic shape:
i
g (y )=1−y /k .
i i i i
The assumptions on the parameters are the following:
Criterionofpositivityforsemilinearproblemswithapplicationsinbiology 5
- d >0 are the coefficients for the space diffusive effect;
i
- a > 0, where a , resp. a , denotes the tumor cell intrinsic growth rate,
i 1 2
resp.thenormaltissueintrinsicgrowthrateanda isthedrugreabsorption
3
coefficient;
- k >0 denote the carraying capacity of the medium;
i
- α >0arecoefficientsthattranslatetheinterspecificcompetitionbetween
i,j
tumor and normal cells;
- κ ≫κ >0 are the degradation rates due to the treatment;
1,3 2,3
- u(x,t)≥0 represents the flux of injected drug over time at position x.
Similarlytothepreviousbiologicalexamples,weaimatprovingwell-posedness
and positivity of the solution.
3 A criterion of positivity and well-posedness
In allthis section,let us consider(W,+,k·k ,≥)a Banachlattice (see [7, p.
W
6]), i.e. an partially ordered Banach space for which any given elements x,y
of W have a supremum sup(x,y) and for all y , y , y ∈W and α≥0,
1 2 3
y ≤y ⇒(y +y ≤y +y and αy ≤αy ),
1 2 1 3 2 3 1 2
(5)
(|y1|W ≤|y2|W ⇒ky1kW ≤ky2kW,
with, for all y ∈ W, |y| = sup(y, −y). We will denote by W+ = {y ∈ W :
W
0 ≤ y} the non-negative cone and for every m > 0 by B the ball of W of
m
radius m.
We consider in this work the system
y′(t)=Ay(t)+f(y(t),t), t≥0 in W,
(6)
(y(0)=y0 in W,
where A : D(A) ⊂ W → W is an infinitesimal generator of a C -semigroup
0
(T (t)) , y′(t) is an element of W and f :W ×R+ →W is continuous in t
A t≥0
and locally Lipschitz continuous in y uniformly in t in the following sense: for
every m>0 there exists a constant k >0 such that for every z ,z ∈B ,
m 1 2 m
kf(z ,t)−f(z ,t)k ≤k kz −z k , ∀t∈R+.
1 2 W m 1 2 W
Finally, let us briefly remind that for a fixed T ∈]0,∞], a mild solution of
Problem (6) on [0,T[ is a function y ∈ C([0,T[;W) that satifies the integral
equation
t
y(t)=T (t)y + T (t−s)f(y(s),s)ds.
A 0 A
Z0
Remark 1 Since W+ is closed (see [7]), we deduce that for all T > 0, the
order ≥ is compatible with the integration in time, more precisely, for all
x,y ∈C([0,T];W),
T T
(x(t)≥y(t) ∀ t∈[0,T])⇒ x(s)ds≥ y(s)ds. (7)
Z0 Z0
6 MichelDuprez,AntoinePerasso
The following theorem, that states well-posedness and positivity property
for the solution of Problem (6), is the main result of the present article:
Theorem 1 Let y ∈W+. We suppose that
0
(i) A is generator of a positive C -semigroup on W, i.e. T (t)W+ ⊂ W+ for
0 A
all t≥0,
(ii) for all m > 0, there exists λ ∈ R such that, for all z ∈ C(R+;W+ ∩
m
B(0,m)),
f(z(t),t)+λ z(t)≥0, ∀t≥0. (8)
m
Then there exists t ∈]0,∞] such that system (6) has an unique positive
max
mild solution y ∈C([0,t [;W). Moreover, if t <∞,
max max
lim ky(t)k =∞.
W
t→tmax
Themainideaoftheproofistoperformavectorialtranslationtotherange
values ofthe non-linearpartf so thatthey remaininW+. This translationis
thencompensatedbythesubstractionofalineartermtothedifferentialoper-
ator, that does not affect its spectral and positivity properties. Consequently,
we shall study the following system in the proof of the theorem:
y′(t)=(A−λI)y(t)+f(y(t),t)+λy(t), t>0in W,
(9)
(y(0)=y0 in W.
Remark 2 Since A is an infinitesimal generator of a positive C -semigroup
0
(T (t)) , then, for every λ ∈ R, A−λI is also an infinitesimal generator
A t≥0
of a positive C -semigroup(T (t)) . Indeed, we remark that T (t)=
0 A−λI t≥0 A−λI
e−λtT (t) for all t≥0.
A
Proof (Proof of Theorem 1.1) Without loss of generality, we can assume that
λ is nonnegative in (8). Since A is generator of a positive C -semigroup
m 0
(T (t)) , there exists ω,M ≥1 such that, for all t∈R+,
A t≥0
kT (t)k ≤Meωt.
A W
Remark 2 then implies that for evey λ ∈ R, A−λI is also generator of a
positive C -semigroup(T (t)) . Moreover,it is easy to check that for all
0 A−λI t≥0
t∈R+,
kT (t)k ≤Meωt, ∀λ∈R+. (10)
A−λI W
Let t ∈ (0,1), m = 2Meωky k and λ that satisfies (8). Consider the
0 0 W m
set Γ = {y ∈ C([0,t ];W+) : y(0) = y ,ky(t)k ≤ m,∀t ∈ [0,t ]}. The
m 0 0 W 0
continuity properties of the lattice operations (see [7]) imply that Γ is a
m
non-empty closed subset of C([0,t ];W).
0
Consider now the mapping ψ, defined on Γ by
m
t
ψ(y)(t)=T (t)y + T (t−s)[f(y(s),s)+λ y(s)]ds, t∈[0,t ].
A−λmI 0 A−λI m 0
Z0
Criterionofpositivityforsemilinearproblemswithapplicationsinbiology 7
We aim at proving that ψ has a unique fixed point in Γ .
m
Letus startbyprovingthatψ preservesΓ .The positivityof(T (t))
m A−λmI t≥0
and the positivity assumption (8) clearly imply that ψ(y) ∈ C([0,t ];W+).
0
Furthermore, from the inequality (10), one deduces that
t0
kψ(y)(t)k ≤Meωtky k +Meωt (kf(y(s),s)−f(0,s)k
W 0 W W
Z0
+kf(0,s)k +λ ky(s)k )ds.
W m W
Thetimecontinuitypropertyonf inducestheexistenceofγ >0(independent
of t <1) such that for every y ∈Γ and every t∈(0,t ),
0 m 0
kψ(y)(t)k ≤Meω(ky k +t (mk +γ+mλ )).
W 0 W 0 m m
Thus, for t =min{1,ky k ×(mk +γ+mλ )−1} we have kψ(y)(t)k ≤
0 0 W m m W
2Meωky k =m and so ψ(y)∈Γ .
0 W m
Wenowprovethatψ iscontractantinthefollowingsense:foreveryy,z ∈Γ ,
m
every n∈N∗ and every t∈[0,t ],
0
[Meωt(k +λ )]n
kψn(y)(t)−ψn(z)(t)k ≤ m m sup ky(t)−z(t)k . (11)
W W
n!
t∈[0,t0]
Let us prove (11) by induction. By definition of Γ , we have
m
ky(t)k , kz(t)k ≤m
W W
for all t∈[0,t ]. Then the Lipschitz assumption on f implies that
0
kψ(y)(t)−ψ(z)(t)k ≤Meω(k +λ )t sup ky(θ)−z(θ)k ,
W m m W
θ∈[0,t0]
and equality (11) holds for n =1. Suppose now that (11) holds for a k ∈N∗.
Then for all t∈[0,t ],
0
kψk+1(y)(t)−ψk+1(z)(t)k
W
t
≤(Meω(k +λ )) kψk(y)(s)−ψk(z)(s)k ds,
m m W
Z0
[Meω(k +λ )]k+1 t
≤ m m sup ky(θ)−z(θ)k skds,
W
k!
θ∈[0,t0] Z0
and (11) is true for k +1 and consequently for every n ∈ N∗ by induction.
Finally,wecanapplytheBanach’sfixedpointtheoremtoconcludethatψ has
auniquefixedpointy¯inΓ .Systems(6)and(9)beingequivalent,y¯isamild
m
solutionof (6). Then some standardtime extending properties of the solution
induce thatthe solutiony¯isdefinedonamaximalinterval[0,t [.Tofinish,
max
weprovetheuniquenessofthesolutiononthewholespaceC([0,t (y¯)[,W+).
max
Ifz¯isanothermildsolutiondefinedon[0,t [witht <t (y¯),then,denoting
1 1 max
R= max {ky¯(θ)k ,kz¯(θ)k }, we obtain for all t∈[0,t ],
W W 1
θ∈[0,t1]
t
ky¯(t)−z¯(t)k ≤Meωt1k ky¯(s)−z¯(s)k ds.
W R W
Z0
8 MichelDuprez,AntoinePerasso
Then ky¯(t) −z¯(t)k = 0 by a standard Gronwall argument and y¯ = z¯ in
W
[0,t ]×W. Furthermore, if t (y¯) < ∞, since kz¯(t)k = ky¯(t)k for all
1 max W W
t < min{t (y¯),t (z¯)} and lim ky¯(t)k = ∞, we deduce that the
max max W
t→tmax(y¯)
maximal intervals of existence of y¯and z¯are equal.
4 Illustrations of the criterion in mathematical biology
In this section, we exhibit the application of well-posedness and positivity
criterion on the three biological examples of Section 2.
Epidemiology Consider the Banach lattice X = R × L1(J), X+ the non-
negative cone of X and y = (S ,I ) ∈ X+. Then it is clear that Problem
0 0 0
(2) can rewrite as (6), where the function f : X → X and the differential
operator A:D(A)⊂X →X are given by
f (u,v) γ−uT(βv) −µ −α 0
f(u,v)= 1 = , A= 0 ,
f (u,v) φuT(βv) 0 −d(νi·)−µ
(cid:18) 2 (cid:19) (cid:18) (cid:19) (cid:18) di (cid:19)
with D(A) = {(x,ϕ) ∈ X,(iϕ) ∈ W1(J) and νκϕ(κ) = αx}. In [10], the au-
1
thors prove that the differential operator (A,D(A)) is an infinitesimal gen-
erator of a positive C -semigroup (T (t)) on X and that function f is
0 A t≥0
locally Lipschitz continuous on X. Moreover, for every m > 0 and every
(S¯,I¯)∈C(R+;X+∩B(0,m)), one gets, denoting λ =mβ,
m
f (S¯(t),I¯(t))+λ S¯(t)≥γ+S¯(t)(λ −βT(I¯(t)))≥0,
1 m m
f (S¯(t),I¯(t))+λ I¯(t)=φS¯(t)T(βI¯(t))+λ I¯(t)≥0.
2 m m
(cid:26)
Thus, condition (8) of Theorem 1 is satisfied and there exists t ∈]0,∞]
max
such that Problem (2) has an unique mild solution (S,I) in C([0,t [,X+).
max
Predator-prey interactions Let X = L1(R+)×R, X+ the non-negative cone
and (x ,y ) ∈ X+. Considering the operator A : D(A) ⊂ X → X and the
0 0
functional f :X →X given by
f (φ,z) −zγφ L 0
f(φ,z)= 1 = ∞ , A= ,
f (φ,z) αz γ(a)φ(a)da 0 −δ
(cid:18) 2 (cid:19) (cid:18) 0 (cid:19) (cid:18) (cid:19)
with D(A) = {(φ,z) ∈ X,φ ∈ WR1(R+) and ϕ(0) = ∞β(a)φ(a)da} and
1 0
Lφ = −φ′ − µφ. The map f is clearly locally Lipschitz continuous on X.
R
Furthermore,under the assumptionthat there exists µ >0 such that µ(a)≥
0
µ f.a.e. a ∈ R+ the operator A is the infinitesimal generator of a positive
0
C -semigroup (T (t)) on X. This is a standard result that we can find for
0 A t≥0
example in [2]. Then, for all m > 0, denoting λ = mγ, we obtain for every
m
(x¯,y¯)∈C(R+;X+∩B(0,m))
f (x¯(t),y¯(t))+λ x¯(t)≥x¯(t)(λ −αmγ)≥0,
1 m m
f (x¯(t),y¯(t))+λ y¯(t)≥0.
2 m
(cid:26)
Criterionofpositivityforsemilinearproblemswithapplicationsinbiology 9
Again, condition (8) of Theorem 1 holds and the existence of t ∈]0,∞]
max
such that system (3) has an unique mild solution (x,y) in C([0,t [,X+) is
max
ensured.
Oncology Let X =L2(Ω;R3), X+ the corresponding non-negative cone, y ∈
0
X+ and u∈L2(Q )+. Then system (4) can be reformulated as (6) where
T
f(y)=(g+h)(y)+(0,0,u)∗,
g(y)=diag(a g (y )y ,a g (y )y ,−a y ),
 1 1 1 1 2 2 2 2 3 3
h(y)=diag(−(α1,2y2+κ1,3y3)y1,−(α2,1y1+κ2,3y3)y2,0),
A=diag(d ∆,d ∆,d ∆).
1 2 3
The existence of the semigroup (T (t)) is a consequence of the Lumer-
A t≥0
Phillips Theorem (see [9, p. 14]) for maximal dissipative operators. Indeed,
in the present case, A is clearly maximal dissipative since it is defined with
Laplacian operators. Using the maximum principle of the heat equation, the
semigroup is positive.
Consequently, when taking λ =max{m(a /k −α −κ ),m(a /k −
m 1 1 1,2 1,3 2 2
α −κ ),a } for m > 0, we obtain the following estimations for all y¯ =
2,1 2,3 3
(y¯ ,y¯ ,y¯ )∈C(R+;X+∩B(0,m))
1 2 3
f (y¯)+λ y¯ = a g (y¯ )y¯ −(α y¯ +κ y¯ )y¯ +λ y¯
1 m 1 1 1 1 1 1,2 2 1,3 3 1 m 1
≥ y¯ [λ −m(a /k −α −κ )]≥0,
 1 m 1 1 1,2 1,3
f2(y¯)+λmy¯2 ≥= ya¯22[gλ2m(y¯−2)y¯m2(−a2(/αk22,1−y¯1α+2,1κ2−,3κy¯32,)3y¯)2]+≥λ0m, y¯2
f (y¯)+λ y¯ = −a y¯ +u+λ y¯ ≥y¯ (λ −a )≥0.
3 m 3 3 3 m 3 3 m 3
Thuscondition(8)issatisfiedand,usingTheorem1,thereexiststmax ∈]0,∞]
such that problem (4) has an unique mild solution (x,y) in C([0,t [,X+).
max
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