Fundamental properties of differential equation with dynamically defined delayed feedback”, preprint

We consider an initial value problem for nonautonomous functional dierential equations where the delay term is defined via the solution of another system of dierential equations. We obtain the general existence and uniqueness re- sult for the initial value problem by showing a Lipschitz property of the dynamically defined delayed feedback function and give conditions for the nonnegativity of solu- tions. We determine the steady-state solution of the autonomous system and obtain the linearized equation about the equilibria. Our work was motivated by biological applications where the model setup leads to a system of dierential equations with such dynamically defined delay terms. We present a simple model from population dynamics with fixed period of temporary separation and an epidemic model with long distance travel and entry screening.


Introduction
The topic of delay differential equations concerns with a type of differential equations in which the derivative of the unknown function at a certain time is determined by the values of the function at previous times.Many studies focus on well known model equations where the delayed feedback function is given explicitly (Mackey-Glass equation, Nicholson's blowflies, Wright's equation etc.), while others only require some important properties for the feedback, for example monotonicity or unimodularity.However, the study of some models from population dynamics and epidemiology leads to differential equations where the delay terms arise as the solution of another system of differential equations.In this work, we consider initial value problems for differential equations with such dynamically defined delayed feedback.Our goal is to obtain fundamental properties of the system to ensure that the model equations coming from biological applications are meaningful.The paper is organized as follows.In Section 2 we introduce the general form of systems of functional differential equations with dynamically defined delay term, and we prove the existence and uniqueness of the solution in Section 3. Due to possible biological interpretations, we give conditions so that the solution preserves nonnegativity.In Section 4 we detail the autonomous case, determine equilibria of the system and formulate the linearized equation, while Section 5 concerns with two examples of applications in population dynamics and the spread of infectious diseases with travel delay.

A system of differential equations with dynamically defined delayed feedback
Consider the initial-value problem for the nonautonomous functional differential equation (1) x (t) = F(t, x t ), x σ = ϕ, where x : R R n , n ∈ Z + , t, σ ∈ R and t ≥ σ.For τ > 0, we define our phase space C = C([−τ, 0], R n ) as the Banach space of continuous functions from [−τ, 0] to R n , equipped with the usual supremum norm || • ||.Let ϕ ∈ C be the state of the system at σ.We use the notation x t ∈ C, x t (θ) = x(t + θ) for θ ∈ [−τ, 0].Let F : R × C R n and let F have the special form F(t, φ) = f (t, φ(0)) + W (t, φ(−τ )) for φ ∈ C, f : R × R n R n , W : R × R n R n .In the sequel we use the notation |v| k for the Euclidean norm of any vector v ∈ R k for k ∈ Z + .For k = 1 we omit lower index 1 for simplicity.We define a Lipschitz condition as follows.For k, l ∈ Z + , we say that a function F : R × R k R l satisfies the Lipschitz condition (Lip) on each bounded subset of R × R k if: (Lip) For all a, b ∈ R and M > 0, there is a K(a, b, M ) > 0 such that: We assume that f : R × R n R n is continuous and satisfies (Lip) on each bounded subset of R × R n .For the definition of W , we make the following preparations.For any s 0 ∈ R and y * ∈ R m , m ∈ Z + , we consider the initial value problem (2) y (s) = g(s, y(s)), y(s 0 ) = y * , where y : R R m , s, s 0 ∈ R, s ≥ s 0 , g : R×R m R m , g is continuous on R×R m and satisfies the Lipschitz condition (Lip) on each bounded subset of R×R m .The Picard-Lindelöf theorem (see Chapter II, Theorem 1.1 and Chapter V, Theorem 2.1 in [2]) states that as g is continuous on a parallelepiped R : with the bound B for |g| m on R and g possesses the Lipschitz property (Lip), there exists a unique solution of (2) y(s; s 0 , y * ) on the interval [s 0 , s 0 + α] for α = α s 0 ,y * ,c,d := min{c, d B }, and the solution continuously depends on the initial data.We make the following additional assumption: ( ) For every s 0 and y * , the solution y(s; s 0 , y * ) of (2) exists at least for τ units of time, i.e. on [s 0 , s 0 + τ ].
Remark 2.1.The reader may notice that ( ) is equivalent to the following assumption: For every s 0 and y * , solution y(s; s 0 , y * ) exists for all s ≥ s 0 .
Remark 2.2.With various conditions on g, we can guarantee that assumption ( ) is fulfilled.For instance, for any s 0 ∈ R and L ∈ R + , we define the constant L g = L g (s 0 , L) as the maximum of |g| m on the set [s 0 , s 0 + τ ] × {v ∈ R m : |v| m ≤ 2L} (continuous functions attain their maximum on every compact set).Then, the condition that for every s 0 ∈ R and L ∈ R + the inequailty The less restrictive condition κ := inf s 0 ,L L Lg > 0 implies the existence of the solution of (2) on [s 0 , s 0 + κ] for any s 0 .Then it follows that for any s 0 the solution exists for all s ≥ s 0 , which is equivalent to ( ).If we assume that a global Lipschitz condition (gLip) holds for g, that is, the Lipschitz constant for g in (Lip) can be chosen independently of a, b and M , then for any s 0 and y * the solution of (2) exists for all s ≥ s 0 , thus also for τ units of time.Now we are ready for the definition of W .For h : R×R n R m , k : R×R m R n , let us assume that h and k are continuous and satisfy the Lipschitz condition (Lip).For simplicity, we use the notation y s 0 ,v (s) = y(s; s 0 , h(s 0 , v)) for the unique solution of system (2) in the case

Basic properties
Our goal is to prove the usual existence and uniqueness theorem for (1).First we obtain the following simple results.
Proof.The Picard-Lindelöf theorem and ( ) guarantee that for every s 0 , y * , there exists a unique solution of system (2) on the interval [s 0 , s 0 + τ ] and the solution y(s; s 0 , y * ) continuously depends on the initial data.Moreover, h and k are continuous which implies the continuity of W .The function f is also continuous, hence we conclude that F is continuous on R × C. Proof.The Picard-Lindelöf theorem and ( ) guarantee that for every s 0 ∈ R and y * ∈ R m , there exists a unique solution y(s; s 0 , y * ) of system (2) on the interval [s 0 , s 0 + τ ], and the solution continuously depends on the initial data.Thus, for any c, d ∈ R where c < d and for any L ∈ R + , the solution y(s; s 0 , y * ) Continuous functions reach their maximum on any compact set, i.e. there exists a constant J(c, d, L) such that |y(s; s 0 , y * )| m ≤ J.The proof is complete.
Now we show that besides continuity, F also satisfies a Lipschitz condition on each bounded subset of R × C: Proof.Fix constants a, b and M , a < b, M > 0. Our aim is to find K(a, b, M ).Due to the continuity of h, there exists a constant L h (a, b, M ) such that for any ||ψ|| ≤ M and s and from Gronwall's inequality we have that for any s as a special case of (6) with s 0 = t−τ .The constant J = J(a, b, L h ) was defined as the bound for so we arrive to the following inequality: where we used ( 4) and (7).
We state the following simple remark.Remark 3.4.If f , g, h and k satisfy a global Lipschitz condition (gLip), that is, if K f , K g , K h and K k in the definition of the Lipschitz condition (Lip) can be chosen independent of a, b and M , then a global Lipschitz condition (gLip C ) for F arises, i.e. there exists a Lipschitz constant K of F which is independent of a, b and M .Now, as we have proved that F is continuous and satisfies the Lipschitz condition (Lip C ), all conditions of Theorem 3.7 in [4] are satisfied.We arrive to the following result.
Assuming stronger conditions on f , g, h and k, we arrive to a more general existence result.We follow Remark 3.8 in [4].
Remark 3.6.If f , g, h and k satisfy condition (gLip), then condition (gLip C ) arises for F and we do not need to make any restrictions on A in Theorem 3.5.More precisely, its statements hold for all A > 0. In this case, the solution exists for all t ≥ σ and the inequality holds for all t ≥ σ.
Most functional differential equations that arise in population dynamics or epidemiology deal only with nonnegative quantities.Therefore it is important to see what conditions ensure that nonnegative initial data give rise to nonnegative solution.We reformulate (1) using the definition of F. Since F(t, x t ) = f (t, x(t)) + W (t, x(t − τ )), we consider the following differential equation system, which is equivalent to (1): We claim that under reasonable assumptions the solution of system (8) preserves nonnegativity for nonnegative initial data.Let us suppose that for each t ∈ R, h and k map nonnegative vectors to nonnegative vectors.We also assume that for every i ∈ {1, . . .n}, j ∈ {1, . . .m}, u ∈ R n + , w ∈ R m + and t, s ∈ R, u i = 0 implies f i (t, u) ≥ 0 and w j = 0 implies g j (s, w) ≥ 0. Then for nonnegative initial value the solution of system (2) is nonnegative, which implies that for every i ∈ {1, . . .n}, v ∈ R n + and t ∈ R, the inequality (k(t, y(t all conditions of Theorem 3.4 in [4] hold and we conclude that nonnegative initial data give rise to nonnegative solution of system (8).Clearly, systems (8) and ( 1) are equivalent, which implies that the result automatically holds for system (1).We summarize our assumptions and consequence.Proposition 3.7.Suppose that h : R × R n R m and k : R × R m R n map nonnegative vectors to nonnegative vectors for each t ∈ R, moreover assume that Then for nonnegative initial data the solution of system (1) preserves nonnegativity i.e. x(t) ≥ 0 for t ≥ σ where it is defined.
4 The autonomous case

Fundamental properties
As a special case of system (1), we may derive similar results for the autonomous system.Let Let us assume that f, g, h and k satisfy the Lipschitz condition (Lip), which can be stated as follows.For k, l ∈ Z + , we say that a function F : R k R l satisfies the Lipschitz condition (Lip) if for all M > 0 there is a There is no need to assume the continuity for f, g, h and k since these functions are independent of t and hence this property follows from the Lipschitz condition (Lip).For τ > 0, let C = C([−τ, 0], R n ) be the phase space as we defined it in Section 2. Then system (1) has the form (9) where t ≥ 0, ϕ ∈ C is the state of the system at t = 0, F : C R n and F has the special form where s ≥ 0. Similarly as in Section 2, the Picard-Lindelöf theorem guarantees the existence and uniqueness of the solution of system (10) on [0, α] for some α > 0. We make the following additional assumption: ( ) For every y * , solution y(s; 0, y * ) of (10) exists at least for τ units of time.
This is equivalent to the assumption that y(s; 0, y * ) exists on [0, ∞) for every y * , which holds if g satisfies (gLip) (see Remark 2.2).We use the notation y 0,v (s) = y(s; 0, h(v)) for the unique solution of system (10) in the case y * = h(v), and we define W : R n R n by ( 11) where v ∈ R n .It is straightforward that the Lipschitz condition (Lip C ) and therefore the continuity hold for F, since this function is only a special case of the F defined in Section 2. Moreover if we assume that f , g, h and k satisfy the global Lipschitz condition (gLip), then we obtain that condition (gLip C ) holds for F (for the definitions of (gLip) and (gLip C ) see Remark 3.4).As an immediate consequence of Theorem 3.5, we state the following corollary.The following remark rises automatically as the autonomous case of Remark 3.6.holds for all t ≥ 0.
Clearly, we can adapt Proposition 3.7 to the autonomous system with similar conditions.Corollary 4.3.Suppose that h : R n R m and k : R m R n map nonnegative vectors to nonnegative vectors, moreover assume that Then for nonnegative initial data the solution of system (9) preserves nonnegativity i.e. x(t) ≥ 0 for t ≥ 0 where it is defined.

Equilibria and linearization
Consider the nonlinear functional differential equation system (9): where F(φ) = f (φ(0)) + W (φ(−τ )) for φ ∈ C. Then x(t) = x ∈ R n is a steady-state solution of (9) if and only if F( x) = 0, where x ∈ C is the constant function equal to x. Suppose there exists such an equilibrium.We formulate the linearized equation system about the equilibrium x.We obtain the linear system (12) where DF( x) : C R n is a bounded linear operator and z : R R n .Due to the special form of F, (12) can be written as where Proof.Theorem 3.3 in Chapter I in [1] states that as g has continuous derivative, the solution y(s; 0, y * ) of system (10) is continuously differentiable with respect to s and y * on its domain of definition.The matrix ∂y(s;0,y * ) ∂y * ∈ R m×m satisfies the linear variational equation where Y : R R m×m (we use slightly different notations as [1]) and ∂y(0;0,y * )  Note that Dk(y(τ ; 0, h(x))) ∈ R n×m , Dg(y(r; 0, h(x))) ∈ R m×m and Dh(x) ∈ R m×n , hence the result of the matrix multiplication is indeed DW (x) ∈ R n×n .The proof is complete.
It follows from (11) that x satisfies the equation −f (x) = k(y(τ ; 0, h(x))).However, x being a steady-state solution of (9) does not necessarily imply that y(s, 0; h(x)) = h(x) for s ∈ [0, τ ] i.e. h(x) is an equilibrium of (10).It is easy to construct examples for f , g, h and k such that such situation occurs.We say that x ∈ R n is a total equilibrium of systems ( 9) and (10) if x(t) = x is a steady-state solution of (9) and y(s) = h(x) is a steady-state solution of (10).The equilibrium solution y(s) = ȳ, ȳ ∈ R m of (10) satisfies the equation g(ȳ) = 0, and since h(x) = ȳ and −f (x) = k(y(τ ; 0, h(x))) should hold for the total equilibrium, we conclude that x arises as the solution of the system It follows from (17) that in the special case when f and g are invertable functions, the total equilibrium can be expressed by x = f −1 (−k(g −1 (0))), and we also obtain that ȳ = h(f −1 (−k(g −1 (0)))).We remark that if functions g, h and k are continuously differentiable and x is the total equilibrium of systems ( 9) and (10), then it follows from Proposition 4.4 that the matrix DW (x) has the form DW (x) = Dk(h(x))e τ Dg(h(x)) Dh(x).

A basic model from population dynamics
A simple model describing the growth of a single population with fixed period of temporary separation is given by ( 18) where t denotes time and functions b, d and q stand for recruitment, mortality and temporary separation (e.g.migration).Let τ > 0 be the fixed period of separation.We define the phase space C + as the nonnegative cone of C = C([−τ, 0], R), let ϕ ∈ C + .We assume that b, d, q : R R satisfy the Lipschitz condition (Lip) on each bounded subset of R, which implies their continuity on R. Since b, d and q denote the recruitment, mortality and separation functions, it should hold that they map nonnegative values to nonnegative values.Function V expresses the inflow of individuals arriving to the population at time t after τ units of time of separation.For the thorough definition of V , the growth of the separated population needs describing.We assume that individuals who left the population due to separation in different times do not make contact to each other.Hence for each time t * , the evolution of the density of the separated population with respect to the time elapsed since the beginning of separation is given by the following differential equation, when separation started at time t * : where s denotes the time elapsed since the beginning of separation and functions b S and d S stand for recruitment and mortality during separation.At s = 0, the density of the separated population is determined by the number of individuals who start separation at time t * , hence the initial value for system (19) is given by m(0; t * ) = q(n(t * )).We assume that b S ,d S : R R satisfy the Lipschitz condition (Lip) on each bounded subset of R, this also means they are continuous on R. The Picard-Lindelöf theorem ensures that for any initial value m * there exists a unique solution y(s; 0, m * ) of system (19) on [0, α] for some α > 0. We make the additional assumption that the unique solution exists at least for τ units of time for every m * , we have seen in Section 2 that this assumption can be fulfilled with some conditions on b S and d S .In order to guarantee that nonnegative initial data give rise to nonnegative solution of (19), we assume that the inequality b S (0) − d S (0) ≥ 0 holds, this condition can be satisfied with many reasonable choices of the recruitment and mortality functions.We assumed that separation lasts exactly for τ units of time, i.e. the feedback in (18) at t * + τ is determined by the solution of ( 19) at s = τ .We define the feedback function V : R R as V (v) := y(τ ; 0, q(v)).
. For a given t * , define y(s) = m(s; t * ) and let g(y) = b S (y) − d S (y), where y : [0, τ ] R, g : R R. Furthermore, for h, k : R R let h(v) = q(v), k(v) = v.Then system (18) can be written in a closed form as (9) and for each t * (10) is a compact form of (19).Clearly, functions f , g, h and k satisfy the Lipschitz condition (Lip) on each bounded subset of R, moreover ( ) also holds by assumption.Hence F, defined by F(φ) = f (φ(0)) + W (φ(−τ )) for φ ∈ C + satisfies the Lipschitz condition (Lip C ), so Corollary 4.1 states that system (18) has a unique solution defined on [−τ, A] for some A > 0. By assuming that condition (gLip) holds for b, d, q, b S and d S , we get that f , g, h and k satisfy the global Lipschitz condition (gLip) and A = ∞.We have assumed that q = h maps nonnegative values to nonnegative values, which obviously holds for k as well, moreover we gave the condition b S (0) − d S (0) ≥ 0. In addition, if we suppose that b(0) − d(0) − q(0) ≥ 0 is satisfied (e.g.b(0) = d(0) = q(0) = 0 holds in many models), then Corollary 4.3 implies that for nonnegative initial data the solution of system (18) preserves nonnegativity, that is, C + is invariant.

Epidemic model with travel delay and entry screening
We formulate a dynamic model describing the spread of an infectious disease in two regions, and also during travel from one region to the other.We assume that the time required to complete travel between the regions is not neglectable, this leads to delay differential equations in the model setup.Several recent works (see e.g.[3] and [5]) considered SIS type transportation models where the delay terms arise explicitely.Here we present an epidemic model where delay is defined via the solution of another system of differential equations.We divide the entire populations of the two regions into the disjoint classes S 1 , I 1 , R 1 , J 1 , S 2 , I 2 , R 2 and J 2 .Lower index denotes the current region, letters S and R represent the compartments of susceptible and recovered individuals.We assume that individuals are traveling between the regions and travelers are requested to undergo an entry screening procedure before entering a region after travel.The purpose of the examination is to detect travelers who are infected with the disease and isolate them in order to minimize the chances of an infected agent spreading the infection in a disease free region.Such interventions were proven to have significant effect in mitigating the severity of epidemic outbreaks.Some individuals who have been infected with the disease get screened out by the arrival to a region and they become isolated and belong to class J. Others whose infection remained hidden by the examination or those who are sick but do not travel are in class I and we simply call them infecteds.Let S 1 (t), I 1 (t), R 1 (t), J 1 (t), S 2 (t), I 2 (t), R 2 (t) and J 2 (t) be the number of individuals belonging to S 1 , I 1 , R 1 , J 1 , S 2 , I 2 , R 2 and J 2 at time t, respectively.Susceptible, infected and recovered individuals of region 1 travel to region 2 by travel rate µ 1 .The travel rate of individuals in classes S 2 , I 2 and R 2 from region 2 to region 1 is denoted by µ 2 .Isolated individuals are not allowed to travel, moreover we assume that they do not make contact with individuals of other classes until they recover.Model parameters γ 1 and γ 2 represent the recovery rate of infected and isolated individuals in region 1 and region 2, we denote the transmission rates in region 1 and region 2 by β 1 and β 2 .Let s 1 , i 1 , r 1 , s 2 , i 2 and r 2 be the classes of susceptible, infected and recovered individuals during travel to region 1 and to region 2. We denote the recovery rate of traveling infecteds by γ T , they can transmit the disease by rate β T during travel.Let τ > 0 denote the time required to complete a one-way travel, which is assumed to be fixed.To describe the disease dynamics during travel, for each t * we define s 1 (s; t * ), i 1 (s; t * ), r 1 (s; t * ), s 2 (s; t * ), i 2 (s; t * ) and r 2 (s; t * ) as the density of individuals with respect to s who started travel at time t * and belong to classes s 1 , i 1 , r 1 , s 2 , i 2 and r 2 , where s ∈ [0, τ ] denotes the time elapsed since the beginning of the travel.The total density of traveling individuals is constant during the travel started at t * , Key model parameters β 1 , β 2 transmission rate in region 1 and in region 2 µ 1 , µ 2 traveling rate of individuals in region 1 and in region 2 γ 1 , γ 2 recovery rate of infected and isolated individuals in region 1 and in region 2 p 1 , p 2 probability of screening out infected travelers arriving to region 1 and to region 2 τ duration of travel between the regions β T transmission rate during travel γ T recovery rate during travel Table 1: Key model parameters that is, express the inflow of susceptible and recovered individuals arriving to region 1 to compartments S 1 , R 1 and region 2 to compartments S 2 , R 2 at time t, respectively.We assume that travelers undergo an examination by the arrival to region 1 and 2, which detects infection by infecteds with probability 0 < p 1 , p 2 < 1.This implies that the densities p 1 i 1 (τ ; t − τ ) and p 2 i 2 (τ ; t − τ ) determine individuals who enter J 1 and J 2 at time t since p 1 and p 2 are the probabilities that infected travelers get screened out by the arrival.However, infected travelers enter classes I 1 and I 2 with probabilities 1 − p 1 and 1 − p 2 by the arrival, hence (1 − p 1 )i 1 (τ ; t − τ ) and (1 − p 2 )i 2 (τ ; t − τ ) give the inflow to classes I 1 and I 2 at time t.
The flow chart of the model is depicted in Figure 1, see Table 1 for the key model parameters.We obtain the following system of differential equations for the disease spread in the regions, where disease transmission is modeled by standard incidence: (20) For each t * , the following system describes the evolution of the densities during the travel which started at time t * : (21) where again we assume standard incidence for the disease transmission.Note that the dimensions of systems ( 20) and ( 21) are different.For s = 0, the densities are determined by the rates individuals start their travels from one region to the other at time t * .Hence, the initial values for system (21) at s = 0 are given by ( 22) Now we turn our attention to the terms s 1 (τ ; t − τ ), (1 − p 1 )i  22).Thus, we set up the initial functions as follows: (23) where θ ∈ [−τ, 0] and for each j ∈ {1, 2}, K ∈ {S, I, R, J}, ϕ K,j is continuous. For For a given t * , let y(s) = (s 1 (s; t * ), i 1 (s; t * ), r 1 (s; t * ), s 2 (s; t * ), i 2 (s; t * ), r 2 (s; t * )) T and let g = (g 1 , g 2 , g 3 , g 4 , g 5 , g 6 ) T , where y : [0, τ ] R 6 and g : R 6 R 6 .Define g as g i (y) equals the right hand side of the equation for y i in system (21).For instance, g 5 (y) = β T y 4 y 5 y 4 + y 5 + y 6 − γ T y 5 .
Then for each t * , (10) is a compact form of (21) with the initial value y * in ( 22) for m = 6.Define h = (h 1 , h 2 , h 3 , h 4 , h 5 , h 6 ) : R 8 R 6 as The feasible phase space is the nonnegative cone to be the right hand side of the equation of x i in (20) without the inflow from travel.For instance, Clearly our system (20) with initial conditions (23) can be written in a closed form as (9) for n = 8.Our aim is to show that there exists a unique positive solution of system (20), moreover nonnegative initial data give rise to nonnegative solution.We showed in the previous sections that these results can be obtained by assuming certain conditions on f , g, h and k.Now we check if these conditions hold for the f, g, h and k defined for the SIRJ model.It is not hard to see that h and k possess the global Lipschitz condition (gLip), now we prove that it holds for f and g as well.
Proposition 5.1.Functions f and g, as defined for the SIRJ model, satisfy the global Lipschitz condition (gLip) on each bounded subset of R 8 + and R 6 + .

Proposition 3 . 2 .
For any c, d ∈ R such that c < d and for any L ∈ R + , there exists a bound J = J(c, d, L) such that for any s 0 ∈ [c, d] and for any y * ∈ R m such that |y * | m ≤ L, the inequality |y(s; s 0 , y * )| m ≤ J holds for s ∈ [s 0 , s 0 + τ ].

Remark 4 . 2 .
If f , g, h and k satisfy the global Lipschitz condition (gLip), then we do not need to make any restrictions on A in Corollary 4.1.More precisely, its statements hold for all A > 0. In this case, the solution exists for all t ≥ 0 and the inequality ||x t (φ) − x t (ψ)|| ≤ ||φ − ψ||e Kt

Figure 1 :
Figure 1: Flow chart of disease transmission and travel dynamics.The disease transmission in the two regions is shown in two different columns, the disease progresses vertically from the top to the bottom (solid arrows).Dashed arrows represent that individuals are traveling, dot-dashed arrows show the dynamics of the disease spread during the course of the travel.