THE ERROR ESTIMATION IN TERMS OF THE FIRST DERIVATIVE IN A NUMERICAL METHOD FOR THE SOLUTION OF A DELAY INTEGRAL EQUATION FROM BIOMATHEMATICS

. The positive, bounded and smooth solution of a delay integral equation, which arise in epidemics spread, can be found in an approximative manner using numerical methods achieved by the sequence of successive approximations and quadrature rules classiﬁed after the properties of the function f , such as Lipschitzian or in the C 1 smoothness case.


INTRODUCTION
The aim of this paper is to present some numerical methods for the approximation of the solution of the delay integral equation: (1) x(t) = t t−τ f (s, x(s))ds, t ∈ R, This equation is a mathematical model for the spread of certain infectious diseases with a contact rate that varies seasonaly.Here, x(t) represents the proportion of the infectivities in the population size at the time t, τ is the length of time in which an individual remains infectious and f (t, x(t)) is the proportion of new infectivities on unit time.
Some important results about the existence and the uniqueness of the solution of the equation (1) were given in [3], [5], [6], [7] and [8].In [8] I.A. Rus give sufficient conditions for the existence and the uniqueness of a periodic continuous nonnegative solution.In [5] R. Precup obtains the existence of a positive solution using the Schauder's fixed point theorem, and the existence and the uniqueness of a positive solution using the Banach's fixed point theorem.In [6] and [7] R. Precup obtains existence result for a positive periodic solution using the Leray-Schauder's continuation principle, and the existence and uniqueness of a positive solution using the monotone iterative technique.Supposing the conditions which involve the existence of an unique positive bounded solution of the equation (1) to be satisfied, C. Iancu obtains in [4] a numerical method for the approximation of this solution using the trapezoidal quadrature formula when the function f is in the C 2 smoothness class, and give an algorithm for effective computation on the knots of this solution, together the error estimation bound.
In the Section 2 of this paper we present a theorem for the existence of an unique positive bounded solution of the equation (1).In the Section3 we obtain a trapezoid quadrature rule for functions with the first derivative Lipschitzian.Afterwards, in the Section 4, we present the error estimation in the numerical method from [4], in the case that the function f is only Lipschitzian.In the last section we present the error estimation in the assumption that f ∈ C 1 ([−τ, T ] × R + ) and when ∂f ∂t and ∂f ∂x are Lipschitzian in each argument.

THE EXISTENCE AND THE UNIQUENESS OF THE POSITIVE, BOUNDED, AND SMOOTH SOLUTION
For the equation ( 1) we consider the initial condition x(t) = Φ(t), ∀t ∈ [−τ, 0], which means that the proportion Φ(t) of infectives in population is known for t ∈ [−τ, 0].Let T > 0 be a fixed real number.We have the initial value problem: (2) where Φ ∈ C[−τ, 0] and satisfy the condition: , using the Leibnitz's formula of derivation into an integral with parameters, we can see that the initial value problem (2) leads to a Cauchy's problem for a delay differential equation: and ( 2) and ( 4) are equivalent.
To obtain the existence of an unique positive, bounded solution we list these assumptions: (iv) M τ ≤ β and there exists an integrable function g(t) such that x, y ∈ [a, +∞).
Theorem 1.If the condition (3) and the above conditions (i)-(v) are satisfied, then the equation (1) has a unique continuous on Proof.As in [5] under the conditions (i), (ii), (iv), (v) follow the existence of an unique positive continuous on [−τ, T ] solution for (1) From (iv) we see that , and the solution is bounded.We know that x is solution for (1) and we have x(t) = t t−τ f (s, x(s))ds, for all t ∈ [0, T ], and because f ∈ C([−τ, T ] × [a, β]) we infer that x is derivable on [0, T ], and x is continuous on [0, T ].From the condition (iii) follow that x is derivable with x continuous on [−τ, 0] (including the continuity in the point t = 0).Then x ∈ C 1 [−τ, T ] and this concludes the proof.
Remark 1.In the hypotheses of Theorem 1 the initial value problem (2) has an unique positive, bounded and smooth solution on the interval [−τ, T ].As in [8], this solution depends continuously by f .

Consider the functions F
Proof.From the condition (v) we have, We can see that Then and so F m is Lipschitzian, ∀m ∈ N.

A TRAPEZOID QUADRATURE RULE FOR FUNCTIONS WITH THE FIRST DERIVATIVE LIPSCHITZIAN
For Lipschitzian functions and for smooth functions ( in the class C 1 ) the trapezoid quadrature rules are presented in [1] and [2].Here we obtain a new quadrature rule for functions Proof.We compute by parts the following integral, b a Corollary 4. In the conditions of the previous lemma, considering a partition of the interval [a, b], Proof.On the each interval [t i , t i+1 ], i = 0, n − 1, we have In this case, the quadrature rule is: with the remainder estimation:

THE ERROR ESTIMATION IN THE LIPSCHITZIAN CASE
Let ϕ be the solution of (2), which, by virtue of Theorem 1, can be obtained by the successive approximations method.
So, we have To obtain the sequence of successive approximations (8), it is necessary to calculate the integrals which appear in the right-hand side, but this problem can be solved only approximatively.In this purpose can be use quadrature formulae rules.This rules will be adapted in correlation with the properties of the function f .First, we consider functions f Lipschitzian, with a constant L > 0. We can apply the following quadrature formula: with the remainder estimation • L, and the generalized quadrature rule obtained using an uniform partition of the interval [a, b] (see [1] and [2]).
Considering an equidistant division, a = x 0 < x 1 < ... < x n = b, n ∈ N * , the quadrature rule becomes where the remainder R n (f ) satisfies the inequality: To compute the integrals from (8) we consider instead the function f, the functions F m , which by virtue of Proposition 2 are Lipschitzian.
We consider an uniform partition of the interval [−τ, 0] by the points: Using the quadrature rule (9) to compute ϕ m (t) from ( 8) for t = t k , k = n + 1, q, we have: where for the remainder r m,k (f ) we have the estimation 4n , ∀k = n + 1, q, ∀m ∈ N * .In [4] C. Iancu presents the following algorithm: and By induction, for m ≥ 3, it follows: To estimate the remainder, he use the classic quadrature rule (for functions f ∈ C 2 ).For the error estimation of the remainders, r m,k (f ), we obtain in this case the estimations: ), ∀k = n + 1, q, and for m ≥ 3, by induction, we obtain: and we get the sequence ( ϕ m (t k )) m∈N * , k = n + 1, q which approximates the sequence of successive approximations (8) on the knots t k , k = n + 1, q, with the error Theorem 5. Consider the initial value problem (2) under the conditions (i)-(v), ( 3), ( 5) and τ L 1 < 1.If the exact solution ϕ is approximated by the sequence ( ϕ m (t k )) m∈N * , k = n + 1, q, on the knots t k , k = n + 1, q, by the successive approximations method (8) combined with the quadrature rule (9), then the error estimation is: Proof.By the Banach's principle and Picard's theorem (Theorem 1.4.4,page 33, from [9]) we have the following estimation: Then, by virtue of the (13) estimation we have: We can approximate also, the values of the derivative of the solution on the knots.Indeed, from (8) we have By differentiation we obtain: For the remainder m,k (f ) we have the estimation For this reason, Remark 3. From these above, we see that it is possible to compute the values ϕ m (t k ) given in (11) and ϕ m (t k ) given in (16) on the knots t k , k = n + 1, q, approximating the solution and his derivative with the error estimations as in (13) and (18), respectively.

Suppose that
), and consequently there exists M 10 ≥ 0, M 01 ≥ 0 such that We can use the following trapezoid quadrature formula (as in [1] and [2]), with the error estimation Considering the algorithm (11) adapted at quadrature rule (20) with the remainder estimation (21) we obtain the following result: Theorem 6.Consider the initial value problem (2) under the conditions ) .If the exact solution ϕ is approximated by the sequence ( ϕ m (t k )) m∈N , k = n + 1, q, on the knots t k , k = n + 1, q, by the successive approximations method (8) combined with the quadrature rule (20) and the remainder estimation (21), then the error estimation is: Proof.We have, analogous with the proof of the Theorem 5, and according to the above quadrature rule, the following: ), using the classic trapezoidal formula, C. Iancu obtains in [4] the result: ) .If the exact solution ϕ is approximated by the sequence ( ϕ m (t k )) m∈N , k = n + 1, q, on the knots t k , k = n + 1, q, by the successive approximations method (8), combined with the trapezoidal rule which give on the knots the values, then the following error estimation holds: We apply now, the quadrature rule ( 6)-( 7) to compute the integrals from the successive approximations (8), in the case that F m are Lipschitzian, using the algorithm (11).
Proposition 8. Suppose that the following conditions holds: Then the functions F m are Lipschitzian on [−τ, T ].
Proof.We have, x m ≤ 2M and Proof.Follows from Remark 2, Proposition 8 and Theorem 5.
For ε = 10 −6 we found m = 6 and in the bellow table there are ϕ m (t k ) and ϕ m−1 (t k ) for k = 0, q − 1.

Table 1 .
Numerical results