PULSE INFECTION. CONTROL FIXING TIME BETWEEN INFECTION EVENTS

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Novemb er 2, 2008 1 8:38 P ro ceedings Trim Size: 9in x 6in epidemics

PULSE INFECTION. CONTROL FIXING TIME BETWEEN

INFECTION EVENTS

F. CORDOVA-LEPE

Universidad Catolica del Maule,

3605 San Miguel Avenue,

Talca, Chile

E-mail: [email protected]

E. GONZALEZ-OLIVARES

Pontificia Universidad Catolica de Valparaso,

2950 Brasil Avenue,

Valparaso, Chile

E-mail: [email protected]

We assumed a population affected by a disease, whose infection process is asso-

ciated to a sequence of social punctual events. The event is a kind of cultural

activity or an economic necessity that happens with some frequency. We formu-

late a generalist mathematical model for determining, with analytic techniques of

the Impulsive Differential Equations, the dynamic behavior of the infectious group.

We introduce diverse conditions on the frequency of the infection events with the

intention of to put control tools in hands of the regulatory authority, for a better

sanitary management. The idea is to avoid a spread of the disease, trying to keep

the amount of infectious under predetermined levels or going towards extinction.

1. Introduction

From the fundamentals of the Mathematical Epidemiology, as the worksof Hamer1(1906), Ross2(1911) and Kermack & Mc. Kendrick3(1927), the

matter it has been centered in establish models that permit the study (de-

scription, understanding and prediction) of the dynamics of infectious dis-

eases principally in human populations. The literature gives account of a

wide variety of classical models with good general perspectives4.

This work is supported by Universidad Catolica del Maule.Work partially supported by UMCE.

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The most used deterministic mathematical tools have been the ordinary

differential systems. Normally, we can think, in presence of an infectious

disease, that the population is divided, for instance, in two subpopulation:

the susceptible group, and the infectious group, according the individuals

may be infected (without immunity or resistance), or those being infected

have the capacity of infect. Usually the population sizes of these groups

are denoted with letters S and I respectively. The ordinary differentiable

models try to give account of some inner characteristics of the disease. Theyexpress, in SI S models, the rates of change of S and I as functions of S

and I. But, there are diseases that for its particularities does not resist a

natural model by the way of the ordinary differential systems. In this case,

for deterministic models it can to opt by different mathematical tools such

as the Difference Equations (DE), Differential Equations with Delay (DDE),

Equations in Time Scale, or Impulsive Differential Equations (IDE), which

are hybrid systems.

This article is aimed at studying the potential of a new type of IDE

proposed in Cordova-Lepe5(2007). Some details are given in Section 2.

The main uses of the traditional IDE in Mathematical Epidemiology it have

been concentrated in the possibility of to exert impulsive preventive control.

It has considered that in the global dynamics of the disease a vaccination

process, with very short duration, represents a real jump (impulse) in the

variables values. An impulse that transports in an instantaneous way, a

part of the susceptible population to the group of removed individuals, this

is, the individuals that get immunity and do not infect. See Meng XZ &

Chen LS22, Aug. 2008 ; Wei CJ & Chen LS7, 2008; Gao SJ, Teng ZD &

Xie DH8, Jul. 2008; and Zhang TL & Teng ZD9, Jan. 2008. This model

class is knew as Pulse Vaccination Models.

Our interest is the construction and analysis of adequate mathematical

SI S models, for studying the dynamics of diseases, where the infection

process is associated with the occurrence of certain events. An event, that

considering time variable, it is feasible of consider punctual, i.e., the eventhas a short duration and it is sporadic. The idea is insert the impulsive effect

principally in the infection process. We suppose that in those mentioned

events a considerable group of susceptible individuals are transported to the

infectious group. This transportation is a process fast enough compared

with the dynamic the rest of the time, so that it is possible to suppose

instantaneous.

The idea of discretization of the instants of contagion is justified be-

cause it is not unreal to suppose that the human populations can have


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associated social activities (cultural or economic) with transmission of dis-

eases individual to individual. Diseases which are impossible of block or

only partially lockable. When do not occur those infection events, we will

assume the process dynamics of continuous type. Immediately we can de-

tect two scenarios. One where the events of infection are uniformly spaced

and represent a usual activity of the community. An other possibility is

that there is a health authority or a sanitary organism, who has the power

for to establish a calendar of activities with a level of flexibility due to cul-tural or economic reasons, although if it can not prohibit the execution of

infectious events. This is, we are in front the possibility of to introduce

an element of control that can down the infection rates. In both cases the

IDE seem to be the kind of mathematical model more appropriated, but

in the second we require a type of IDE in which the time between impulse

instants, this is, the time between the events of infection, is a function of

the impulse size, or at least of the amount of infected individuals.

This work is organized as follow. In Section 2 we do a summary of

some aspect of the IDE, specially those about the new type mentioned. In

Section 3 we present the epidemiological model (an IDE) of our interest.

Finally in Section 4 we have three results, they show conditions to obtain

stable equilibria that could allow exercise control.

2. The Impulsive Differential Equations

In the literature it is possible find several forms for modelling phenomena

of reality that show an impulsive effect in its evolution. A form is repre-

senting the evolutive trajectories as solution of IDE. This equations were

introduced by Milman & Myshkis10(1960), point of view revived in eight-

ies, see Perestyuk & Samoilenko11 12 13(1977, 1981, 1987). We note the two

books of Bainov & Simeonov14 15(1989, 1993), because they have meant a

significant role diffuser.

We are in the domain of the IDE at Fixed Times, IDE-FT, when ina system, the impulsive instants are known before the development of the

dynamics. If the impulses appear when, a priori, a determined relation

there exists between the state and the time, we are in the field of the

IDE at Variable Times, IDE-VT. The theory of IDE-FT is widely de-

veloped and presents a development based in direct analogies with the

ODE. But, comparatively the IDE-VT has exhibited a smaller develop-

ment, because they have phenomena as: solution with infinity pulses

in finite time, the not existence of the uniqueness toward the past in


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the initial value problem, lost of autonomy, among other difficulties.

Other classical texts are Bainov & Covachev16(1994); Lakshmikantham,

Bainov & Simeonov17(1989); Lakshmikantham & Liu18(1993); Samoilenko

& Perestyuk19(1995); and Yang20(2001). In the last time, they have ap-

peared diverse types of hybrid equations, which combine discrete techniques

and continuous for working the state variable. Some examples are the ad-

vanced and delayed differential equations with piecewise constant argument,

see Cook & Wiener21

(1987). The use of IDE in biomathematics is veryextended. In pulse vaccination see the works of Meng & Chen22(2008),

Gakkhar & Negi23(2008) or Zhang & Teng24(2008). In the treatment in

chemotherapy of diseases, see Lakmeche & Arino25(2001). Impulsive strate-

gies of control and pest management, see Zhang, Jianjun & Lansun26(2007)

or Pang & Chen27(2008). Impulsive Harvest in management of renewable

resources, see Allegretto & Papini28(2008) or Negi & Gakkhar29(2007).

The type of IDE that we need for expressing mathematically the evo-

lution law, which is associated to the process of infection of our objective,

can not be expressed by means of traditional IDE-FT or IDE-VT.

In fact we are in front of a necessity of a new impulsive model, as

the proposed by Cordova-Lepe5(2007) for the biomathematical community,

and by Cordova-Lepe, Pinto & Gonzalez-Olivares30(2008) for mathematical

community.

Now we will review some introductory elements of IDE-ITD necessary

for the formulation of models. Let n, be the set of all possible states

of the process. We denote by x : the function that relates each

instant t with the state x(t). To formulate the dynamics of model we

require: First, an ordinary differential equations system

x = f(t, x), x , t , (1)

for some f : n, a function that describes the state almost

always, except for {tk}k1, recursively determined during the course of the

dynamics.Second, a law regulating the impulses, it act in {tk}k1, as follows

x(t+) = It(x(t)), t {tk}k1, (2)

For each t , the function It : transfers instantaneously x(t)

towards a new state I(x(t)). Finally, from t0 , the first impulse instant,

the sequence {tk}k1 is generated, by the recurrence

tk+1 = tk + Gtk(x(t+

k), x(tk)), (3)

where Gt : +, for each t .

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PULSE INFECTION. CONTROL FIXING TIME BETWEEN INFECTION EVENTS