Open Access

On a Generalized Time-Varying SEIR Epidemic Model with Mixed Point and Distributed Time-Varying Delays and Combined Regular and Impulsive Vaccination Controls

  • M. De la Sen1,
  • Ravi P. Agarwal2, 3Email author,
  • A. Ibeas4 and
  • S. Alonso-Quesada5
Advances in Difference Equations20102010:281612

DOI: 10.1155/2010/281612

Received: 17 August 2010

Accepted: 2 December 2010

Published: 14 December 2010

Abstract

This paper discusses a generalized time-varying SEIR propagation disease model subject to delays which potentially involves mixed regular and impulsive vaccination rules. The model takes also into account the natural population growing and the mortality associated to the disease, and the potential presence of disease endemic thresholds for both the infected and infectious population dynamics as well as the lost of immunity of newborns. The presence of outsider infectious is also considered. It is assumed that there is a finite number of time-varying distributed delays in the susceptible-infected coupling dynamics influencing the susceptible and infected differential equations. It is also assumed that there are time-varying point delays for the susceptible-infected coupled dynamics influencing the infected, infectious, and removed-by-immunity differential equations. The proposed regular vaccination control objective is the tracking of a prescribed suited infectious trajectory for a set of given initial conditions. The impulsive vaccination can be used to improve discrepancies between the SEIR model and its suitable reference one.

1. Introduction

Important control problems nowadays related to Life Sciences are the control of ecological models like, for instance, those of population evolution (Beverton-Holt model, Hassell model, Ricker model, etc. [15]) via the online adjustment of the species environment carrying capacity, that of the population growth or that of the regulated harvesting quota as well as the disease propagation via vaccination control. In a set of papers, several variants and generalizations of the Beverton-Holt model (standard time-invariant, time-varying parameterized, generalized model, or modified generalized model) have been investigated at the levels of stability, cycle-oscillatory behavior, permanence, and control through the manipulation of the carrying capacity (see, e.g., [15]). The design of related control actions has been proved to be important in those papers at the levels, for instance, of aquaculture exploitation or plague fighting. On the other hand, the literature about epidemic mathematical models is exhaustive in many books and papers. A nonexhaustive list of references is given in this manuscript compare [614] (see also the references listed therein). The sets of models include the following most basic ones [6, 7]:
  1. (i)

    SI-models where not removed-by-immunity population is assumed. In other words, only susceptible and infected populations are assumed,

     
  2. (ii)

    SIR-models, which include susceptible, infected, and removed-by-immunity populations,

     
  3. (iii)

    SEIR models where the infected populations are split into two ones (namely, the "infected" which incubate the disease but do not still have any disease symptoms and the "infectious" or "infective" which do exhibit the external disease symptoms).

     

The three above models have two possible major variants, namely, the so-called "pseudo-mass action models," where the total population is not taken into account as a relevant disease contagious factor or disease transmission power, and the so-called "true mass action models," where the total population is more realistically considered as being an inverse factor of the disease transmission rates. There are other many variants of the above models, for instance, including vaccination of different kinds: constant [8], impulsive [12], discrete-time, and so forth, by incorporating point or distributed delays [12, 13], oscillatory behaviors [14], and so forth. On the other hand, variants of such models become considerably simpler for the disease transmission among plants [6, 7]. In this paper, a mixed regular continuous-time/impulsive vaccination control strategy is proposed for a generalized time-varying SEIR epidemic model which is subject to point and distributed time-varying delays [12, 13, 1517]. The model takes also into account the natural population growing and the mortality associated to the disease as well as the lost of immunity of newborns, [6, 7, 18] plus the potential presence of infectious outsiders which increases the total infectious numbers of the environment under study. The parameters are not assumed to be constant but being defined by piecewise continuous real functions, the transmission coefficient included [19]. Another novelty of the proposed generalized SEIR model is the potential presence of unparameterized disease thresholds for both the infected and infectious populations. It is assumed that a finite number of time-varying distributed delays might exist in the susceptible-infected coupling dynamics influencing the susceptible and infected differential equations. It is also assumed that there are potential time-varying point delays for the susceptible-infected coupled dynamics influencing the infected, infectious, and removed-by-immunity differential equations [2022]. The proposed regulation vaccination control objective is the tracking of a prescribed suited infectious trajectory for a set of given initial conditions. The impulsive vaccination action can be used for correction of the possible discrepancies between the solutions of the SEIR model and that of its reference one due, for instance, to parameterization errors. It is assumed that the total population as well as the infectious one can be directly known by inspecting the day-to-day disease effects by directly taking the required data. Those data are injected to the vaccination rules. Other techniques could be implemented to evaluate the remaining populations. For instance, the infectious population is close to the previously infected one affected with some delay related to the incubation period. Also, either the use of the disease statistical data related to the percentages of each of the populations or the use of observers could be incorporated to the scheme to have either approximate estimations or very adjusted asymptotic estimations of each of the partial populations.

1.1. List of Main Symbols

SEIR epidemic model, namely, that consisting of four partial populations related to the disease being the susceptible, infected, infectious, and immune.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq1_HTML.gif : Susceptible population, that is, those who can be infected by the disease

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq2_HTML.gif : Infected population, that is, those who are infected but do not still have external symptoms

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq3_HTML.gif : Infectious population, that is, those who are infected exhibiting external symptoms

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq4_HTML.gif : Immune population

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq5_HTML.gif : Total population

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq6_HTML.gif : Function associated with the infected floating outsiders in the SEIR model

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq7_HTML.gif : Disease transmission function

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq8_HTML.gif : Natural growth rate function of the population

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq9_HTML.gif : Natural rate function of deaths from causes unrelated to the infection

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq10_HTML.gif : Takes into account the potential immediate vaccination of new borns

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq11_HTML.gif : Functions that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq12_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq13_HTML.gif are, respectively, the instantaneous durations per populations averages of the latent and infectious periods at time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq14_HTML.gif .

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq15_HTML.gif : the rate of lost of immunity function

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq16_HTML.gif : related to the mortality caused by the disease

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq17_HTML.gif : Thresholds of infected and infectious populations

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq18_HTML.gif : Different point and impulsive delays in the epidemic model

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq19_HTML.gif : Functions associated with the regular and impulsive vaccination strategies

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq20_HTML.gif : Weighting functions associated with distributed delays in the SEIR model.

2. Generalized True Mass Action SEIR Model with Real and Distributed Delays and Combined Regular and Impulsive Vaccination

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq21_HTML.gif be the "susceptible" population of infection at time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq22_HTML.gif the "infected" (i.e., those which incubate the illness but do not still have any symptoms) at time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq23_HTML.gif the "infectious" (or "infective") population at time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq24_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq25_HTML.gif the "removed-by-immunity" (or "immune") population at time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq26_HTML.gif . Consider the extended SEIR-type epidemic model of true mass type
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ1_HTML.gif
(2..1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ2_HTML.gif
(2..2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ3_HTML.gif
(2..3)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ4_HTML.gif
(2..4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ5_HTML.gif
(2..5)
for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq27_HTML.gif subject to initial conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq28_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq29_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq30_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq31_HTML.gif which are absolutely continuous functions with eventual bounded discontinuities on a subset of zero measure of their definition domain and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ6_HTML.gif
(2..6)
is the maximum delay at time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq32_HTML.gif of the SEIR model (2.1)–(2.4) subject to (2.5) under a potentially jointly regular vaccination action https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq33_HTML.gif and an impulsive vaccination action https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq34_HTML.gif at a strictly ordered finite or infinite real sequence of time instants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq35_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq36_HTML.gif being bounded and piece-wise continuous real functions used to build the impulsive vaccination term and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq37_HTML.gif being the indexing set of the impulsive time instants. It is assumed
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ7_HTML.gif
(2..7)

and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq38_HTML.gif which give sense of the asymptotic limit of the trajectory solutions.

The real function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq39_HTML.gif in (2.5) is a perturbation in the susceptible dynamics (see, e.g., [18]) where function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq40_HTML.gif , subject to the point wise constraint https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq41_HTML.gif , takes into account the possible decreasing in the susceptible population while increasing the infective one due to a fluctuant external infectious population entering the investigated habitat and contributing partly to the disease spread. In the above SEIR model,
  1. (i)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq42_HTML.gif is the total population at time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq43_HTML.gif .

     
The following functions parameterize the SEIR model.
  1. (i)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq44_HTML.gif is a bounded piecewise-continuous function related to the natural growth rate of the population. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq45_HTML.gif is assumed to be zero if the total population at time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq46_HTML.gif is less tan unity, that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq47_HTML.gif , implying that it becomes extinguished.

     
  2. (ii)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq48_HTML.gif is a bounded piecewise-continuous function meaning the natural rate of deaths from causes unrelated to the infection.

     
  3. (iii)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq49_HTML.gif is a bounded piecewise-continuous function which takes into account the immediate vaccination of new borns at a rate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq50_HTML.gif .

     
  4. (iv)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq51_HTML.gif is a bounded piecewise-continuous function which takes into account the number of deaths due to the infection.

     
  5. (v)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq52_HTML.gif is a bounded piecewise-continuous function meaning the rate of losing immunity.

     
  6. (vi)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq53_HTML.gif is a bounded piecewise-continuous transmission function with the total number of infections per unity of time at time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq54_HTML.gif .

     
  7. (vii)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq55_HTML.gif is a transmission term accounting for the total rate at which susceptible become exposed to illness which replaces https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq56_HTML.gif in the standard SEIR model in (2.1)–(2.2) which has a constant transmission constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq57_HTML.gif . It generalizes the one-delay distributed approach proposed in [20] for a SIRS-model with distributed delays, while it describes a transmission process weighted through a weighting function with a finite number of terms over previous time intervals to describe the process of removing the susceptible as proportional to the infectious. The functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq58_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq59_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq60_HTML.gif nonnegative weighting real functions being everywhere continuous on their definition domains subject to Assumption 1(1) below, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq61_HTML.gif are the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq62_HTML.gif relevant delay functions describing the delay distributed-type for this part of the SEIR model. Note that a punctual delay can be modeled with a Dirac-delta distribution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq63_HTML.gif within some of the integrals and the absence of delays is modeled with all the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq64_HTML.gif functions being identically zero.

     
  8. (viii)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq65_HTML.gif are bounded continuous functions defined so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq66_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq67_HTML.gif are, respectively, the instantaneous durations per populations averages of the latent and infective periods at time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq68_HTML.gif .

     
  9. (ix)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq69_HTML.gif are piecewise-continuous functions being integrable on any subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq70_HTML.gif which are threshold functions for the infected and the infectious growing rates, respectively, which take into account (if they are not identically zero) the respective endemic populations which cannot be removed. This is a common situation for some diseases like, for instance, malaria, dengue, or cholera in certain regions where they are endemic.

     
  10. (x)
    The two following coupling infected-infectious dynamics contributions:
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ8_HTML.gif
    (2..8)

    are single point-delay and two-point delay dynamic terms linked, respectively, to the couplings of dynamics between infected-versus-infectious populations and infectious-versus-immune populations, which take into account a single-delay effect and a double-delay effect approximating the real mutual one-stage and two-staged delayed influence between the corresponding dynamics, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq71_HTML.gif are the gain and their associate infected and infectious delay functions which are everywhere continuous in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq72_HTML.gif . In the time-invariant version of a simplified pseudomass-type SIRS-model proposed in [21], the constant gains are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq73_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq74_HTML.gif .

     
  11. (xi)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq75_HTML.gif in (2.1) and (2.4) are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq76_HTML.gif nonnegative nonidentically zero vaccination weighting real functions everywhere on their definition domains subject to distributed delays governed by the functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq77_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq78_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq79_HTML.gif is a vaccination function to be appropriately normalized to the day-to-day population to be vaccinated subject to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq80_HTML.gif . As for the case of the transmission term, punctual delays could be included by using appropriate Dirac deltas within the corresponding integrals.

     
  12. (xii)

    The SEIR model is subject to a joint regular vaccination action https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq81_HTML.gif plus an impulsive one https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq82_HTML.gif at a strictly ordered finite or countable infinite real sequence of time instants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq83_HTML.gif . Specifically, it is a single Dirac impulse of amplitude https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq84_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq85_HTML.gif and zero if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq86_HTML.gif . The weighting function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq87_HTML.gif can be defined in several ways. For instance, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq88_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq89_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq90_HTML.gif , otherwise, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq91_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq92_HTML.gif and it is zero, otherwise. Thus, the impulsive vaccination is proportional to the total population at time instants in the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq93_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq94_HTML.gif , then the impulsive vaccination is proportional to the susceptible at such time instants. The vaccination term https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq95_HTML.gif in (2.1) and (2.4) is related to a instantaneous (i.e., pulse-type) vaccination applied in particular time instants belonging to the real sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq96_HTML.gif if a reinforcement of the regular vaccination is required at certain time instants, because, for instance, the number of infectious exceeds a prescribed threshold. Pulse control is an important tool in controlling certain dynamical systems [15, 23, 24] and, in particular, ecological systems, [4, 5, 25]. Pulse vaccination has gained in prominence as a result of its highly successfully application in the control of poliomyelitis and measles and in a combined measles and rubella vaccine. Note that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq97_HTML.gif , then neither the natural increase of the population nor the loss of maternal lost of immunity of the newborns is taken into account. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq98_HTML.gif , then some of the newborns are not vaccinated with the consequent increase of the susceptible population compared to the case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq99_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq100_HTML.gif , then such a lost of immunity is partly removed by vaccinating at birth a proportion of newborns.

     
Assumption.
  1. (1)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq101_HTML.gif .

     
  2. (2)

    There exist continuous functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq102_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq103_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq104_HTML.gif for some prefixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq105_HTML.gif and any given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq106_HTML.gif .

     

Assumption 1(1) for the distributed delay weighting functions is proposed in [20]. Assumption 1(2) implies that the infected and infectious minimum thresholds, affecting to the infected, infectious, and removed-by-immunity time derivatives, may be negative on certain intervals but their time-integrals on each interval on some fixed nonzero measure is nonnegative and bounded. This ensures that the infected and infectious threshold minimum contributions to their respective populations are always nonnegative for all time. From Picard-Lindelöff theorem, it exists a unique solution of (2.1)–(2.5) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq107_HTML.gif for each set of admissible initial conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq108_HTML.gif and each set of vaccination impulses which is continuous and time-differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq109_HTML.gif for time instant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq110_HTML.gif , provided that it exists, being such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq111_HTML.gif , or on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq112_HTML.gif , if such a finite impulsive time instant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq113_HTML.gif does not exist, that is, if the impulsive vaccination does not end in finite time. The solution of the generalized SEIR model for a given set of admissible functions of initial conditions is made explicit in Appendix A.

3. Positivity and Boundedness of the Total Population Irrespective of the Vaccination Law

In this section, the positivity of the solutions and their boundedness for all time under bounded non negative initial conditions are discussed. Summing up both sides on (2.1)–(2.4) yields directly
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ9_HTML.gif
(3.1)
The unique solution of the above scalar equation for any given initial conditions obeys the formula
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ10_HTML.gif
(3.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq114_HTML.gif is the mild evolution operator which satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq115_HTML.gif is the forcing function in (3.1). This yields the following unique solution for (3.1) for given bounded initial conditions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ11_HTML.gif
(3.3)
Consider a Lyapunov function candidate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq116_HTML.gif whose time-derivative becomes
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ12_HTML.gif
(3.4)
Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq117_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ13_HTML.gif
(3.5)

if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq118_HTML.gif .

Decompose uniquely any nonnegative real interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq119_HTML.gif as the following disjoint union of subintervals
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ14_HTML.gif
(3.6)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq120_HTML.gif are all numerable and of nonzero Lebesgue measure with the finite or infinite real sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq121_HTML.gif of all the time instants where the time derivative of the above candidate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq122_HTML.gif changes its sign which are defined by construction so that the above disjoint union decomposition of the real interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq123_HTML.gif is feasible for any real https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq124_HTML.gif , that is, if it consists of at least one element), as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ15_HTML.gif
(3.7)
Note that the identity of cardinals of sets https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq125_HTML.gif holds since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq126_HTML.gif is the indexing set of ST and, furthermore,
  1. (a)
    the sequence ST trivially exists if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq127_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq128_HTML.gif with at least one of the real interval unions being nonempty, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq129_HTML.gif are disjoint subsets of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq130_HTML.gif satisfying,
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ16_HTML.gif
    (3.8)
    and defined as follows:
    1. (i)

      for any given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq131_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq132_HTML.gif ,

       
    2. (ii)

      for any given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq133_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq134_HTML.gif , and define also https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq135_HTML.gif ,

       
     
  1. (b)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq136_HTML.gif , where unit cardinal means that the time-derivative of the candidate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq137_HTML.gif has no change of sign and infinite cardinal means that there exist infinitely many changes of sign in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq138_HTML.gif ,

     
  2. (c)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq139_HTML.gif if it exists a finite https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq140_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq141_HTML.gif , and then, the sequence ST is finite (i.e., the total number of changes of sign of the time derivative of the candidate is finite) as they are the sets https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq142_HTML.gif ,

     
  3. (d)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq143_HTML.gif if there is no finite https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq144_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq145_HTML.gif and, then, the sequence ST is infinite and the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq146_HTML.gif has infinite cardinal.

     
It turns out that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ17_HTML.gif
(3.9)

The following result is obtained from the above discussion under conditions which guarantee that the candidate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq147_HTML.gif is bounded for all time.

Theorem 3.1.

The total population https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq148_HTML.gif of the SEIR model is nonnegative and bounded for all time irrespective of the vaccination law if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ18_HTML.gif
(3.10)

Remark 3.2.

Note that Theorem 3.1 may be validated since both the total population used in the construction of the candidate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq149_HTML.gif and the infectious one (exhibiting explicit disease symptoms) can be either known or tightly estimated by direct inspection of the disease evolution data. Theorem 3.1 gives the most general condition of boundedness through time of the total population. It is allowed for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq150_HTML.gif to change through time provided that the intervals of positive derivative are compensated with sufficiently large time intervals of negative time derivative. Of course, there are simpler sufficiency-type conditions of fulfilment of Theorem 3.1 as now discussed. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq151_HTML.gif . Thus, from (3.4):
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ19_HTML.gif
(3.11)
leads to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq152_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq153_HTML.gif , irrespective of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq154_HTML.gif since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq155_HTML.gif is bounded, so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq156_HTML.gif and then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq157_HTML.gif cannot diverge what leads to a contradiction. Thus, a sufficient condition for Theorem 3.1 to hold, under the ultimate boundedness property, is that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq158_HTML.gif if the infectious population is non negative through time. Another less tighter bound of the above expression for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq159_HTML.gif is bounded by taking into account that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq160_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq161_HTML.gif since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq162_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq163_HTML.gif . Then,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ20_HTML.gif
(3.12)

what leads to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq164_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq165_HTML.gif which again contradicts that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq166_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq167_HTML.gif and it is a weaker condition than the above one.

Note that the above condition is much more restrictive in general than that of Theorem 3.1 although easier to test.

Since the impulsive-free SEIR model (2.1)–(2.5) has a unique mild solution (then being necessarily continuous) on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq168_HTML.gif , it is bounded for all finite time so that Theorem 3.1 is guaranteed under an equivalent simpler condition as follows.

Corollary 3.3.

Theorem 3.1 holds if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ21_HTML.gif
(3.13)
and, equivalently,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ22_HTML.gif
(3.14)

Corollary 3.3 may also be simplified to the light of more restrictive alternative and dependent on the parameters conditions which are easier to test, as it has been made in Theorem 3.1. The following result, which is weaker than Theorem 3.1, holds.

Theorem 3.4.

Assume that

there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq170_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq171_HTML.gif ,

and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ23_HTML.gif
(3.15)

for some constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq173_HTML.gif and some prefixed finite https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq174_HTML.gif . Then, the total population https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq175_HTML.gif of the SEIR model is nonnegative and bounded for all time, and asymptotically extinguishes at exponential rate irrespective of the vaccination law.

If the second condition is changed to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ24_HTML.gif
(3.16)

then the total population https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq176_HTML.gif of the SEIR model is nonnegative and bounded for all time.

The proof of Theorem 3.4 is given in Appendix B. The proofs of the remaining results which follow requiring mathematical proofs are also given in Appendix B. Note that the extinction condition of Theorem 3.4 is associated with a sufficiently small natural growth rate compared to the infection propagation in the case that the average immediate vaccination of new borns (of instantaneous rate https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq177_HTML.gif ) is less than zero. Another stability result based on Gronwall's Lemma follows.

Theorem 3.5.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq178_HTML.gif exists such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ25_HTML.gif
(3.17)

for some prefixed finite https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq179_HTML.gif . Then, the total population https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq180_HTML.gif of the SEIR model is nonnegative and bounded for all time irrespective of the vaccination law. Furthermore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq181_HTML.gif converges to zero at an exponential rate if the above second inequality is strict within some subinterval of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq182_HTML.gif of infinite Lebesgue measure.

Remark 3.6.

Condition (3.17) for Theorem 3.5 can be fulfilled in a very restrictive, but easily testable fashion, by fulfilling the comparisons for the integrands for all time for the following constraints on the parametrical functions:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ26_HTML.gif
(3.18)
which is achievable, irrespective of the infectious population evolution provided that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq183_HTML.gif , by vaccinating a proportion of newborns at birth what tends to decrease the susceptible population by this action compared to the typical constraint https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq184_HTML.gif . See Remark 3.2 concerning a sufficient condition for Theorem 3.1 to hold. Another sufficiency-type condition, alternative to (3.17), to fulfil Theorem 3.5, which involves the infectious population is
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ27_HTML.gif
(3.19)

Note that the infectious population is usually known with a good approximation (see Remark 3.2).

4. Positivity of the SEIR Generalized Model (2.1)–(2.5)

The vaccination effort depends on the total population and has two parts, the continuous-time one and the impulsive one (see (2.1) and (2.4)).

4.1. Positivity of the Susceptible Population of the Generalized SEIR Model

The total infected plus infectious plus removed-by-immunity populations obeys the differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ28_HTML.gif
(4.1)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ29_HTML.gif
(4.2a)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ30_HTML.gif
(4.2b)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ31_HTML.gif
(4.2c)

The non-negativity of any considered partial population is equivalent to the sum of the other three partial populations being less than or equal to the total population. Then, the following result holds from (3.3) and (4.1) concerning the non negative of the solution of the susceptible population for all time.

Assertion 1.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq185_HTML.gif in the SEIR generalized model (2.1)–(2.5) if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ32_HTML.gif
(4.3)

4.2. Positivity of the Infected Population of the Generalized SEIR Model

The total susceptible plus infectious plus removed obeys the differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ33_HTML.gif
(4.4)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ34_HTML.gif
(4.5)

Then, the following result holds concerning the non negativity of the infected population.

Assertion 2.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq186_HTML.gif if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ35_HTML.gif
(4.6)

4.3. Positivity of the Infectious Population of the Generalized SEIR Model

The total susceptible plus infected plus removed population obeys the following differential equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ36_HTML.gif
(4.7)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ37_HTML.gif
(4.8)

Thus, we have the following result concerning the non negativity of the infectious population.

Assertion 3.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq187_HTML.gif in the SEIR generalized model (2.1)–(2.5) if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ38_HTML.gif
(4.9)

4.4. Positivity of the Removed by Immunity Population of the Generalized SEIR Model

The total numbers of susceptible, infected, and infectious populations obey the following differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ39_HTML.gif
(4.10)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ40_HTML.gif
(4.11)

Then, the following result holds concerning the non negativity of the immune population.

Assertion 4.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq188_HTML.gif in the SEIR generalized model (2.1)–(2.5) if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ41_HTML.gif
(4.12)

Assertions 1–4, Theorem 3.1, Corollary 3.3, and Theorems 3.4-3.5 yield directly the following combined positivity and stability theorem whose proof is direct from the above results.

Theorem 4.1.

The following properties hold.
  1. (i)

    If Assertions 1–4 hold jointly, then, the populations https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq189_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq190_HTML.gif in the generalized SEIR model (2.1)–(2.5) are lower bounded by zero and upper bounded by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq191_HTML.gif . If, furthermore, either Theorem 3.1, or Corollary 3.3, or Theorem 3.4 or Theorem 3.5 holds, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq192_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq193_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq194_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq195_HTML.gif are bounded https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq196_HTML.gif .

     
  2. (ii)

    Assume that

     

for each time instant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq198_HTML.gif , any three assertions among weakly formulated Assertions 1–4 hold jointly in the sense that their given statements are reformulated for such a time instant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq199_HTML.gif instead for all time,

the three corresponding inequalities within the set of four inequalities (4.3), (4.6), (4.9), and (4.12) are, furthermore, upper bounded by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq201_HTML.gif for such a time instant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq202_HTML.gif ,

either Theorem 3.1, or Corollary 3.3, or Theorem 3.4 or Theorem 3.5 holds, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq204_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq205_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq206_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq207_HTML.gif are bounded https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq208_HTML.gif .

Then, the populations https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq209_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq210_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq211_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq212_HTML.gif of the generalized SEIR model (2.1)–(2.5) are lower bounded by zero and upper bounded by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq213_HTML.gif what is, in addition, bounded, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq214_HTML.gif .

4.5. Easily Testable Positivity Conditions

The following positivity results for the solution of (2.1)–(2.4), subject to (2.5), are direct and easy to test.

Assertion 5.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq215_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq216_HTML.gif if and only if the conditions below hold:
  1. (a)

    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq217_HTML.gif and

     
  2. (b)
    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ42_HTML.gif
    (4.13)
     

for some sufficiently small https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq218_HTML.gif .

Remark 4.2.

The positivity of the susceptible population has to be kept also in the absence of vaccination. In this way, note that if Assertion 5 holds for a given vaccination function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq219_HTML.gif and a given impulsive vaccination distribution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq220_HTML.gif , then it also holds if those vaccination function and distribution are identically zero.

Assertion 6.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq221_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq222_HTML.gif if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ43_HTML.gif
(4.14)

for some sufficiently small https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq223_HTML.gif .

Assertion 7.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq224_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq225_HTML.gif if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ44_HTML.gif
(4.15)

for some sufficiently small https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq226_HTML.gif .

The following result follows from (2.3) and it is proved in a close way to the proof of Assertions  5–7.

Assertion 8.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq227_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq228_HTML.gif for any given vaccination law satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq229_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq230_HTML.gif if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ45_HTML.gif
(4.16)

for some sufficiently small https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq231_HTML.gif .

The subsequent result is related to the first positivity interval of all the partial susceptible, infected, infectious, and immune populations under not very strong conditions requiring the (practically expected) strict positivity of the susceptible population at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq232_HTML.gif , the infected-infectious threshold constraint https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq233_HTML.gif and a time first interval monitored boundedness of the infectious population which is feasible under the technical assumption that the infection spread starts at time zero.

Assertion 9.

Assume that

the set of absolutely continuous with eventual bounded discontinuities functions of initial conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq235_HTML.gif satisfy, furthermore, the subsequent constraints:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ46_HTML.gif
(4.17)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq237_HTML.gif and, furthermore, it exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq238_HTML.gif such that the infectious population satisfies the integral inequality,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ47_HTML.gif
(4.18)

Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq239_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq240_HTML.gif irrespective of the delays and vaccination laws that satisfy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq241_HTML.gif (even if the SEIR model (2.1)–(2.5) is vaccination free). Furthermore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq242_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq243_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq244_HTML.gif irrespective of the delays and vaccination law even if the SEIR model (2.1)–(2.5) is vaccination-free.

Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq245_HTML.gif , that is, the set of impulsive time instants in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq246_HTML.gif is identical to that in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq247_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq248_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq249_HTML.gif includes https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq250_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq251_HTML.gif . Note also that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq252_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq253_HTML.gif , in particular, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq254_HTML.gif . A related result to Assertion 9 follows.

Assertion 10.

Assume that the constraints of Assertion 9 hold except that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq255_HTML.gif is replaced by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq256_HTML.gif . Then, the conclusion of Assertion 9 remains valid.

A positivity result for the whole epidemic model (2.1)–(2.5) follows.

Theorem 4.3.

Assume that the SEIR model (2.1)–(2.5) under any given set of absolutely continuous initial conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq257_HTML.gif , eventually subject to a set of isolated bounded discontinuities, is impulsive vaccination free, satisfies Assumptions 1, the constraints (4.14)–(4.16) and, furthermore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ48_HTML.gif
(4.19)

Then, its unique mild solution is nonnegative for all time.

Theorem 4.3 is now directly extended to the presence of impulsive vaccination as follows. The proof is direct from that of Theorem 4.3 and then omitted.

Theorem 4.4.

Assume that the hypotheses of Theorem 4.3 hold and, furthermore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq258_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq259_HTML.gif . Then, the solution of the SEIR model (2.1)–(2.5) is nonnegative for all time.

5. Vaccination Law for the Achievement of a Prescribed Infectious Trajectory Solution

A problem of interest is the calculation of a vaccination law such that a prescribed suitable infectious trajectory solution is achieved for all time for any given set of initial conditions of the SEIR model (2.1)–(2.5). The remaining solution trajectories of the various populations in (2.1)–(2.4) are obtained accordingly. In this section, the infected trajectory is calculated so that the infectious one is the suitable one for the given initial conditions. Then, the suited susceptible trajectory is such that the infected and infectious ones are the suited prescribed ones. Finally, the vaccination law is calculated to achieve the immune population trajectory such that the above suited susceptible trajectory is calculated. In this way, the whole solution of the SEIR model is a prescribed trajectory solution which makes the infectious trajectory to be a prescribed suited one (for instance, exponentially decaying) for the given delay interval-type set of initial condition functions. The precise mathematical discussion of this topic follows through Assertions 11–13 and Theorem 5.1 below.

Assertion 11.

Consider any prescribed suitable infectious trajectory https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq260_HTML.gif fulfilling https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq261_HTML.gif and assume that the infected population trajectory is given by the expression:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ49_HTML.gif
(5.1)
which is in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq262_HTML.gif for any susceptible trajectory https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq263_HTML.gif under initial conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq264_HTML.gif , where the desired total population https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq265_HTML.gif is calculated from (3.3) as the desired population https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq266_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ50_HTML.gif
(5.2)

with initial conditions being identical to those of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq267_HTML.gif . Then, the infected population trajectory (5.1) guarantees the exact tracking of the infectious population of the given reference infectious trajectory https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq268_HTML.gif which furthermore satisfies the differential equation (2.3).

Assertion 12.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq269_HTML.gif and that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq270_HTML.gif . Consider the prescribed suitable infectious trajectory https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq271_HTML.gif of Assertion 11 and assume also that the infected population trajectory is given by (5.1). Then, the susceptible population trajectory given by the expression
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ51_HTML.gif
(5.3)

is in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq272_HTML.gif under initial conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq273_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq274_HTML.gif being given by (5.2) with initial conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq275_HTML.gif . Then, the susceptible population trajectory (5.3), subject to the infected one (5.1), guarantees the exact tracking of the infectious population of the given reference infectious trajectory https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq276_HTML.gif with the suited reference infected population differential equation satisfying (2.2).

Assertion 13.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq277_HTML.gif and that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq278_HTML.gif fulfils in addition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq279_HTML.gif . Assume also that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ52_HTML.gif
(5.4)
Consider the prescribed suitable infectious trajectory https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq280_HTML.gif of Assertions 11–12 under initial conditions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq281_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq282_HTML.gif being given by (5.2) with initial conditions. Then, the vaccination law
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ53_HTML.gif
(5.5)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ54_HTML.gif
(5.6)
makes the immune population trajectory to be given by the expression
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ55_HTML.gif
(5.7)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Equ56_HTML.gif
(5.8)

which follows from (2.1) and (A.8), such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq283_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq284_HTML.gif . Then, the immune population trajectory (5.7)-(5.8), subject to the infected one (5.1) and the susceptible one (5.3), guarantees the exact tracking of the infectious population of the given reference infectious trajectory https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq285_HTML.gif with the infected, susceptible, and immune population differential equations satisfying their reference ones (2.1), (2.3), and (2.4).

Note that the regular plus impulsive vaccination law (5.5)-(5.6) ensures that a suitable immune population trajectory (5.7)-(5.8) is achieved. The combination of Assertions 11–13 yields the subsequent result.

Theorem 5.1.

The vaccination law (5.5)–(5.6) makes the solution trajectory of the SEIR model (2.1)–(2.5), to be identical to the suited reference one for all time provided that their functions of initial conditions are identical.

The impulsive part of the vaccination law might be used to correct discrepancies between the SEIR model (2.1)–(2.5) and its suited reference solution due, for instance, to an imperfect knowledge of the functions parameterizing (2.1)–(2.4) which are introduced with errors in the reference model. The following result is useful in that context.

Corollary 5.2.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq286_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq287_HTML.gif and any given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq288_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq289_HTML.gif (prescribed) if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq290_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq291_HTML.gif so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq292_HTML.gif .

Note that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq293_HTML.gif guarantees the existence of a unique solution of (2.1)–(2.4) for each set of admissible initial conditions and a vaccination law. Corollary 5.2 is useful in practice in the following situation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq294_HTML.gif due to errors in the SEIR model (2.1)–(2.5) for some prefixed unsuitable sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq295_HTML.gif . Then, an impulsive vaccination at time https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq296_HTML.gif may be generated so that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq297_HTML.gif .

Extensions of the proposed methodology could include the introduction of hybrid models combining continuous-time and discrete systems and resetting systems by jointly borrowing the associate analysis of positive dynamic systems involving delays [15, 16, 2628].

6. Simulation Example

This section contains a simulation example concerning the vaccination policy presented in Section 5. The free-vaccination evolution and then vaccination policy given in (5.5)-(5.6) are studied. The case under investigation relies on the propagation of influenza with the elementary parameterization data previously studied for a real case in [7, 29], for time-invariant delay-free SIR and SEIR models without epidemic threshold functions. In the first subsection below, the ideal case when the parameterization is fully known is investigated while in the second subsection, some extra simulations are given for the case where some parameters including certain delays are not fully known in order to investigate the robustness against uncertainties of the proposed scheme.

6.1. Ideal Case of Perfect Parameterization

The time-varying parameters of the system described by equations (2.1)–(2.4) are given by: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq354_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq355_HTML.gif , which represent periodic oscillations around fixed values given by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq298_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq299_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq300_HTML.gif days, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq301_HTML.gif (the natural growth rate is larger than the natural death rate), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq302_HTML.gif days, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq303_HTML.gif days, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq304_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq305_HTML.gif , which means that the 80% of newborns are immediately vaccinated. The frequency is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq306_HTML.gif , where the period https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq307_HTML.gif , in the case of influenza, is fixed to one year. The remaining parameters are given by: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq308_HTML.gif per day, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq309_HTML.gif per day, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq310_HTML.gif per day, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq311_HTML.gif (a constant total contribution of external infectious is assumed), and the delays https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq312_HTML.gif days, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq313_HTML.gif days, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq314_HTML.gif days, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq315_HTML.gif days, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq316_HTML.gif days, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq317_HTML.gif days. The weighting functions are given by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq318_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq319_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq320_HTML.gif . The initial conditions are punctual at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq321_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq322_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq323_HTML.gif individuals and remain constant during the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq324_HTML.gif days. The population evolution behavior without vaccination is depicted in Figure 1 while the total population is given by Figure 2.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Fig1_HTML.jpg
Figure 1

Evolution of the populations without vaccination.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Fig2_HTML.jpg
Figure 2

Evolution of the total population.

As it can be appreciated from Figure 2, the total population increases slightly with time as it corresponds to a situation where the natural growth rate is larger than the combination of the natural and illness-associated death rates. As Figure 1 points out, the infectious trajectory possesses a peak value of 2713 individuals and then it stabilizes at a constant value of 1074 individuals. The goals of the vaccination policy are twofold, namely, to decrease the trajectory peak and to reduce the number of infected individuals at the steady-state.

The vaccination policy of (5.5)-(5.6) is implemented to fulfil those objectives. The desired infectious trajectory to be tracked by the vaccination law is selected as shown in Figure 3. Note that the shape of the desired trajectory is similar to the vaccination-free trajectory but with the above-mentioned goals incorporated: the peak and the steady-state values are much smaller. The partial populations are depicted in Figure 4 when the vaccination law (5.5)-(5.6) is implemented.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Fig3_HTML.jpg
Figure 3

Desired infectious trajectory.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Fig4_HTML.jpg
Figure 4

Populations under vaccination.

On one hand, the populations reach the steady-state very quick. This occurs since the desired infectious trajectory reaches the steady-state in only 10 days. On the other hand, the above-proposed goals are fulfilled as Figure 5 following on the infectious trajectory shows.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Fig5_HTML.jpg
Figure 5

Comparison between real and desired trajectories.

The peak in the infectious reaches only 607 individuals while the steady-state value is 65 individuals. These results are obtained with the vaccination policy depicted in Figure 6.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Fig6_HTML.jpg
Figure 6

Vaccination law.

The vaccination effort is initially very high in order to make the system satisfies the desired infectious trajectory. Afterwards, it converges to a constant value. Moreover, note that with this vaccination strategy, the immune population increases while the susceptible, infected, and infectious reduces in comparison with the vaccination-free case. However, since the total population increases in time (Figure 2), the number of susceptible and infected individuals would also increase through time as the infectious population remains constant. In order to reduce this effect, an impulse vaccination strategy is considered. The vaccination impulses according to the law (5.6) are injected in order to increase the immune population by 100 individuals while removing the same number of individuals from the susceptible. Figures 7 and 8 display a zoom on the immune and susceptible populations when the impulsive effect is considered. The vaccination law is shown in Figure 9.
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Fig7_HTML.jpg
Figure 7

Immune population evolution with impulse vaccination.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Fig8_HTML.jpg
Figure 8

Susceptible population evolution with impulse vaccination.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Fig9_HTML.jpg
Figure 9

Vaccination law with impulses.

Note that the impulsive vaccination allows to improve the numbers of the immune population at chosen time instants, for instance, in cases when the total population increases through time while the disease tends to spread rapidly.

6.2. Simulations with Uncertainties

This subsection contains some numerical examples concerning the case when small uncertainties in some of the parameters of the system are present. In particular, the new values for the parameters are: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq325_HTML.gif days, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq326_HTML.gif , and especially, the modified delays are: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq327_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq328_HTML.gif days. Furthermore, a small uncertainty in the initial susceptible and infected populations is considered with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq329_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_IEq330_HTML.gif instead of 9172 and 150, respectively, taken as initial nominal values. The following Figures 10, 11, and 12 show the ideal responses for the infectious, infective, and immune and the ones obtained when the real system possesses different parameters (i.e., system with uncertainties).
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Fig10_HTML.jpg
Figure 10

Infectious evolution in the presence of small uncertainties.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Fig11_HTML.jpg
Figure 11

Infected evolution in the presence of small uncertainties.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F281612/MediaObjects/13662_2010_Article_1271_Fig12_HTML.jpg
Figure 12

Immune evolution in the presence of small uncertainties.

As it can be deduced from Figures 10, 11, and 12, the proposed vaccination strategy is robust to small uncertainties in its parameters, especially in the delays. Also, the impulsive vaccination possesses the same effect as in the example of the ideal case, that is, it increases the immune by 100 individuals at each impulsive instant and could be used to mitigate any potential deviation of the immune population due to the parameters mismatch. More technical solutions could be made for the case of presence of uncertainties, as for instance, the use of observers to estimate the state and the use of estimation-based adaptive control for the case of parametrical uncertainties.

7. Concluding Remarks

This paper has dealt with the proposal and subsequent investigation of a time-varying SEIR-type epidemic model of true mass-action type. The model includes time-varying point delays for the infected and infectious populations and distributed delays for the disease transmission effect in the model. The model also admits a potential mortality associated with the disease, a potential lost of immunity of newborns at birth, the presence of threshold population residuals in the infected and infectious populations as well as the contribution to the disease propagation in the local population of potential outsiders taking part of a floating population. A combined regular plus impulsive vaccination strategy has been proposed to remove the disease effects, the second one being used to correct major discrepancies with respect to the suitable population trajectories. The main issues have been concerned with the stability, positivity, and model-following of a suitable reference strategy via vaccination. Also, an example for the influenza disease has been given.

Declarations

Acknowledgments

The authors thank the Spanish Ministry of Education support by Grant no. DPI2009-07197. They are also grateful to the Basque Government by its support through Grants nos. IT378-10, SAIOTEK S-PE08UN15, and SAIOTEK SPE07UN04 and SPE09UN12. The authors are very grateful to the reviewers for their useful suggestions to improve the manuscript.

Authors’ Affiliations

(1)
Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country
(2)
Department of Mathematical Sciences, Florida Institute of Technology
(3)
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals
(4)
Department of Telecommunications and Systems Engineering, Autonomous University of Barcelona
(5)
Department of Electricity and Electronics, Faculty of Science and Technology, University of the Basque Country

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© M. De la Sen et al. 2010

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