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 Sen et al. 2010
Received: 17 August 2010
Accepted: 2 December 2010
Published: 14 December 2010
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.
SI-models where not removed-by-immunity population is assumed. In other words, only susceptible and infected populations are assumed,
SIR-models, which include susceptible, infected, and removed-by-immunity populations,
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 , impulsive , discrete-time, and so forth, by incorporating point or distributed delays [12, 13], oscillatory behaviors , 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, 15–17]. 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 . 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 [20–22]. 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.
: Susceptible population, that is, those who can be infected by the disease
: Infected population, that is, those who are infected but do not still have external symptoms
: Infectious population, that is, those who are infected exhibiting external symptoms
: Immune population
: Total population
: Function associated with the infected floating outsiders in the SEIR model
: Disease transmission function
: Natural growth rate function of the population
: Natural rate function of deaths from causes unrelated to the infection
: Takes into account the potential immediate vaccination of new borns
: Functions that and are, respectively, the instantaneous durations per populations averages of the latent and infectious periods at time .
: the rate of lost of immunity function
: related to the mortality caused by the disease
: Thresholds of infected and infectious populations
: Different point and impulsive delays in the epidemic model
: Functions associated with the regular and impulsive vaccination strategies
: 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
and which give sense of the asymptotic limit of the trajectory solutions.
is the total population at time .
is a bounded piecewise-continuous function related to the natural growth rate of the population. is assumed to be zero if the total population at time is less tan unity, that is, , implying that it becomes extinguished.
is a bounded piecewise-continuous function meaning the natural rate of deaths from causes unrelated to the infection.
is a bounded piecewise-continuous function which takes into account the immediate vaccination of new borns at a rate .
is a bounded piecewise-continuous function which takes into account the number of deaths due to the infection.
is a bounded piecewise-continuous function meaning the rate of losing immunity.
is a bounded piecewise-continuous transmission function with the total number of infections per unity of time at time .
is a transmission term accounting for the total rate at which susceptible become exposed to illness which replaces in the standard SEIR model in (2.1)–(2.2) which has a constant transmission constant . It generalizes the one-delay distributed approach proposed in  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 , with are nonnegative weighting real functions being everywhere continuous on their definition domains subject to Assumption 1(1) below, and are the 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 within some of the integrals and the absence of delays is modeled with all the functions being identically zero.
are bounded continuous functions defined so that and are, respectively, the instantaneous durations per populations averages of the latent and infective periods at time .
are piecewise-continuous functions being integrable on any subset of 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.
- (x)The two following coupling infected-infectious dynamics contributions:(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 are the gain and their associate infected and infectious delay functions which are everywhere continuous in . In the time-invariant version of a simplified pseudomass-type SIRS-model proposed in , the constant gains are and .
in (2.1) and (2.4) are nonnegative nonidentically zero vaccination weighting real functions everywhere on their definition domains subject to distributed delays governed by the functions where , with is a vaccination function to be appropriately normalized to the day-to-day population to be vaccinated subject to . As for the case of the transmission term, punctual delays could be included by using appropriate Dirac deltas within the corresponding integrals.
The SEIR model is subject to a joint regular vaccination action plus an impulsive one at a strictly ordered finite or countable infinite real sequence of time instants . Specifically, it is a single Dirac impulse of amplitude if and zero if . The weighting function can be defined in several ways. For instance, if when , and , otherwise, then when and it is zero, otherwise. Thus, the impulsive vaccination is proportional to the total population at time instants in the sequence . If , then the impulsive vaccination is proportional to the susceptible at such time instants. The vaccination term 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 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 , then neither the natural increase of the population nor the loss of maternal lost of immunity of the newborns is taken into account. If , then some of the newborns are not vaccinated with the consequent increase of the susceptible population compared to the case . If , then such a lost of immunity is partly removed by vaccinating at birth a proportion of newborns.
There exist continuous functions with such that for some prefixed and any given .
Assumption 1(1) for the distributed delay weighting functions is proposed in . 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 for each set of admissible initial conditions and each set of vaccination impulses which is continuous and time-differentiable on for time instant , provided that it exists, being such that , or on , if such a finite impulsive time instant 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
- (a)the sequence ST trivially exists if and only if . Then, with at least one of the real interval unions being nonempty, are disjoint subsets of satisfying,(3.8)and defined as follows:
for any given if and only if ,
for any given if and only if , and define also ,
, where unit cardinal means that the time-derivative of the candidate has no change of sign and infinite cardinal means that there exist infinitely many changes of sign in ,
if it exists a finite such that , 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 ,
if there is no finite such that and, then, the sequence ST is infinite and the set has infinite cardinal.
The following result is obtained from the above discussion under conditions which guarantee that the candidate is bounded for all time.
what leads to if which again contradicts that as 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 , it is bounded for all finite time so that Theorem 3.1 is guaranteed under an equivalent simpler condition as follows.
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.
there exists such that ,
for some constants and some prefixed finite . Then, the total population of the SEIR model is nonnegative and bounded for all time, and asymptotically extinguishes at exponential rate irrespective of the vaccination law.
then the total population 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 ) is less than zero. Another stability result based on Gronwall's Lemma follows.
for some prefixed finite . Then, the total population of the SEIR model is nonnegative and bounded for all time irrespective of the vaccination law. Furthermore, converges to zero at an exponential rate if the above second inequality is strict within some subinterval of of infinite Lebesgue measure.
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 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.
4.2. Positivity of the Infected Population of the Generalized SEIR Model
Then, the following result holds concerning the non negativity of the infected population.
4.3. Positivity of the Infectious Population of the Generalized SEIR Model
Thus, we have the following result concerning the non negativity of the infectious population.
4.4. Positivity of the Removed by Immunity Population of the Generalized SEIR Model
Then, the following result holds concerning the non negativity of the immune population.
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.
If Assertions 1–4 hold jointly, then, the populations , and in the generalized SEIR model (2.1)–(2.5) are lower bounded by zero and upper bounded by . If, furthermore, either Theorem 3.1, or Corollary 3.3, or Theorem 3.4 or Theorem 3.5 holds, then , , , and are bounded .
for each time instant , 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 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 for such a time instant ,
either Theorem 3.1, or Corollary 3.3, or Theorem 3.4 or Theorem 3.5 holds, then , , and are bounded .
Then, the populations , , , and of the generalized SEIR model (2.1)–(2.5) are lower bounded by zero and upper bounded by what is, in addition, bounded, .
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.
for some sufficiently small .
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 and a given impulsive vaccination distribution , then it also holds if those vaccination function and distribution are identically zero.
for some sufficiently small .
for some sufficiently small .
The following result follows from (2.3) and it is proved in a close way to the proof of Assertions 5–7.
for some sufficiently small .
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 , the infected-infectious threshold constraint 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.
Then, , and irrespective of the delays and vaccination laws that satisfy (even if the SEIR model (2.1)–(2.5) is vaccination free). Furthermore, , , irrespective of the delays and vaccination law even if the SEIR model (2.1)–(2.5) is vaccination-free.
Note that , that is, the set of impulsive time instants in is identical to that in if and only if and includes if and only if . Note also that if and only if , in particular, if . A related result to Assertion 9 follows.
Assume that the constraints of Assertion 9 hold except that is replaced by . Then, the conclusion of Assertion 9 remains valid.
A positivity result for the whole epidemic model (2.1)–(2.5) follows.
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.
Assume that the hypotheses of Theorem 4.3 hold and, furthermore, such that . 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.
with initial conditions being identical to those of . Then, the infected population trajectory (5.1) guarantees the exact tracking of the infectious population of the given reference infectious trajectory which furthermore satisfies the differential equation (2.3).
is in under initial conditions with being given by (5.2) with initial conditions . 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 with the suited reference infected population differential equation satisfying (2.2).
which follows from (2.1) and (A.8), such that and . 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 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.
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.
Assume that for some and any given . Then, (prescribed) if with so that .
Note that 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 due to errors in the SEIR model (2.1)–(2.5) for some prefixed unsuitable sufficiently large . Then, an impulsive vaccination at time may be generated so that .
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, 26–28].
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
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.
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
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.
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.
- De La Sen M, Alonso-Quesada S: A control theory point of view on Beverton-Holt equation in population dynamics and some of its generalizations. Applied Mathematics and Computation 2008,199(2):464-481. 10.1016/j.amc.2007.10.021MathSciNetView ArticleMATHGoogle Scholar
- De La Sen M, Alonso-Quesada S: Control issues for the Beverton-Holt equation in ecology by locally monitoring the environment carrying capacity: Non-adaptive and adaptive cases. Applied Mathematics and Computation 2009,215(7):2616-2633. 10.1016/j.amc.2009.09.003MathSciNetView ArticleMATHGoogle Scholar
- De La Sen M, Alonso-Quesada S: Model-matching-based control of the Beverton-Holt equation in ecology. Discrete Dynamics in Nature and Society 2008, -21.Google Scholar
- De La Sen M: About the properties of a modified generalized Beverton-Holt equation in ecology models. Discrete Dynamics in Nature and Society 2008, -23.Google Scholar
- De La Sen M: The generalized Beverton-Holt equation and the control of populations. Applied Mathematical Modelling 2008,32(11):2312-2328. 10.1016/j.apm.2007.09.007MathSciNetView ArticleMATHGoogle Scholar
- Mollison D (Ed): Epidemic Models: Their Structure and Relation to Data. Newton Institute, Cambridge University Press, New York, NY, USA; 1995.MATHGoogle Scholar
- Keeling MJ, Rohani P: Modeling Infectious Diseases in Humans and Animals. Princeton University Press, Princeton, NJ, USA; 2008:xvi+368.MATHGoogle Scholar
- Yildirim A, Cherruault Y: Anaytical approximate solution of a SIR epidemic model with constant vaccination strategy by homotopy perturbation method. Kybernetes 2009,38(9):1566-1575. 10.1108/03684920910991540View ArticleGoogle Scholar
- Erturk VS, Momani S: Solutions to the problem of prey and predator and the epidemic model via differential transform method. Kybernetes 2008,37(8):1180-1188. 10.1108/03684920810884973MathSciNetView ArticleMATHGoogle Scholar
- Ortega N, Barros LC, Massad E: Fuzzy gradual rules in epidemiology. Kybernetes 2003,32(3-4):460-477. 10.1108/03684920310463876View ArticleMATHGoogle Scholar
- Khan H, Mohapatra RN, Vajravelu K, Liao SJ: The explicit series solution of SIR and SIS epidemic models. Applied Mathematics and Computation 2009,215(2):653-669. 10.1016/j.amc.2009.05.051MathSciNetView ArticleMATHGoogle Scholar
- Song X, Jiang Y, Wei Hg: Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays. Applied Mathematics and Computation 2009,214(2):381-390. 10.1016/j.amc.2009.04.005MathSciNetView ArticleMATHGoogle Scholar
- Zhang T, Liu J, Teng Z: Dynamic behavior for a nonautonomous SIRS epidemic model with distributed delays. Applied Mathematics and Computation 2009,214(2):624-631. 10.1016/j.amc.2009.04.029MathSciNetView ArticleMATHGoogle Scholar
- Mukhopadhyay B, Bhattacharyya R: Existence of epidemic waves in a disease transmission model with two-habitat population. International Journal of Systems Science 2007,38(9):699-707. 10.1080/00207720701596417MathSciNetView ArticleMATHGoogle Scholar
- Barreiro A, Baños A: Delay-dependent stability of reset systems. Automatica 2010,46(1):216-221. 10.1016/j.automatica.2009.10.029MathSciNetView ArticleMATHGoogle Scholar
- De La Sen M: On positivity of singular regular linear time-delay time-invariant systems subject to multiple internal and external incommensurate point delays. Applied Mathematics and Computation 2007,190(1):382-401. 10.1016/j.amc.2007.01.053MathSciNetView ArticleMATHGoogle Scholar
- De La Sen M: Quadratic stability and stabilization of switched dynamic systems with uncommensurate internal point delays. Applied Mathematics and Computation 2007,185(1):508-526. 10.1016/j.amc.2006.07.048MathSciNetView ArticleMATHGoogle Scholar
- Daley DJ, Gani J: Epidemic Modelling: An Introduction, Cambridge Studies in Mathematical Biology. Volume 15. Cambridge University Press, Cambridge, UK; 1999:xii+213.View ArticleGoogle Scholar
- Piccardi C, Lazzaris S: Vaccination policies for chaos reduction in childhood epidemics. IEEE Transactions on Biomedical Engineering 1998,45(5):591-595. 10.1109/10.668749View ArticleGoogle Scholar
- Zhang T, Liu J, Teng Z: Dynamic behavior for a nonautonomous SIRS epidemic model with distributed delays. Applied Mathematics and Computation 2009,214(2):624-631. 10.1016/j.amc.2009.04.029MathSciNetView ArticleMATHGoogle Scholar
- Gao S, Teng Z, Xie Dehui: The effects of pulse vaccination on SEIR model with two time delays. Applied Mathematics and Computation 2008,201(1-2):282-292. 10.1016/j.amc.2007.12.019MathSciNetView ArticleMATHGoogle Scholar
- Khan QJA, Krishnan EV: An epidemic model with a time delay in transmission. Applications of Mathematics 2003,48(3):193-203. 10.1023/A:1026002429257MathSciNetView ArticleMATHGoogle Scholar
- Boichuk A, Langerová M, Škoríková J: Solutions of linear impulsive differential systems bounded on the entire real axis. Advances in Difference Equations 2010, -10.Google Scholar
- Nieto JJ, Tisdell CC: On exact controllability of first-order impulsive differential equations. Advances in Difference Equations 2010, -9.Google Scholar
- Yu HG, Zhong SM, Agarwal RP, Xiong LL: Species permanence and dynamical behavior analysis of an impulsively controlled ecological system with distributed time delay. Computers & Mathematics with Applications 2010,59(12):3824-3835. 10.1016/j.camwa.2010.04.018MathSciNetView ArticleMATHGoogle Scholar
- Marchenko VM, Zachkevich Z: Representation of solutions of control hybrid differential-difference impulse systems. Differential Equations 2009,45(12):1811-1822. 10.1134/S0012266109120118MathSciNetView ArticleMATHGoogle Scholar
- Marchenko VM, Luazo ZZ: On the stability of hybrid differential-difference systems. Differential Equations 2009,45(12):1811-1822. 10.1134/S0012266109120118MathSciNetView ArticleMATHGoogle Scholar
- De La Sen M: A method for general design of positive real functions. IEEE Transactions on Circuits and Systems. I. Fundamental Theory and Applications 1998,45(7):764-769. 10.1109/81.703845MathSciNetView ArticleMATHGoogle Scholar
- De La Sen M, Alonso-Quesada S: On vaccination control tools for a general SEIR-epidemic model. Proceedings of the 18th Mediterranean Conference on Control and Automation (MED '10), 2010 1: 1322-1328.View ArticleGoogle Scholar
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