An observer-based vaccination control law for an SEIR epidemic model based on feedback linearization techniques for nonlinear systems
© Alonso-Quesada et al.; licensee Springer 2012
Received: 30 March 2012
Accepted: 29 August 2012
Published: 12 September 2012
This paper presents a vaccination strategy for fighting against the propagation of epidemic diseases. The disease propagation is described by an SEIR (susceptible plus infected plus infectious plus removed populations) epidemic model. The model takes into account the total population amounts as a refrain for the illness transmission since its increase makes the contacts among susceptible and infected more difficult. The vaccination strategy is based on a continuous-time nonlinear control law synthesised via an exact feedback input-output linearization approach. An observer is incorporated into the control scheme to provide online estimates for the susceptible and infected populations in the case when their values are not available from online measurement but they are necessary to implement the control law. The vaccination control is generated based on the information provided by the observer. The control objective is to asymptotically eradicate the infection from the population so that the removed-by-immunity population asymptotically tracks the whole one without precise knowledge of the partial populations. The model positivity, the eradication of the infection under feedback vaccination laws and the stability properties as well as the asymptotic convergence of the estimation errors to zero as time tends to infinity are investigated.
KeywordsSEIR epidemic models vaccination nonlinear control stability positivity nonlinear observers design
A relevant area in the mathematical theory of epidemiology is the development of models for studying the propagation of epidemic diseases in a host population [1–20]. The epidemic mathematical models analysed in such an exhaustive list of books and papers include the most basic ones [1–9], namely (i) SI models where only susceptible and infected populations are assumed to be present in the model, (ii) SIR models which include susceptible plus infected plus removed-by-immunity populations and (iii) SEIR models where the infected population is split into two ones, namely the ‘infected’ (or ‘exposed’) which incubate the disease but do not still have any disease symptoms and the ‘infectious’ (or ‘infective’) which do have the external disease symptoms. Those models can be divided into two main classes, namely the so-called ‘pseudo-mass action models’, where the total population is not taken into account as a relevant disease contagious factor and the so-called ‘true-mass action models’, where the total population is more realistically considered as an inverse factor of the disease transmission rates. There are many variants of the above models as, for instance, the SVEIR epidemic models which incorporate the dynamics of a vaccinated population in comparison with the SEIR models [10–12] and the SEIQR-SIS model which adds a quarantine population . Other variant consists of the generalisation of such models by incorporating point and/or distributed delays [8, 10–12, 14]. All of the aforementioned models are so-called compartmental models since host individuals are classified depending on their status in relation to the infectious disease. However, there are diseases where some factors such as the disease transmission, the mortality rate and so on are the functions of age. Then such diseases are described more precisely by means of the so-called compartmental models with age structure . Moreover, although the dynamics of infectious diseases transmission through a host population is continuous-time, some researchers have proposed models composed of difference equations to describe the dynamics of epidemics and develop treatments to minimise its effects within the population . On the one hand, a key point in such research works is the choice of an optimal-time step in order to obtain an acceptable discrete-time model from the discretisation of the continuous-time ones. On the other hand, an advantage is that discrete-time models are easier to analyse than continuous-time ones, and then the effectiveness of a potential treatment to eradicate the disease from the host population can be easier to derive.
The analysis of the existence of equilibrium points, relative to either the persistence (endemic equilibrium point) or extinction (disease-free equilibrium point) of the epidemics in the host population [6, 9, 11–14], the constraints for guaranteeing the positivity and the boundedness of the solutions of such models [11, 12, 17] and the conditions that generate an oscillatory behaviour in such solutions [11, 18] have been some of the main objectives in the literature about epidemic mathematical models. Other important aim is that relative to the design of control strategies in order to eradicate the persistence of the infection in the host population [2, 5, 11, 12, 17]. In this context, an explicit vaccination function of many different kinds may be added to all aforementioned epidemic models, namely constant [5, 12], continuous-time [2, 17], impulsive , mixed constant/impulsive , mixed continuous-time/impulsive , discrete-time and so on. Concretely, the research in  exhaustively analyses the equilibrium points of an SEIR epidemic model under a vaccination strategy based on a state feedback control law with respect to the model parameters and/or the controller gains. The conditions for the eradication of the diseases from the host population, the extinction of the host population or the persistence of the disease in a non-extinguished host population are derived form such a study. Other alternative approaches, as those based on fuzzy rules  or networks framework [13, 20], have been also proposed for modelling the epidemics transmission through a host population. In this way, the influence of certain social network parameters such as visiting probability, hub radius and contact radius on the epidemics propagation has been investigated . Moreover, there are studies about the influence of the immigration on the persistence or extinction of the epidemics in a population subject to immigration from other regions . Also, the influence of epidemic diseases on the dynamics of prey-predator models has been considered in ecoepidemic models .
In this paper, an SEIR epidemic model which includes susceptible (S), infected or exposed (E), infectious (I) and removed-by-immunity (R) populations is considered. The dynamics of susceptible and immune populations are directly affected by a vaccination function , which also has an indirect influence on the time evolution of exposed and infectious population. In fact, such a vaccination function has to be suitably designed in order to eradicate the infection from the population. This model has already been studied in  from the viewpoint of equilibrium points in the controlled and free-vaccination cases. A vaccination auxiliary control law proportional to the susceptible population was proposed in order to achieve the whole population being asymptotically immune. Such an approach assumed that the SEIR model was of the aforementioned true-mass action type, its parameters were known and the illness transmission was not critical. Moreover, some important issues of positivity, stability and tracking of the SEIR model were discussed. The main drawback of such a control strategy is the need of online measures of the susceptible, infected, infectious and immune populations. However, the precise online measures of susceptible and infected populations are not always feasible in some real situations, while only true measures of the infectious and whole populations are available. The main motivation of the present paper is to provide a control solution to overcome such a drawback. In this sense, the use of a switching control law coupled with a state observer to synthesise the vaccination function under no precise knowledge of the exact partial populations which are online estimated by the observer is proposed. Such a law only switches once and in this way the control process is divided into two stages. In the first stage, the so-called observation stage, the control function is identically zero and only the observer is working to reduce the initial difference between the true infectious population measure and the estimated one provided by the own observer below a prescribe threshold. In the second stage, related to a combined observation/control stage, the vaccination function is synthesised by means of an input-output exact feedback linearization technique while the observer is maintained active providing the estimates of the true partial populations. In both stages, the state observer provides online estimations of susceptible and infected populations through time overcoming the unfeasibility of obtaining true measures of such partial populations. Such a combination of a linearization control strategy with a nonlinear observer to online estimate all the partial populations constitutes the main contribution of the paper. Moreover, mathematical proofs about the epidemics eradication based on such a controlled SEIR model coupled with the nonlinear observer are presented while maintaining the non-negativity of all the partial populations for all time. The exact feedback linearization can be implemented by using a proper nonlinear coordinate transformation and a static-state feedback control. The use of such a linearization strategy is motivated by three main facts, namely (i) it is a power tool for controlling nonlinear systems which is based on well-established technical principles [22, 23], (ii) the given SEIR model is highly nonlinear and (iii) such a control strategy has not been yet applied in epidemic models.
On the one hand, approaches based on switching control laws have been broadly dealt with in the control theory and its applications . On the other hand, the combination of exact feedback linearization techniques with state observers has been widely used in many control applications, for instance, in biological systems and chemical engineering [25, 26]. The exact linearization technique requires the system to satisfy some structural and regularity conditions, like the existence of relative degree, the minimum phase property and the integrability condition [27, 28]. The SEIR epidemic model satisfies such assumptions, and the aforementioned linearization technique can be applied without any modification. Otherwise, alternative approaches developed to approximately linearize nonlinear systems violating one or more of such assumptions could be used [29, 30].
The paper is organised as follows. Section 2 describes the set of differential equations which compound the SEIR model for the propagation of an epidemic disease through a host population. A result related to the positivity property of such a model is proven. Section 3 presents a control action based on an input-output linearization technique, guaranteeing the positivity and stability properties of the system while asymptotically achieving the eradication of the infection from the host population and, simultaneously, the whole population becoming immune. The positivity property is required from the own nature of the system which forbids the existence of negative populations at any time instant. The control strategy requires the knowledge of the susceptible, infected, infectious and whole population for all time. In this context, the knowledge of the infectious and whole population for all time is feasible, but the knowledge of the susceptible and infected population for all time is not a realistic assumption. As a consequence, such partial populations have to be estimated by means of an observer dynamic system. Then a control action based on such estimates, instead of the corresponding true partial populations, is carried out in Section 4. These theoretical results and the effectiveness of the feedback input-output linearizing controller combined with the observer are illustrated by means of some simulation results in Section 5.
Notation is the first open n th real orthant and is the first closed n th real orthant. is a positive real n- vector in the usual sense that all its components are non-negative. Also, and are, respectively, used instead of and for scalars. denotes the identity matrix and the determinant of the matrix M.
2 SEIR epidemic model
so that the total population is constant . As a consequence, this model is suitable for epidemic diseases with very small mortality incidence caused by infection and for populations with equal birth and death rates so that the total population may be considered constant for all time. The following result relative to the positivity of the SEIR model in the absence of vaccination is proven. It is relevant since positivity is required for the model validity in real cases.
Lemma 2.1 Assume the SEIR model (2.1)-(2.4) with an initial condition subject to and under no vaccination action before a finite time instant , i.e. . Then .
Proof Let eventually existing finite time instants , , and with being such that:
If , then and .
If , then and .
If , then and .
If , then and .
Proceed by contradiction by assuming that there exists a finite such that , and (where is the value of the function at the time instant which is infinitesimally close to by the right-hand side) with . Thus, from (2.1) since . The facts that and imply that since the solution of the SEIR model (2.1)-(2.4) is continuous for all time. The result contradicts the assumption that and the time instant does not exist.
Proceed by contradiction by assuming that there exists a finite such that , and with . Thus, from (2.2). The facts that and imply that since the solution of the SEIR model (2.1)-(2.4) is continuous for all time. Such a result contradicts the assumption that and the time instant does not exist.
Proceed by contradiction by assuming that there exists a finite such that , and with . Thus, from (2.3). The facts that and imply that since the solution of the SEIR model (2.1)-(2.4) is continuous for all time. Such a result contradicts the assumption that and the time instant does not exist.
Proceed by contradiction by assuming that there exists a finite such that , and with . Thus, from (2.4) since . The facts that and imply that since the solution of the SEIR model (2.1)-(2.4) is continuous for all time. Such a result contradicts the assumption that and the time instant does not exist.
As a result, if and the vaccination function then follows directly since a time instant , for which any of the four partial populations reaches a zero value with its first-time derivative being simultaneously negative at such a time instant, does not exist. □
Remark 2.1 The result is implied by Lemma 2.1, combined with the equation (2.5), provided that and the SEIR model is initialised such that .
3 Vaccination strategy
An ideal control objective is that the removed-by-immunity population asymptotically tracks the whole population. In this way, the joint infected plus infectious population asymptotically tends to zero as time tends to infinity, so the infection is eradicated from the population. A vaccination control law based on a static-state feedback linearization strategy is developed for achieving such a control objective. This technique requires a nonlinear coordinate transformation, based on the theory of Lie derivatives , in the system representation.
where . The first step to apply a coordinate transformation based on the Lie derivation is to determine the relative degree of the system. For such a purpose, the following definitions are taken into account: (i) The k th-order Lie derivative of along is with and (ii) the relative degree r of the system is the number of times that the system output (i.e. the infectious population) must be differentiated in order to obtain the input explicitly, i.e. the number r such that for and .
Both transformations and are smooth mappings, i.e. they have continuous partial derivatives of any order. Then defines a diffeomorphism on D. The feature that the relative degree of the system is equal to the system order allows to change it into a linear and controllable one around any point via the coordinate transformation (3.3) and an exact linearization feedback control [23, 28]. The following result being relative to the input-output linearization of the system is established.
around any point .
with ℓ denoting the order of the differentiation of . Finally, the dynamics of the closed-loop system (3.8) is obtained by direct calculations from (3.12). □
- (ii)The control (3.7) may be rewritten as follows:(3.13)
- (iii)The control law (3.7) is well defined for all except in the surface . However, the infection may be considered eradicated from the population once the infectious population strictly exceeds zero while it is smaller than one individual. So the vaccination strategy may be switched off when . This fact implies that the singularity in the control law is not going to be reached, i.e. such a control law is well defined by the nature of the system. In this sense, the control law given by(3.15)
The linear system (3.8) is strictly identical to the SEIR model (2.1)-(2.4) under the transformation (3.3) and the control law (3.15) for , i.e. until the time instant at which the epidemics is eradicated.
The implementation of the control law (3.15) requires online measurement of the susceptible, infected and infectious population. In a practical situation, only online measures of the infectious and whole populations may be feasible, so the populations of susceptible and infected can only be estimated. In this context, a complete state observer is going to be designed for such a purpose in Section 4.
3.1 Controller tuning parameters choice
The application of the control law (3.7), obtained from the exact input-output linearization strategy, makes the closed-loop dynamics of the infectious population be given by (3.8). Such a dynamics depends on the control parameters for . Such parameters have to be appropriately chosen in order to guarantee the following suitable properties: (i) the stability of the controlled SEIR model, (ii) the eradication of the infection, i.e. the asymptotic convergence of and to zero as time tends to infinity and (iii) the positivity property of the controlled SEIR model under a vaccination based on such a control strategy. The following theorems related to the choice of the controller tuning parameter values for are proven, in order to meet such properties under an eventual vaccination effort.
Theorem 3.2 Assume that the initial condition is bounded, and all roots for of the characteristic polynomial associated with the closed-loop dynamics (3.8) are of strictly negative real part via an appropriate choice of the free-design controller parameters for . Then the control law (3.7) guarantees the exponential stability of the transformed controlled SEIR model (3.1)-(3.6) while achieving the eradication of the infection from the host population as time tends to infinity. Moreover, the SEIR model (2.1)-(2.4) has the following properties: , , and are bounded for all time, , , and exponentially as , and .
Proof The dynamics of the controlled SEIR model (3.8) can be equivalently rewritten with the state equation (3.11) and the output equation , where , by taking into account that , and . The initial condition in such a realization is bounded since it is related to via the coordinate transformation (3.3), and is assumed to be bounded. The controlled SEIR model is exponentially stable since the eigenvalues of the matrix A are the roots for of which are assumed to be in the open left-half plane. Then the state vector exponentially converges to zero as time tends to infinity while being bounded for all time. Moreover, and are also bounded and converge exponentially to zero as from the boundedness and exponential convergence to zero of as according to the first and second equations of the coordinate transformation (3.3). Then the infection is eradicated from the host population. Furthermore, the boundedness of follows from that of and , and the fact that the total population is constant for all time. Also, the exponential convergence of to the total population as is derived from the exponential convergence to zero of and as , and the fact that . Finally, from the third equation of (3.3), it follows that is bounded and it converges exponentially to zero as from the boundedness and convergence to zero of , and as . The facts that and as imply directly that . □
Remark 3.2 Theorem 3.2 implies the existence of a finite time instant after which the epidemics is eradicated if the vaccination control law (3.15) is used instead of that in (3.7). Concretely, such an existence derives from the convergence of to zero as via the application of the control law (3.7).
, and , so that ,
- (b)and satisfy the inequalities:
the application of the control law (3.7) to the SEIR model guarantees that the epidemics is asymptotically eradicated from the host population while , and , and
the application of the control law (3.15) guarantees the epidemics eradication after a finite time , the positivity of the controlled SEIR epidemic model and that so that ,
provided that the controller tuning parameters for are chosen such that for are the roots of the characteristic polynomial associated with the closed loop dynamics (3.8).
- (ii)On the one hand, if the control law (3.15) is used instead of that in (3.7), then the time evolution of the infectious population is also given by (3.17) while the control action is active. Thus, the exponential convergence of to zero as in (3.17) implies directly the existence of a finite time instant at which the control (3.15) switches off. Obviously, the non-negativity of , and is proven by following the same reasoning used in the part (i) of the current theorem. The non-negativity of is proven by using continuity arguments. In this sense, if reaches negative values for some starting from an initial condition , then passes through zero, i.e. there exists at least a time instant such that . Then it follows from (2.4) that(3.31)
Then by taking into account (3.34) in (3.32). The facts that , and imply that via complete induction. Finally, the positivity of the controlled SEIR model follows from the non-negativity of , , and and Lemma 2.1.
by taking into account the third equation in (3.34) and the non-negativity of , and . Finally, it follows that from (3.15) and (3.37). □
In summary, this section has dealt with a vaccination strategy based on linearization control techniques for nonlinear systems. The proposed control law satisfies the main objectives required in the field of epidemics models, namely the stability, the positivity and the eradication of the infection from the population. Such results are proven formally in Theorems 3.2 and 3.3. In Section 5, some simulation results illustrate the effectiveness of such a vaccination strategy. However, such a strategy has a main drawback, namely the control law needs the knowledge of the true values of the susceptible, infected and infectious populations at all time instants which are not available in certain real situations. An alternative approach useful to overcome such a drawback is dealt with in the following section where an observer to estimate all the partial populations is proposed.
4 Vaccination control strategy based on the use of a state observer
for some real constants , and so that . Note that and . The signal denotes the estimation error corresponding to the infectious population, i.e. the deviation between the infectious population estimated by the observer and the true one. Note that from , and then also , by taking into account the coordinate change (3.3) or (3.6). In other words, the estimation error associated to the infectious population is identical in both system representations, defined, respectively, by (3.1)-(3.2) and (3.4)-(3.5). The signal (4.2) has the same structure as the control law (3.14) used for linearizing the SEIR model in the case where the measures of all partial populations were available. Note that the control law (4.1)-(4.3) is expressed in terms of the variables of since the observer design is developed based on the system representation (3.4)-(3.5). Moreover, the observer has to be designed in such a way that the estimation error converged rapidly to zero while maintaining the stability, positivity and epidemics eradication objectives.
The following result relative to the existence of a finite time instant defined as in (4.3) is proven.
The SEIR model parameters are such that ,
in the definition of the switching time instant satisfies ,
the control parameter in (4.2) and the constant in the definition of are such that , and
the observer gain is large enough for the estimation error , with and , to converge asymptotically to zero as .
Then a finite time instant at which the control law (4.1) switches for the first time exists.
for some definite positive function if is large enough since . Then there exists a finite time instant such that for , with being the eventual time instant at which the control law switches for the first time and any real constant. Note that and . Furthermore, the convergence rate of depends on the value of the observer gain θ in the sense that such a convergence rate is increased as the observer gain increases.
provided that the parameters and satisfy the constraint (iii) and by taking into account that , , and from (3.3), (3.5) and (4.9). Thus, (4.10) shows the existence of a finite time instant such that . As a consequence, there exists a finite time instant at which the control law switches, which contradicts the starting hypothesis. □
Remark 4.1 Lemma 4.1 requires the SEIR model parameters to fulfil the condition so that the endemic equilibrium point exists. Otherwise, i.e. if the parameters are such that , the SEIR model converges to the disease-free equilibrium point at which all the population is susceptible and no individual is either infected or infectious so a control action is not necessary to eradicate the epidemic. Then such a situation is not interesting from the control theory viewpoint.
The following result, supported by Lemma 2.1, Theorem 3.3 and Lemma 4.1, is relative to the eradication of the epidemic via the application of the control law (4.1)-(4.3) while guaranteeing the non-negativity of the control signal for all time.
The SEIR model parameters are such that ,
its initial condition satisfies ,
in the definition of the switching time instant and the SEIR model parameters are such that , with large enough,
in the definition of the switching time instant is small enough for and , and ,
the gain observer is large enough, and
the controller parameters for , which are related via (3.33) to the roots for of the closed-loop dynamics characteristic polynomial , are chosen such that the conditions (a)-(b) of Theorem 3.3 are fulfilled while for some sufficient large real constant .
Then the control law (4.1)-(4.3) combined with the observer (4.4)-(4.7) leads to the eradication of the epidemics while guaranteeing the non-negativity of the control signal .
where the control law (4.1)-(4.3) has been introduced in (4.11) and denotes the eventual time instant at which the epidemics is eradicated.
with , if the observer gain θ is large enough for both and to be guaranteed . In this context, on the one hand, note that all terms of in (4.15), except the one with , are strictly negative whenever , and for any observer gain . Such a term may be made small enough, in comparison with the absolute value of the contribution of the rest of the terms in , by means of a large enough observer gain so that . On the other hand, can be also ensured with a large enough observer gain . In this case, the term depending on in (4.21) is strictly negative since in such a time interval. Note also that , which is non-negative , may be made small in comparison with the absolute value of the rest of the terms in (4.21) by using a sufficiently large so that .
provided that . Note that if satisfies , then any also satisfies it. From (4.23), the population estimates vector would converge asymptotically to zero as . As a consequence, if the time interval is sufficiently large for the conditions (4.17), (4.18) and , (4.21) be satisfied until reaching a time instant at which , then the propagation of the infection will be eradicated from the population by taking into account that is very small, while maintaining the positivity of the controlled SEIR model. □
The control strategy described by (4.1)-(4.3) is composed of two consecutive stages. The first one is an observation stage at which no control action is applied to the SEIR model and the main objective is that the observer variables go converging to the true partial populations. Once the estimation error is sufficiently small, such that the observer variables track the true population with a suitable precision, this observation stage ends. Then the second stage at which a control action is applied begins. Such a control action linearizes the closed-loop observer so that the estimated variables converge asymptotically to zero as time grows while also guaranteeing the convergence to zero of the estimation error. This implies that the true infectious and infected populations decrease toward zero as time grows until reaching a finite time instant when the epidemic disease is considered eradicated from the host population. After such an instant, the control action is removed from the system.
On the one hand, the length of the observation stage may be short enough with a suitable choice of the observer gain. In this sense, such a length decreases as the value of the observer gain θ increases. Then a large value for θ seems to be appropriate. On the other hand, a large value of θ makes the contribution of the perturbation term in (4.16) considerable. As a consequence, a tradeoff value for θ has to be searched. In this context, in the limiting case that θ was a very small positive real number (i.e. ) the closed-loop dynamics of the observer system would be with A defined in (3.11), i.e. such a dynamics would be identical to that of the closed loop SEIR model without the observer. As a consequence, if the controller parameters for were chosen according to the conditions of Theorem 3.3, the estimates in would be non-negative and would tend asymptotically to zero as time tends to infinity. However, the estimation error and would not converge asymptotically to zero as time tends to infinity with such a value of , so the infection would not be eradicated. On the contrary, if the value for were very large, then a very small real constant would be necessary for the term , which acts as a perturbation for the observer closed-loop system (4.16), to be sufficiently small. Such a fact would delay the beginning of the control action.
- (iv)Once the observer has been designed in the state representation , associated to the state vector , the coordinate transformation (3.6) has to be applied to obtain the estimates in , corresponding to the original state vector , from the vector . The following dynamics equations for the estimated populations are obtained from (4.11) by applying (3.6)(4.24)
The equation (4.21) is a key result in the mathematical proof of stability. Note that once the control law switches at time instant , and the system passes from the observation stage to the observation/control one, it is ensured that the observation error still exhibits an asymptotically decaying behaviour on a sufficiently large time interval provided that the size of the control signal is large enough at such a switching time instant. This may be guaranteed by means of a sufficient large real constant , and simultaneously and with small enough and large enough so that and on in order to ensure the validity of (4.18) on .
In view of (3.33), the assumption (vi) of Theorem 4.1 can be satisfied by choosing a root for the closed-loop system characteristic polynomial with as large as it is necessary for the constraints (a) and (b) of Theorem 3.3 to be fulfilled.
5 Simulation results
An example based on an outbreak of influenza in a British boarding school in early 1978  is used to illustrate the theoretical results presented in the paper. Such an epidemic can be described by the SEIR mathematical model (2.1)-(2.4) with the parameter values: , , and . A total population of is considered with the initial conditions , , and . Three sets of simulation results are presented to compare the evolution of the SEIR mathematical model populations in three different situations, namely (i) when no vaccination control actions are applied (ii) if a control action based on the feedback input-output linearization approach is applied and (iii) if a nonlinear observer dynamics is coupled with a switching control law, as that defined by (4.1)-(4.3), so as to use population estimates in the vaccination control.
5.1 Epidemic evolution without vaccination
5.2 Epidemic evolution with the feedback control law without an observer
5.3 Epidemic evolution with the feedback control law with an observer
5.3.1 Influence of the observer gain in the time evolution of the system
6 Concluding remarks
A vaccination control strategy based on feedback input-output linearization techniques has been proposed to fight against the propagation of epidemic diseases within a host population. An SEIR model with known parameters is used to describe the propagation of the disease. The stability and positivity properties of the closed-loop system have been proven in the case where true data of the susceptible, infected and infectious populations are available. Otherwise, the control law may be combined with a complete state non-linear observer which provides online estimates of such populations used in the vaccination controls. These theoretical results are complemented with some simulation results to illustrate the effectiveness of the proposed approach. Future research into the subject is going to deal with the application of the current approach and similar non-linear techniques to other disease propagation models.
The authors thank the Spanish Ministry of Education for its support of this work through grants DPI2009-07197 and DPI2012-30651 and to the Basque Government for its support through grants IT378-10, SAIOTEK SPE07UN04 and SAIOTEK SPE09UN12.
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