Research Article | Open Access
On Vaccination Strategies for a SISV Epidemic Model Guaranteeing the Nonexistence of Endemic Solutions
A vaccination strategy based on the state feedback control theory is proposed. The objective is to fight against the propagation of an infectious disease within a host population. This propagation is modelled by means of a SISV (susceptible-infectious-susceptible-vaccinated) epidemic model with a time-varying whole population and with a mortality directly associated with the disease. The vaccination strategy adds four free-design parameters, with three of them being the feedback gains of the vaccination control law. The other one is used to switch off the vaccination if the proportion of susceptible individuals is smaller than a prescribed threshold. The paper analyses the positivity of such a model under the proposed vaccination strategy as well as the conditions for the existence of the different equilibrium points of its normalized model. The fact that an appropriate adjustment of the control gains avoids the existence of endemic equilibrium points in the normalized SISV model while guaranteeing the existence of a unique disease-free equilibrium point being globally exponentially stable is proved. This is a relevant novelty dealt with in this paper. The persistence of the infectious disease within a host population irrespective of the growing properties of the whole population can be avoided in this way. Such theoretical results are mathematically proved and, also, they are illustrated by means of simulation examples. Moreover, the performance of the proposed vaccination strategy in several real situations is studied in some simulation examples. One of them deals with the presence of uncertainties, which affects the synthesis of the vaccination control law, in the measures of the subpopulations of the model.
The use of mathematical models to describe the propagation of epidemic diseases has been broadly carried out for several decades [1, 2]. The dynamics of such models can be used to take a decision about the convenience of applying control strategies, based on vaccination campaigns, quarantines, effective isolation measures in hospitals, and/or use of antiviral drugs among others in order to avoid the persistence of the disease or at least to minimize its effects within the host population [3–10]. The vaccination strategies can be governed by a signal synthesized using some control techniques such as state variables feedback, adaptive control, sliding mode control, or optimal control among others [3, 11–14]. Moreover, the control signal can be constant, time-varying function, impulsive, and so on [15–18].
A great variety of models have been used to study the propagation of infectious diseases. An important kind of such models is referred to as the class of compartmental models . Such models split the total population in different categories depending on the status of the individuals with respect to the infection. The simplest one is the SIS epidemic model . Such a model splits the population in two categories: susceptible (S) and infectious (I) subpopulations. There are two transitions in this model, one from the susceptible to the infectious category, which occurs with certain probability when contacts between susceptible and infectious individuals happen, and the other one from the infectious to the susceptible category once the infectious individual has passed an infectious period. Such a period as well as the probability of contagion of a susceptible individual after contacting with an infectious one is particular for each infectious disease. This kind of models assumes that the infectious individuals lose immunity and become susceptible immediately after recovering as it happens for sexually transmitted infections like chlamydia and gonorrhoea . Also, the SIS model is appropriate for bacterial diseases such as meningitis and cholera. Moreover, there can be a mortality associated with each disease so that some of the infectious individuals may not recover from the disease. In this sense, the total population of the model can be considered as a constant if the mortality caused by the infectious disease and/or the disease time duration are small enough so that such an assumption be acceptable. Otherwise, a time-varying population has to be considered to properly study the dynamics of the infectious disease [17, 22]. There are other compartmental models, such as SIR and SEIR ones, which are widely used in the literature [11, 12, 14–16, 18, 19, 22–24]. The SIR one adds a subpopulation of recovered (R) by immunization individuals. Such a category includes individuals who have recovered from the disease after passing the infectious period and the immunization can be for life or for a certain limited period. Once such an immunization period has ended, the recovered individuals come back to the susceptible category. This model is properly referred to as SIRS model . The SEIR model adds a new subpopulation with respect to the SIR one, namely, the exposed (E) one. Such a category includes the susceptible individuals who have been infected after contacts with infectious individuals but they do not present symptoms and they do not transmit the disease after contacting with susceptible individuals. A vaccination (V) subpopulation can be added to all of the aforementioned epidemic models. Such models are known, respectively, as SISV, SVIR, and SVEIR models [26–30]. The vaccination subpopulation includes the individuals who have received a vaccine to prevent the contagion of the disease. Other models consider the application of treatment by drugs to a fraction of the infective individuals. Then, the infective subpopulation is split into two categories, namely, the infective without treatment (I) and the infective under treatment (T). The individuals within the first one can transmit the disease after contacts, so that they are infectious, whereas the individuals in the other one cannot propagate the disease . All the aforementioned models only consider the direct transmission of the disease after direct contacts between the susceptible and infectious individuals. However, there are also models describing the propagation of vector-transmitted diseases as well as models for indirectly transmitted diseases via pathogens shed in the environment by the infectious individuals [32, 33].
In this paper, the analysis of a SISV epidemic model for directly transmitted diseases, with a time-varying population and mortality directly associated with the disease, under a vaccination strategy based on a state feedback control law is carried out. The model assumes that the susceptible individuals who receive a vaccine pass directly to the vaccinated subpopulation and they maintain the immunity for life although they have contacts with infectious individuals. Also, an efficiency of 100% of the vaccines is assumed, which implies that all of the susceptible individuals who receive a vaccine pass to the vaccinated category. The vaccination strategy provides some free-design parameters, namely, the three constant control gains, with each one associated with each state variable of the model, and an additional one which switches off the vaccination when the proportion of susceptible individuals is smaller than a prescribed threshold. A similar SISV model is analysed in [26, 30] under a vaccination rate being proportional to the number of susceptible individuals. The main differences and novelties of the current paper with respect to them are focused on the synthesis of the vaccination control law, namely, the following:(a)The control law is based on the feedback of all the variables of the model so that there are three control constant gains to be tuned instead of just one used in [26, 30].(b)Some of the control gains can take negative values but if the feedback control signal becomes negative at some time instant, then such a signal is zeroed. Precisely, an appropriate choice for the values of such gains is key to guarantee the inexistence of endemic equilibrium (EE) points and in this way to achieve the eradication of the infection. This is an important novelty of this paper.(c)Another difference with respect to  is that the current paper deals with a time-varying whole population and a mortality directly induced by the disease.(d)A different with respect to , where the whole population is also time-varying with a mortality related to the infection, is that in the current paper the growing properties of the total population and the infection are separated by the control design. In this sense the infection can be asymptotically removed even if the total population is unbounded as time increases due to the higher rate of births related to the natural death rate. Note that this is the current situation for the whole entire world population. However, the model proposed in  under vaccination converges to a disease-free equilibrium (DFE) point or an EE one where the total population is a constant value.
The analysis of the current paper includes the proof of the positivity of the model under the proposed vaccination strategy. Also the influence of the control gains on the dynamics of the disease transmission within the host population is studied by means of a normalized SISV model. Such a normalized model has two independent state variables, instead of the three of the original SISV model, which simplifies the analysis. The conditions for the existence of the equilibrium points of this normalized model depending on the assigned values to the control gains are analysed. In this context, the existence of appropriate choices for the control gains guaranteeing the nonexistence of EE points jointly with the existence of a unique globally exponentially stable DFE point for the normalized SISV model is proved. Such a fact is a key result to achieve the main objective of the paper, namely, the eradication of the disease propagation within the host population, while guaranteeing the persistence of the whole population, by means of the application of a vaccination strategy based on the proposed state feedback control law with an appropriate adjustment of the control gains. Furthermore, the proportion of the vaccinated subpopulation in the aforementioned globally exponentially stable DFE point depends on the values of the control gains. Then, the number of vaccines to be used during the vaccination campaign can be prefixed by adjusting such gains. In a practical situation, this fact can be used to choose the control gains according to the number of available vaccines while guaranteeing the unique existence of the globally asymptotically stable DFE point.
Several simulation examples are carried out to complement the theoretical results of the paper. The first one shows that an infectious disease, with a mortality directly associated with the illness, can lead to the extinction of the host population in absence of some prevention action. This fact motivates the application of a vaccination campaign to avoid such an extinction. In this context, the second example shows that the application of a vaccination campaign based on the proposed state feedback control law, with an appropriate adjustment of the control gains, achieves the main objective of eradicating the infectious disease while guaranteeing the growing of the host population. Such an example illustrates the main theoretical results of the paper. The third example points out that some specifications of the vaccination campaign, such as the number of required vaccines and the duration of the campaign, depend on the values assigned to the control gains. Finally, the two last examples show that a vaccination campaign based on the proposed strategy could be implemented in a real situation. The fourth example takes into account that the number of vaccines to be injected at each day has to be an integer number. In this context, two alternative ways for designing the vaccination campaign with such a restriction are dealt with. On the other hand, the fifth example takes into account the fact that the measures of the state variables of the model can be subject to uncertainties since the knowledge of the exact number of susceptible, infectious, and vaccinated individuals at each time instant is not possible in a real situation [34, 35]. In this context, this example considers the lack of precision in the measures of the subpopulations affecting the synthesis of the control law. The result shows that the control law is robust under the presence of small uncertainties in the measures. In summary, the main contributions of the paper are as follows: (i) the design of a control law via the feedback of all the state variables of the SISV epidemic model where some of the control gains can take negative values while maintaining the control signal nonnegative for all time, (ii) the proof that an appropriate adjustment of the control gains guarantees the nonexistence of EE points and the existence of a unique globally exponentially stable DFE point in the normalized SISV model which leads to the eradication of the infectious disease, (iii) the study of the influence of the control gains values in the proportions of the subpopulations in the globally exponentially stable DFE point as well as in the transient behavior of the controlled model by means of a simulation example, and (iv) the presentation of a method to implement a vaccination campaign based on the proposed control technique in a realistic situation with uncertainties in the subpopulations measures needed to synthesize the control law and a simulation example to illustrate it.
The paper is organized as follows. Section 2 describes the SISV model and the vaccination strategy based on a state feedback control law. Section 3 deals with the analysis of the positivity of the SISV model under the proposed vaccination. Also, the analysis of the equilibrium points of its normalized model related to the control gains is carried out. Such a study points out that a vaccination strategy based on feedback of the model variables can achieve the eradication of the disease with an appropriate choice of the values for the control gains. Section 4 deals with numerical examples which illustrate the theoretical results and proposes several ways of implementing a vaccination campaign based on the presented strategy in a realistic situation. Finally, Section 5 ends the paper with some concluding remarks.
2. The SISV Epidemic Model
The SISV epidemic model splits the host population into three different categories: susceptible, infectious, and vaccinated subpopulations. The transitions between the subpopulation categories of this epidemic model are given by the following differential equations:subject to , , , and , where , , and denote, respectively, the susceptible, infectious, and vaccinated subpopulations at the instant and denotes the whole population. By summing up the equations of (1) one obtains the dynamics of the whole population given byThe parameters of the model are all strictly positive. Namely, and denote, respectively, the birth and the mortality by natural causes rates of the host population. The transmission of the infection from mothers to sons/daughters is not considered so the new births are included directly in the susceptible category. The rest of the parameters are associated with the infectious disease: denotes the infection transmission rate, is the recuperation rate, and is the mortality by causes directly related to the disease. The inverse of denotes the average time interval that an infectious individual spends within the infectious category before passing to the susceptible category. The transition of an individual from the susceptible category to the infectious one can happen, with certain probability, when such a susceptible individual contacts an infectious one. In this sense, the factor is the per capita probability of acquiring the infection at the instant and the term represents the total rate of transmissions of the infection at the instant .
The function denotes a control signal based on the application of a vaccination strategy. It is assumed that the application of a vaccine to a susceptible individual transfers such an individual to the vaccinated category and he/she/it is maintained in such a status until he/she/it dies by natural causes. Such a fact is taken into account by the inclusion of the term in the dynamics of the vaccinated subpopulation. In other words, an efficiency of 100% in the vaccination and that the vaccinated individuals acquire immunization for life are assumed.
Remark 1. (i)The dynamics of the whole population depends on the net growth rate in the absence of disease. It implies that the whole population exponentially increases if , it holds constant to if , or it exponentially decreases to zero and the host population is extinguished if . This form of population dynamics is called exponential birth and deaths. An alternative way of modelling the dynamics of the whole population consists in replacing the term by a constant recruitment rate . Such a form is referred to as constant immigration with exponential death rate .(ii)The propagation of the infectious disease is described by a SIS epidemic model composed by the two first equations of (1) in the absence of vaccination, i.e., if , and with an initial condition without vaccinated individuals, i.e., if . Direct calculations show that such a model has two possible equilibrium points depending on the value of the basic reproduction number . If , the model has a unique equilibrium point at which all the population is susceptible. Such an equilibrium point is referred to as DFE point since there are not infectious individuals. Furthermore, such a point is globally asymptotically stable so that the disease is eradicated from the host population irrespective of the initial condition. Otherwise, if , the model has two equilibrium points: the DFE point, which is unstable, and an EE point at which there is a proportion of the whole population in the susceptible status, , and the other one in the infectious category, . Furthermore, such an EE point is globally asymptotically stable so that the disease persists within the host population irrespective of the initial condition. Such properties are proved in  for the case of whole population dynamics with constant immigration and exponential death rate. However, such properties can be also proved for a whole population dynamics with exponential birth and deaths, as proposed in the current paper, by using a normalized version of the SIS model built from applying the variable changes and to such a model.
3. Design of Vaccination Strategies Based on Control Theory
The application of vaccination strategies can be considered in two situations: (i) when the propagation of the infectious disease leads to an EE point or (ii) when the propagation dynamics converges to a DFE point but an improvement in the transient behavior is required. The first situation is going to be dealt with in this section. In this sense, an infectious disease whose propagation is described by the epidemic model (1), where its parameters satisfy the condition , is considered. This means that the basic reproduction number of the model without considering the vaccination subpopulation (SIS model) fulfills . The vaccination is applied to a fraction of the susceptible individuals. Such individuals pass to the category referred to as vaccinated subpopulation when they receive the vaccine.
A vaccination based on a control signal defined byis considered where , , and are real constants, namely, the controller gains. Such a law is based on the feedback of the variables of the SISV epidemic model while it is nonnegative definite. Furthermore, the vaccination is suspended while the proportion of susceptible subpopulation is strictly smaller than a prescribed strictly positive threshold .
Theorem 2 (positivity of the SISV model). The SISV epidemic model (1) is positive under the application of the control signal (3) irrespective of the values assigned to the control gains since , , and provided that , , and .
Proof. Assume that there exists a finite time instant such that and with . The continuity of the SISV model together with , and implies that there exists some time instant such that with and . It implies that and then from (1). Then, the starting assumption at a finite time instant is not true. Now, assume that there exists a finite time instant such that and with . The continuity of the SISV model together with , , and implies that there exists some time instant such that with and . This implies that from (1) and (3). Then, the starting assumption at a finite time instant is not true. Finally, assume that there exists a finite time instant such that and with . The continuity of the SISV model together with the conditions , , and implies that there exists some time instant such that with and . It follows that from (1) and (3). Then the starting assumption at some finite time instant is not true. In summary, there are not time instants at which , , or take negative values provided that , , and . The result of the theorem is proved.
3.1. Normalized SISV Model
A variable change lets us obtain a normalized SISV epidemic model useful to analyse the dynamics of the propagation of the disease under the proposed vaccination strategy. Such a variable change is given by where the resulting new variables , , and represent, respectively, the proportion of susceptible, infectious, and vaccinated individuals within the host population. Note that , , and are derived from Theorem 2 if a vaccination strategy based on the control law (3) is applied in the epidemic model.
One obtains the following normalized SISV model:by applying the aforementioned variable change in (1) wheredenotes the normalized control signal. Model (5) under the control law (6) can be written asby taking into account the fact that . The following subsection analyses the equilibrium points for model (5) under the control law (6) depending on the values of the controller gains.
3.1.1. Equilibrium Points
The normalized SISV model (5) under the control law (6) asymptotically reaches an equilibrium point given by , , and when and . Thenwhere denotes the value of the control signal at some potential equilibrium point. Such a value can be or by taking into account the control law (6).
Assume that . Then, the second equation of (8) has two solutions, namely, either or . The first equation of (8) has two solutions when , namely, and . Both of them satisfy that from the fact that so that they are compatible with the positivity of the SISV model established in Theorem 2. Note that such a property implies that a feasible equilibrium point of the normalized SISV model has to have . On the other hand, the first equation of (8) has a unique solution in when , namely, . From the positivity of the SISV model, such a solution is only valid if the model parameters fulfil the conditions and so that and , respectively, as the positivity of the SISV model requires. In summary, the normalized SISV model has potentially three equilibrium points assuming that , one of them is a DFE point (DFE1) and the others are EE points (EE1 and EE2); namely,The feasibility of such points requires that at them. In this context, the point DFE1 exists if the control gain is chosen such that so that at it. Moreover, such an existence is irrespective of the values chosen for the control gains and . Such facts are deduced from (6) by taking into account the fact that . On the other hand, the existence of the point EE1 requires that or, otherwise, that the controller gains and are chosen such that so that and then at such a point by taking into account (6). Finally, the existence of the point EE2 requires that or, otherwise, that the controller gains , , and are chosen such that so that and then at such a point by taking into account (6).
Now, assume that . The equation system (8) has three potential solutions. One of them with and the others with where , , denotes the potential solutions of with the coefficients and , depending on the controller gains, defined as follows.The substitution of in the second equation of (8) implies that so that the model has a potential DFE point, namely, the point DFE2 given byif . Otherwise, i.e., if , such an equilibrium point does not exist except in the particular case that simultaneously and .
In the general case that , the feasibility of the point DFE2 requires that , , and . Such conditions are simultaneously satisfied if the control gains and are chosen so that . This implies that the point DFE2 exists if (i) and , (ii) and , or (iii) while . Note that the point DFE2 is the same as DFE1 in the last case.
On the other hand, each one of the solutions of , if they exist, is feasible equilibrium point for the model if . In such a case, the model has one or two EE points given byThe feasibility of the point EE3 and, respectively, EE4, requires that , , and for and, respectively, , while the value of the control signal at such a point fulfils and, respectively, . The four conditions for the feasibility of the point EE3 and, respectively, EE4, are jointly fulfilled if and the controller gains , , and are chosen such that for and, respectively, , as it can be deduced by direct calculations. Note that there are not solutions for satisfying simultaneously the four conditions , , , and if the free-design control parameter is chosen such that . Then, the points EE3 and EE4 do not exist in such a situation. However, the normalized model has at least one EE point, namely, the point EE2, under such a choice for the parameter .
In the particular case that and simultaneously, the system of equations (8) becomes as follows.The second equation of (13) has two solutions, namely, for any and for any . By introducing such solutions in the first equation of (13), one obtains the following equilibrium points for the model.The feasibility of the equilibrium points defined by DFE3 requires that the values of the control signal at such points fulfil . Such a condition is fulfilled if by taking into account the fact that from the control law (6). Then the set of feasible points DFE3 is as follows.Note that the point DFE1 is included in the set of points DFE3. On the other hand, the feasibility of the point EE5 requires that , , and as well as that the value of the control signal at such a point fulfils . Such conditions are jointly fulfilled if the control gain is chosen such that while . The existence of a nonzero range for the values of so that the point EE5 is feasible requires that the model parameters satisfy while the parameter of the control law satisfies . Moreover, the point EE5 is the same as EE1 if and its feasibility is guaranteed irrespective if either or since at the point EE1 (or EE5) while , , and .
Table 1 summarizes the equilibrium points of the model and their feasibility conditions for the different assignments of the controller gains. The following theorem establishes the conditions that the control parameters have to satisfy for guaranteeing the inexistence of EE points in the normalized SISV model. In this way, the infectious disease can be eradicated from the host population.
Theorem 3 (conditions for inexistence of EE points in the normalized SISV model). The normalized SISV model (5) under the control law (6) does not have EE points if the parameter of the control law (3), or equivalently (6), satisfies the condition(c1) and the control gains , , and simultaneously satisfy(c2),(c3) or ,(c4)if then or ,(c5),(c6)at least one of the following conditions:(i) and ,(ii) and ,(iii),(iv), whereAs a consequence, only the points DFE1, DFE2, and DFE3 are feasible under such conditions.
Proof. The results summarized in the feasibility conditions column of Table 1 are used for this proof. In this way, the points EE1 and EE5 do not exist under the conditions (c1), (c2), (c3), and (c4), where (c4) is only necessary if the model parameters are such that . Moreover, the conditions (c1) and (c5) avoid the existence of the point EE2. Finally, the points EE3 and EE4 do not exist if at least one of the conditions of (c6) is satisfied. Concretely, any of such conditions imply that as it is proved in the following way. The proportions of infectious subpopulation at the potential equilibrium points EE3 and EE4 are the solutions of the equation with and defined in (10). Note that the function , for any given values of , , and , corresponds to a parabola which is opening to the top and its intersecting points with the abscissas axis are the solutions of . In this context, direct calculations prove that such intersecting points are not within the domain under any of the conditions (c6). First, note thatwhere is the value of at which the parabola reaches its minimum value. Under the condition (i) of (c6) one obtains thatwhere the condition is necessary to guarantee and then the existence of values for such that is possible, as one can deduce by direct calculations. The result implies that is monotonically decreasing . Such a fact, together with , implies that since the parabola is opening to the top. Then, cannot have solutions within except under such a condition (i) of (c6) and that solution corresponds to a DFE point.
Under the condition (ii) of (c6) one obtains thatwhere the condition is necessary to guarantee and then the existence of values for such that is possible, as one can deduce by direct calculations. The result