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 [
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 [
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 [ 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 [ 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. Another difference with respect to [ A different with respect to [
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,
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 [
The paper is organized as follows. Section
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:
The function
The dynamics of the whole population depends on the net growth rate The propagation of the infectious disease is described by a SIS epidemic model composed by the two first equations of (
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 (
A vaccination based on a control signal defined by
The SISV epidemic model (
Assume that there exists a finite time instant
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
One obtains the following normalized SISV model:
The normalized SISV model (
Assume that
Now, assume that
In the general case that
On the other hand, each one of the solutions of
In the particular case that
Table
Equilibrium points and their feasibility conditions.
| | |
---|---|---|
DFE1 | | |
| ||
DFE2 | | |
| ||
DFE3 | | |
| ||
EE1 | | (i) |
| ||
EE2 | | (i) |
| ||
EE3 | | |
| ||
EE5 | | |
The normalized SISV model (
and the control gains if at least one of the following conditions:
where
The results summarized in the feasibility conditions column of Table
Under the condition (ii) of (c6) one obtains that
Under the condition (iii) of (c6) one obtains that
Finally, under the condition (iv) of (c6) one obtains that
In summary, there are not feasible solutions for EE points under the conditions established in the theorem and the result is proved.
The following theorem analyses the local stability of the DFE points of the normalized SISV model under the control law (
The point DFE1 is locally exponentially unstable whenever it exists, i.e., when The point DFE2 is locally exponentially stable if the control gains while The point DFE3 is locally unstable if
(i) The feasibility of the point DFE1 requires that
(ii) The feasibility of the point DFE2 requires that either
(iii) The feasibility of the point DFE3 requires that
The following results are derived from Theorems
(i) Assume that the free-design parameter
(ii) Note that some of the control gains can take negative values. Moreover, an appropriate choice of such values guarantees the existence of a unique equilibrium point, namely, the point DFE2 defined in (
(iii) The time evolution of the whole population is given by
Model (
Proportions of susceptible and infectious subpopulations in the SISV model without vaccination when the basic reproduction number of its corresponding normalized SISV model is
Figure
Susceptible, infectious, and whole population in the SISV model without vaccination when the basic reproduction number of its corresponding normalized SISV model is
Zoom of Figure
Another example with the same initial condition and the same values for the parameters considered in the previous example, except that
Infectious subpopulation in the SISV model without vaccination when the basic reproduction number of its corresponding normalized SISV model is
Susceptible subpopulation and whole population in the SISV model without vaccination when the basic reproduction number of its corresponding normalized SISV model is
Whole population in the SISV model without vaccination when the basic reproduction number of its corresponding normalized SISV model is
Model (
Normalized subpopulations of susceptible, infectious, and vaccinated in the SISV model with a vaccination strategy based on the control law (
Susceptible, infectious, vaccinated, and whole population in the SISV model with a vaccination strategy based on the control law (
Figure
Evolution of the vaccines applied during the vaccination campaign.
The transient behavior of the evolution of the infection from the beginning of the vaccination until its eradication, whenever the control parameters are appropriately chosen in order to achieve such an objective as it has been pointed out in Remark
Evolution of the infectious subpopulation for different values of
Figure
Evolution of the vaccinated subpopulation for different values of
The application of the vaccines to the susceptible subpopulation is distributed through time as Figure
Specifications of the vaccination campaign for the different values of
| | | | | |
---|---|---|---|---|---|
3000 | 589 | 763 | 46.7 | 89 | 6 |
4000 | 582 | 780 | 60 | 68 | 6 |
6000 | 518 | 813 | 82 | 47 | 5.5 |
10000 | 450 | 874 | 119 | 31 | 5 |
Evolution of the vaccines applied during the vaccination campaign for different values of
The same values used in Section
Susceptible subpopulation in the SISV model with two different vaccination signals: (i) piecewise constant signal taking integer values and (ii) continuous-time signal.
Infectious subpopulation in the SISV model with two different vaccination signals: (i) piecewise constant signal taking integer values and (ii) continuous-time signal.
Vaccinated subpopulation in the SISV model with two different vaccination signals: (i) piecewise constant signal taking integer values and (ii) continuous-time signal.
Evolution of the vaccination signals: (i) piecewise constant signal taking integer values and (ii) continuous-time signal.
One can see that the differences in Figures
The implementation of a piecewise control strategy online can be done with a delay of one day by applying each day the number of vaccines corresponding to the average of the values of the control signal stored in the previous day. Figures
Susceptible subpopulation in the SISV model with two different vaccination signals: (i) a delayed piecewise constant signal taking integer values and (ii) continuous-time signal.
Infectious subpopulation in the SISV model with two different vaccination signals: (i) a delayed piecewise constant signal taking integer values and (ii) continuous-time signal.
Vaccinated subpopulation in the SISV model with two different vaccination signals: (i) a delayed piecewise constant signal taking integer values and (ii) continuous-time signal.
Evolution of the vaccination signals: (i) a delayed piecewise constant signal taking integer values and (ii) continuous-time signal.
The exact measure of the amount of individuals in each category of the model at each time instant is unviable in a real situation. This fact impedes the synthesis of the proposed vaccination control signal by means of (
A simulation example is carried out in this realistic situation, i.e., with a vaccination campaign supervised by the control law (
Susceptible subpopulations with (i) the vaccination strategy designed if there are uncertainties in the measures of the subpopulations and (ii) the basic vaccination strategy without uncertainties.
Infectious subpopulations with (i) the vaccination strategy designed if there are uncertainties in the measures of the subpopulations and (ii) the basic vaccination strategy without uncertainties.
Vaccinated subpopulations with (i) the vaccination strategy designed if there are uncertainties in the measures of the subpopulations and (ii) the basic vaccination strategy without uncertainties.
Applied vaccines with (i) the vaccination strategy designed if there are uncertainties in the measures of the subpopulations and (ii) the basic vaccination strategy without uncertainties.
One can see that the results obtained in the current example are quite similar to those obtained in that of Section
Subpopulations measures used to synthesize the vaccination control signal in comparison with the real time evolution of such subpopulations and whole population.
A vaccination strategy based on the state feedback control theory has been designed in order to eradicate the persistence of an infectious disease within a host population. Such a vaccination is applied in a SISV epidemic model. The positivity property of such a model is analysed as well as the existence and stability of the equilibrium points of their corresponding normalized model under such a vaccination strategy. The vaccination provides four free-design control parameters. Three of them are the gains to synthesize the state feedback control signal. The eradication of the infection is achieved by adjusting the control parameters to appropriate values such that there are not EE points in the normalized SISV model while a globally exponentially stable DFE is its unique equilibrium point. In such a case, the control gains multiplying to the susceptible and vaccinated subpopulations, respectively,
An extension of the current paper might be the design of other types of control signals, as, for instance, impulsive vaccination strategies adjusting the gain of the impulses and/or the interval between consecutive impulses. Another extension relies on the consideration of delays in the SISV model since, for instance, the effect of a vaccine in the vaccinated individuals cannot be immediate but it could be subject to a certain delay.
The underlying data to support this study are included within the references.
The authors declare that they have no conflicts of interest.
All the authors contribute equally to all the parts of the manuscript.
This research is supported by the Spanish Government through Grant DPI2015-64766-R and by UPV/EHU through Grant PGC 17/33.