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Basic Vaccination Control Techniques for a General True -Mass Action SEIR Model

This paper presents several simple linear vaccination-based control strategies for a SEIR (susceptible plus infected plus infectious plus removed populations) propagation disease model. The model takes into account the total population amounts as a refrain for the illness transmission since its increase makes more difficult contacts among susceptible and infected. The vaccination control objective is the asymptotically tracking of the removed-by-immunity population to the total population while achieving simultaneously that the remaining populations (i.e. susceptible plus infected plus infectious) tend asymptotically to zero.


INTRODUCTION
Important control problems nowadays related to Life Sciences are the control of ecological models like, for instance, those of population evolution (Beverton-Holt model, Hassell model, Ricker model etc.) via the online adjustment of the species environment carrying capacity, that of the population growth or that of the regulated harvesting quota [1][2][3][4][5], as well as those of disease propagation via vaccination control. Several variants and generalizations of the Beverton-Holt model (standard timeinvariant, time-varying parameterized, generalized model or modified generalized model) have been investigated at the levels of stability, cycle-oscillatory behaviour, permanence and control through the manipulation of the carrying capacity, [1,4]. The design of related control actions has been proved to be relevant at the levels, for instance, of aquaculture exploitation or plague fighting. At the same time, properties of the discrete Beverton-Holt equation solutions when the species environment carrying capacity and/or the intrinsic growth rate of the species are periodic functions of time have been studied, [5]. Such a particular case is of interest since periodic fluctuations are quite common in Biology and Ecology. For instance, the habitat resource availability, temperature, humidity and so on, which influence the environment carrying capacity, experiment fluctuation during the year. In the same way, the birth and/or the survivorship rates of the species may suffer fluctuations along the year, so a periodical intrinsic growth rate is an appropriate choice in such ecology models. Furthermore, this kind of difference equations for modelling real life situations in population biology, ecology, economics and so on from a control theory point of view has been recently dealt with in a generalized way in [6]. The boundedness properties of the positive solutions of this type of nonlinear difference equations have been established in such a paper.
On the other hand, the literature about epidemic mathematical models is exhaustive in many books and papers (see, for instance, [7][8][9][10][11][12][13][14][15][16]). The sets of models include the most basic ones: • SI-models, where only susceptible and infected populations are assumed, • SIR models, which include susceptible plus infected plus removed-by-immunity populations, and • SEIR-models, where the infected populations is 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 have the external disease symptoms).
Those models have also two 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 and the socalled "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, for instance, including vaccination of different kinds: constant [9], impulsive [13], discrete-time etc., incorporating point or distributed delays [13,14], oscillatory behaviors [15] and so on. On the other hand, variants of such models become considerably simpler for the illness transmission among plants [7,8].
In this paper, two continuous-time vaccination control strategies are given for a SEIR epidemic model. One of them takes directly the susceptible population to design the vaccination strategy while the second one uses the total and removed-by-immunity populations for such a purpose. It is assumed that the total population remains constant through time, so that the illness transmission is not critical, and the SEIR-model is of the above mentioned true-mass action type. The positivity of the mathematical model, which reflects the real problem at hand, is proved. Such a property is crucial to guarantee the boundedness for all time of all the partial populations.

Let
be the "susceptible" population of infection, the "infected" population, the "infectious" population and the "removed-by-immunity" (or "immune") population at time .
Consider the true-mass action type SEIR epidemic model: (5) irrespective of the vaccination strategy.
Proof: It follows immediately by summing-up both sides of (1) to (4) what leads to: (6) so that: (7) *** Remark 1. The ideal vaccination mechanism objective is to reduce to zero the numbers of susceptible, infected and infectious independent of their initial numbers so that the total population becomes equal to the removed-by-immunity population after a certain time. After inspecting (1) and (4), it becomes obvious that the constraint is necessary to decrease the time variation of the susceptible and to increase simultaneously that of the removed by immunity for all time. However, Assertion 1 proves that the constant population through time is independent of the vaccination strategy so that it is independent of the ideal vaccination objective constraint as a result. For instance, in a biological war, the objective would be to increase the numbers of the infected plus the infectious population for all time. For that purpose, the appropriate vaccination strategy would be negative. *** model is unreachable. It is always trivially uncontrollable to the origin for arbitrary initial conditions for the total population. ***

About the positivity of the SEIR epidemic model
The vaccination strategy has to be implemented so that the SEIR model be positive in the usual sense that none of the populations, namely, susceptible, infected, infectious and immune be negative at any time. This requirement follows directly from the nature of the problem at hand. The following assumption is made: for any given real constant . Such a constraint is guaranteed with if for all . It is possible to rewrite (8)-(10) in a compact form as a dynamic system of state , output and whose input is appropriately related to the vaccination function as . This leads to the following set of identities: (11) where is the i-th unit Euclidean column vector with its i-th component being equal to one and the other two components being zero, and (12) Note that: is a Metzler matrix [17] for any given .
(iii) From Assumption 1 (see also Then, one has that, for any admissible initial condition , the unique solution on of (11) is: since and (14) due to, on one hand: (15) and, on the other hand, from (2): Since and , it exists such that , and (so that ) . The above properties extend to from the structures of (13)- (16). Furthermore, and . These relations also imply from (5) that . *** Remark 3. Note that the mathematical SEIR-model is not guaranteed to be positive according to Theorem 1 in the sense of [17] since Assumption 1 establishes conditions on the initial conditions. ***

Corollary 1. Theorem 1 still holds if .
Proof: It follows from the proof of Theorem 1 since from (15) under this modified vaccination constraint. ***

Vaccination-free case equilibrium points and stability
The equilibrium points of (8)-(10) under identically zero vaccination strategy satisfy the set of constraints: (19), it follows that: By introducing (20) in (18), one obtains that: Note that the condition guarantees both , , and as the real system requires. Finally, note that in the case that both feasible equilibrium points degenerate in a unique one since is derived.
The following result concerning with the local stability of the vaccination-free SEIR mathematical model around its equilibrium points is proven: (i) The vaccination-free SEIR model (8) Proof: The linearized model (8)-(10) about its equilibrium points is: Then: (i) At the equilibrium point , the linearized system (27) becomes: The characteristic equation of (28) becomes: The characteristic zeros are and . As a result, the equilibrium point of (27)  to the complex half-plane , [19].
Finally, note that the global Lyapunov stability is automatically guaranteed for the SEIR model (1)-(4) since the total population is assumed to be constant for all time. *** Remark 4. The equilibrium points in the vaccination-free case are not suitable since one of them is concerned with the whole population being susceptible while the other one is concerned with not all the population being asymptotically converging to the removed-by-immunity, in general. Therefore, a suitable vaccination strategy is necessary to avoid the persistence of the disease in the population.
Also, note that if then the equilibrium point does not exists, i.e., is the unique equilibrium point and it is stable. Otherwise, i.e., if then both equilibrium points exist and is an unstable attractor while the stability around depends on the fulfillment of the condition in Theorem 2 (ii). ***

VACCINATION STRATEGY
A control strategy may be defined in several ways involving the vaccination function which is really the manipulated variable. The control goal is decreasing appropriately the numbers of susceptible, infected and infectious while increasing the removed-by-immunity population. In this sense, two alternative vaccination strategies are proven to be appropriate from a health point of view.
The following result is concerned with this matter.
, and , (iii) the whole population becomes asymptotically removed-by-immunity at an exponential rate, and (iv) there exist values for the control parameter guaranteeing that .

Proof:
(i) Rewrite (1) in the equivalent form: with an auxiliary control being defined as follows: through the vaccination function given by (34). One gets from (35) and (36) that, since and then is bounded . From (2)- (3) and (37), it follows that: where . Note that the eigenvalues of are strictly negative, namely, and . From (38), one gets that: where . By direct calculations, it follows that: where the condition for the control parameter in (33) has been taking into account. From the fact that the right hand side of (40) is a monotonically increasing function, it follows that: provided that satisfies the condition in (33 (iii) From (2) and (37), it follows that: • denotes a vector with at least one component being negative.
(ii) Note that if the control parameter is such that the same results of Theorem 3 are satisfied irrespective of the initial condition for the susceptible population. *** An alternative vaccination strategy is dealt with in below:

Theorem 4 (Control law 2).
Assume that the vaccination strategy is as follows: irrespective of the initial conditions. In particular, if then and , i.e., the removed-by-immunity population equalizes asymptotically the total population at exponential rate while the sum of the infected, infectious and susceptible populations converges asymptotically to zero at exponential decay rate.
(iii) If exists then and .
(iv) In the case that and provided that , the vaccination function is nonnegative for all time if or if and .

Proof:
(i) Suppose that does not exist (or ), then provided that from the definition of . By introducing the vaccination control law (47) in (4) one obtains: Combining the two relations and using Assertion 1 one has for all time: (56) Then, the proof is derived by combining (55)  i.e, the infectious disease would not be eradicated from the population. As a consequence, a control action could be applied in order to eradicate the disease.

Epidemic evolution with the vaccination control law 1
The vaccination strategy associated with the control law 1 defined by (33)-(34) is considered. The control parameter is applied. The time evolution of the respective populations (percentage respect the total population) is displayed in Figure 1. It can be seen that the model tends to a suitable equilibrium point as time goes to infinity since all the population becomes asymptotically removedby-immunity. i.e. the infection would be eradicated from the population with such a vaccination practice in a relative short time period, approximately 50 days. Figure 2 displays the time evolution of the vaccination effort (in terms of percentage with respect to the total population) to be applied to eradicate the disease. with the control law 1.

Epidemic evolution with the vaccination control law 2
A vaccination strategy based on the control law 2 defined by (47) with the parameters and is applied. The time evolution of the partial populations (percentage respect the total population) is displayed in Figure 3. Again a suitable equilibrium point is reached as time tends to infinity since all the population becomes asymptotically removed-by-immunity. As a result, the infection is eradicated from the population in 50 days, approximately. Figure 4 displays the time evolution of the vaccination effort (in terms of percentage with respect to the total population) to be applied to eradicate the disease. with the control law 2.

Design of a vaccination campaign
The time evolution of the vaccination effort may be used to prescribe a suitable vaccination campaign in the population. For such a purpose, note that all the population becomes removed-byimmunity at the equilibrium point with both vaccination laws (see figures 1 and 3) while 8.4% of the population did not reach such a status at the equilibrium point of the vaccination-free case, approximately. The later percentage determines the population ( habitants) to be vaccinated in order to eradicate the disease. In view of figures 2 and 4, the vaccination effort depends on time through a uniformly decreasing function. Then, the vaccination cadence will not be uniform during the disease evolution (approximately 50 days) but it presents a maximum along the first day and will be uniformly decreasing to a zero value at the 20 th day, approximately. In this sense, the vaccination cadence associated to any of the control laws may be given by: where the normalization factor is added to relate as recommendation the vaccination effort with the number of vaccines to be applied. For the purpose of calculating , the contribution of the vaccination effort to the removed-by-immunity population is given by from (4), so that:

Control parameters influence in the infection time evolution
In one hand, the disease propagation experiments a peak in the infectious population (2.5% of the total population) at the 12 th day with the application of the vaccination control law 1 as it can be seen from Figure 1. On the other hand, the use of the vaccination control law 2 makes the disease reaches a peak at the 15 th day when the 5.4% of the total population is infectious as it can be seen from

Example 2: An influenza infection
An example based on an outbreak of influenza in a British boarding school in early 1978, [8], is used. Such an epidemic can be described by the SEIR mathematical model (1)

Epidemic time evolution without vaccination
The time evolution of the respective populations is displayed in Figure 8

Epidemic time evolution with the vaccination control law 1
A feedback control law defined by (33)-(34) with the control parameter is applied. The time evolution of the respective populations is displayed in Figure 9 so in the case that as . Note that the use of such a control law makes the dynamics of the susceptible population be given by , i.e. its time evolution does not depend on the loss of immunity rate . Then, the time evolution of the infected and infectious population is also independent of the loss of immunity rate from (2) and (3). Finally, the removed-by-immunity population does not depend on such a parameter from Assertion 1. In summary, the time evolution of all partial populations is independent of due to the vaccination control rule (33)-(34). Furthermore, it can be seen that the model tends to a suitable equilibrium point as time goes to infinity since all population become removed-by-immunity. i.e. the infection would be eradicated from the population with such a vaccination practice in a relative short time period, approximately 49 days (7 weeks).  The time evolution of the vaccination effort may be used to prescribe a suitable vaccination campaign in the population. For such a purpose, note that the removed-by-immunity population is boys at the steady state day 49 with the application of such a vaccination law (see Figure 9) and it is boys, if , or boys, if , without a vaccination effort as it can be seen from figures 8(a) and 8(b), respectively. Then, the number of boys to be vaccinated is given by if or by if , provided that each boy receives at most one vaccine. In view of Figure 10, the vaccination cadence is not uniform during the disease evolution (7 weeks) but it presents a minimum during the second week and it reaches a maximum during the last four weeks. In this sense, the vaccination cadence may be given by: for if or if . Again, the normalization factor is added to relate as recommendation the vaccination effort with the number of vaccines to be applied. In this case, from Figure 10 and are obtained so that: for . Figure 11 displays the vaccination cadence per day, i.e. the number of vaccines to be applied each day, for both values of . However, a vaccination cadence per week may be recommended to be used in practice. Then, the vaccination cadence per day may be given by: where and . In this way, the vaccines to be applied each week are uniformly distributed in the days of such a week. Figure 12 displays the vaccination cadence if this method for the vaccines distribution is used.

Epidemic time evolution with the vaccination control law 2
A feedback control law defined by (47) with the control parameters and  Finally, Figure 14 displays the time evolution of the vaccination effort to be applied to eradicate the disease, for both values of . By using this result, a similar study to that carried out with the control law 1 could be done to establish an appropriated vaccination campaign through the population.