Citation: Beth Ravit, Frank Gallagher, James Doolittle, Richard Shaw, Edwin Muñiz, Richard Alomar, Wolfram Hoefer, Joe Berg, Terry Doss. Urban wetlands: restoration or designed rehabilitation?[J]. AIMS Environmental Science, 2017, 4(3): 458-483. doi: 10.3934/environsci.2017.3.458
[1] | Bruno Buonomo . A simple analysis of vaccination strategies for rubella. Mathematical Biosciences and Engineering, 2011, 8(3): 677-687. doi: 10.3934/mbe.2011.8.677 |
[2] | Maria do Rosário de Pinho, Filipa Nunes Nogueira . On application of optimal control to SEIR normalized models: Pros and cons. Mathematical Biosciences and Engineering, 2017, 14(1): 111-126. doi: 10.3934/mbe.2017008 |
[3] | Holly Gaff, Elsa Schaefer . Optimal control applied to vaccination and treatment strategies for various epidemiological models. Mathematical Biosciences and Engineering, 2009, 6(3): 469-492. doi: 10.3934/mbe.2009.6.469 |
[4] | Islam A. Moneim, David Greenhalgh . Use Of A Periodic Vaccination Strategy To Control The Spread Of Epidemics With Seasonally Varying Contact Rate. Mathematical Biosciences and Engineering, 2005, 2(3): 591-611. doi: 10.3934/mbe.2005.2.591 |
[5] | M. H. A. Biswas, L. T. Paiva, MdR de Pinho . A SEIR model for control of infectious diseases with constraints. Mathematical Biosciences and Engineering, 2014, 11(4): 761-784. doi: 10.3934/mbe.2014.11.761 |
[6] | Manoj Kumar, Syed Abbas, Abdessamad Tridane . Optimal control and stability analysis of an age-structured SEIRV model with imperfect vaccination. Mathematical Biosciences and Engineering, 2023, 20(8): 14438-14463. doi: 10.3934/mbe.2023646 |
[7] | Amira Bouhali, Walid Ben Aribi, Slimane Ben Miled, Amira Kebir . Impact of immunity loss on the optimal vaccination strategy for an age-structured epidemiological model. Mathematical Biosciences and Engineering, 2024, 21(6): 6372-6392. doi: 10.3934/mbe.2024278 |
[8] | Siyu Liu, Mingwang Shen, Yingjie Bi . Global asymptotic behavior for mixed vaccination strategy in a delayed epidemic model with interim-immune. Mathematical Biosciences and Engineering, 2020, 17(4): 3601-3617. doi: 10.3934/mbe.2020203 |
[9] | Rama Seck, Diène Ngom, Benjamin Ivorra, Ángel M. Ramos . An optimal control model to design strategies for reducing the spread of the Ebola virus disease. Mathematical Biosciences and Engineering, 2022, 19(2): 1746-1774. doi: 10.3934/mbe.2022082 |
[10] | Hamed Karami, Pejman Sanaei, Alexandra Smirnova . Balancing mitigation strategies for viral outbreaks. Mathematical Biosciences and Engineering, 2024, 21(12): 7650-7687. doi: 10.3934/mbe.2024337 |
Infectious diseases have been feared by mankind for a long time, just remember the black death in the 14th century, killing about one third of the European population. Today humans can be vaccinated against many diseases, but still not against all of them, as can be seen from the recent outbreak of the ebola virus disease. Hence, it is hardly surprising that mathematical modelling of spreading of diseases has found much interest in the relevant literature. In this paper, we restrict ourselves to the textbooks of [15], [16], [1], and [4]. More references can be found in Biswas, Paiva and de Pinho [2], on which the present work is mainly based and the results of which we are going to extend.
As in [2], we consider a population of (
Exponential population growth is realistic for a quite young or fast increasing population, but in a highly developed country the population growth is known to stagnate more and more. In this case, the dynamical behaviour may be better represented by logistic population growth. The numerical results of this paper will show that certain unrealistic growth as shown by the results of [2] can be avoided.
The basic model is assumed to have a continuous flow between the compartments according to proportionality laws as indicated in Fig. 1. In the following model the population growth is, in the first instance, modelled according to an exponential growth law, where the total population is assumed to satisfy
˙N(t)=(b−d)⋅N(t) | (1) |
while the complete SEIR model is given by
˙S(t)=bN(t)−dS(t)−cS(t)I(t)−u(t)S(t), | (2) |
˙E(t)=cS(t)I(t)−(e+d)E(t), | (3) |
˙I(t)=eE(t)−(g+a+d)I(t), | (4) |
˙R(t)=gI(t)−dR(t)+u(t)S(t), | (5) |
˙N(t)=(b−d)N(t)−aI(t) | (6) |
with proportionality parameters given in the upper block of Table 1; cp. [2], p. 11. See also Fig. 1. In this system of ordinary differential equations the variable
Parameters | Definitions | Units | Values |
natural birth rate | unit of time | 0.525 | |
| natural death rate | unit of time | 0.5 |
| incidence coefficient | | 0.001 |
| exposed to infectious rate | unit of time | 0.5 |
| natural recovery rate | unit of time | 0.1 |
| disease induced death rate | unit of time | 0.1 |
| maximum vaccination rate | unit of time | 1 |
| maximum available vaccines | unit of capita | various |
| upper bound in Eq. 34 | | various |
| upper bound in Eq. 38 | unit of capita | various |
| weight parameter | | 0.1 |
| weight parameter | unit of money | 1 |
| initial time | unit of time (years) | 0 |
| final time | unit of time (years) | 20 |
| initial susceptible population | unit of capita | 1000 |
| initial exposed population | unit of capita | 100 |
| initial infected population | unit of capita | 50 |
| initial recovered population | unit of capita | 15 |
| initial total population | unit of capita | 1165 |
| initial vaccinated population | unit of capita | 0 |
1Note that the birth and death rates are far from reality if human populations are considered.
A continuous flow between the compartments may not be realistic, but may be accepted for qualitative conclusions for a sufficiently large population. To indicate this, we introduce unit of capita for the real valued variables
The above model is exactly the one investigated in [2] and traces back to [17]. Our paper follows the organization of [2] in order to facilitate a comparison of the results between exponential and logistic growth. We firstly present the new SEIR model with logistic growth. Then a short excursus shows how analytical control laws can be obtained from Pontryagin's minimum principle. These laws are used to validate our numerical results at least approximately. Finally, numerical results for six optimal control scenarios are obtained and discussed including those with a mixed control state constraint, also called state dependend control constraint or zeroth-order state constraint, as well as a pure state constraint which turns out to be of first order, first each constraint of these constraints separately then both in combination.
In addition to [2] we investigate an optimal control problem with a discounted objective function, too.
The Belgian mathematician Verhulst published in 1838 a logistic growth law [23], which sets the growth rate
˙N(t)=rK⋅N(t)⋅[K−N(t)] | (7) |
with
Comparing exponential and logistic population growth, i.e.,
˙Nlog(t)=r⋅N(t)−rKN(t)⋅N(t),˙Nexp(t)=b⋅N(t)−d⋅N(t), |
we see the following equivalences:
b↔randd↔rK⋅N(t). | (8) |
The birth term must only appear in the susceptible population, since every one is susceptible of birth and can be infected. However, the death terms must be split up over the compartments, for exponential growth by
d⋅N(t)=d⋅[S(t)+E(t)+I(t)+R(t)], | (9) |
and for logistic growth by
rKN(t)⋅N(t)=rKN(t)[S(t)+E(t)+I(t)+R(t)]. | (10) |
Note that the additional parameters for the logistic growth model have been chosen according to the equivalences (8) despite the unsoundness of the birth and death rates.2
2The unrealistic data for the birth and death rates transfer here consequently to r and K and lead to the pronounced logistic behaviour with a considerable increase of the entire population over the chosen time interval; see Figs. 2 and 3: a price to be paid for a plain comparability between the two growth models.
Hence, the original SEIR model turns into the new model
˙S(t)=rN(t)−rKN(t)S(t)−cS(t)I(t)−u(t)S(t), | (11) |
˙E(t)=cS(t)I(t)−[e+rKN(t)]E(t), | (12) |
˙I(t)=eE(t)−[g+a+rKN(t)]I(t), | (13) |
˙R(t)=gI(t)−rKN(t)R(t)+u(t)S(t) | (14) |
with the algebraic equation
N(t)=S(t)+E(t)+I(t)+R(t). | (15) |
To complete the system of ordinary differential algebraic equations we choose appropriate (consistent) initial conditions
S(0)=S0,E(0)=E0,I(0)=I0 and R(0)=R0; | (16) |
see the lower block in Table 1.
Remark 1. In [14], a slightly modified exponential-growth model is investigated, where the term
In the Kermack-McKendrick model, the force of infection, i. e. the the probability per unit time for a susceptible to become infected, is assumed to be proportional to
The Kermack-McKendrick model may be adequate, if huge crowds (demonstrations, open air concerts, sport events, etc.) are infected by so-called super-spreaders. An example is the SARS infection of sixteen guests on the nineth floor of the Metropole Hotel in Kowloon, Hong Kong, by one person on February 21,2003. Because of the subsequent continuation of the infected guests' journeys, the infection has spread out worldwide.3 This behaviour of the spread of a disease can be compared to the motion of gas molecules in random walk.
3See Super-spreader, http://en.wikipedia.org/wiki/Super-spreader, Feb. 28,2017. Other pandemic outbreaks of certain viral diseases are depicted on this web page, too.
The term
For the sake of comparison of the two different growth models, we therefore refrain from following the modified model of [14], moreover, since this paper deals with
The dynamics of the flow between the compartments are given by the above four ordinary differential equations for the state variables
The cost of an epidemic for an economy is assumed to consist of the costs for vaccination and for medical treatment of those who are infected. The latter term may also include the loss of benefit of an economy due to sick individuals. The vaccination costs usually amount to only a fraction of the costs for curing patients. Since this turns out to be a multi-objective performance index, we have to obey the scaling of these quantities when scalarizing the multi-objective functional; see below.
If a region with only few physicians is considered, every doctor has to medicate quite a large area and the costs may therefore depend quadratically on the vaccination rate.4 So the cost functional can be modelled by
4Instead of this so-called
T∫t0A1⋅I(t)+A2u(t)2dt; | (17) |
here we investigate a period of 20 years, thus we set
Remark 2. Therefore, the term quadratic in the control plays more the role of a regularization term, to guarantee the existence of unique optimal solutions and to get solutions of higher regularity, which approximate the
Another senceful term for including in the Pareto functional (17) is
Obviously, a box constraint to the vaccination rate must be imposed for practical reasons, say
u∈Uad:={u:[t0,T]→[0,umax] a.e.} | (18) |
which defines the set
Furthermore, it may be interesting how many vaccines have to be used over the considered period of time. Therefore, we introduce an additional state variable
˙W(t)=u(t)⋅S(t), W(0)=0, and W(T)≤WM. | (19) |
We will consider values of
To complete the optimal control problem we have to mention that all other state variables are unspecified at terminal time (except with respect to a trivial non-negativity condition,
(S(T),E(T),I(T),R(T))∈R4≥0. | (20) |
In summary, the optimal control problem reads as follows:
Minimize
T∫0A1⋅I(t)+A2⋅u(t)2dt | (21) |
subject to
˙S(t)=rN(t)−[rKN(t)+cI(t)+u(t)]S(t), | (22) |
˙E(t)=cS(t)I(t)−[e+rKN(t)]E(t), | (23) |
˙I(t)=eE(t)−[g+a+rKN(t)]I(t), | (24) |
˙R(t)=gI(t)−rKN(t)R(t)+u(t)S(t), | (25) |
˙N(t)=rKN(t)[K−N(t)]−aI(t), | (26) |
˙W(t)=u(t)S(t), | (27) |
u(t)∈[0,umax] a.e. | (28) |
Note that the state variable
Remark 3. Note that the cancellation of one of the differential equations stabilizes the numerical computations. Otherwise constraint qualifications would be violated when optimizing.
In the following, we will firstly investigate five optimal control scenarios differing with respect to certain inequality constraints that are additionally imposed. These optimal control problems are solved numerically by a standard first-discretize-then-optimize (direct) method based on the modelling language AMPL of [5], providing exact derivatives via automatic differentiation, and the well-known interior point algorithm for solving large-scale nonlinear programming problems of [24], resp. [25] called IPOPT. All computations were performed using the implicit Euler method with a constant discretization stepsize of
For the first four of the five following scenarios, the existence of optimal solutions can be guaranteed, as for the SEIR model with exponential growth in [2]; see Theorem 2.1 there and its application to that SEIR model. This is due to the facts, that we, too, have a Lagrangian functional with an integrand quadratically dependent on the control and a constrained set of admissible controls as well as dynamics affine in the control although nonlinear in the state variables.
In this paper we will additionally verify the discrete optimal control values approximately via an analytically determined optimal control law obtained from Pontryagin's maximum, resp.minimum principle5, in which we substitute the discrete state and costate variables (multipliers). This consistency condition shows how accurate the discrete solution of the finite dimensional optimization problem fulfills the optimality condition of the maximum principle. For this purpose, we compute an a posteriori error estimate defined by the mean square error
5For remarks on the interesting history of the maximum principle against the background of the beginning Cold War, see [20].
For recipes on how to apply of the necessary conditions of optimal control theory to real-life applications, see [18].
A limitation of the entire amount of vaccines used can be modelled by the aforementioned terminal constraint,
0≤W(T)≤WM. | (29) |
If there is no serum available, the constraint (29) causes
The overall costs are ca. 3 173.8 units, which will be the reference point for making up the balance of the gain of optimization.
All subsequent scenarios will now be optimal control problems. For this purpose we make a short aside how to apply the minimum principle.
We firstly have to define
Definition 4.1 (The Hamiltonian). Let
H(x,u,λ)=λ0⋅f0(x,u)+λ⊤⋅f(x,u) |
with
In general
The core of the minimum principle says that for all
u∗=argminu∈UadH(x∗,u,λ). | (30) |
For a precise and complete formulation of the minimum principle in the most general form see [7], Informal Theorem 4.1 and Theorem 4.2.
This control law will be used to verify the discrete numerical results
6We want to point out that the discrete multipliers provided by IPOPT as by-product may have the opposite sign ---depending whether a maximum or a minimum principle is applied ---and may need to be scaled appropriately. For this, one can exploit the homogenity of the Lagrange multipliers.
Since the Hamiltonian must be constant for autonomous problems, this property can be used to check whether the implementation of the ODEs is correct.
We now apply the core of the minimum principle and its associated verification to five different optimal control problems differing in the kind and number of additional inequality constraints.
The normal case for, let's say, a vaccination against flu is that there are sufficiently many vaccines available, i. e.,
W(T)<WM:=∞ |
or in other words there is no terminal inequality constraint for the state variable
Taking into account the above mentioned redundancy, the Hamiltonian function can be defined for example by
H(S,E,I,R,W,u,λ)=A1I(t)+A2u(t)2++λS˙S(t)+λE˙E(t)+λI˙I(t)+λR˙R(t)+λW˙W(t). | (31) |
In case the minimizer of the here convex optimization problem lies in the interior of the admissible set of control values, it is uniquely given by
u∗(t):=uint(t)=S(t)⋅[λS−λR−λW]2. | (32) |
If
u∗(t)=min{umax;max{0;uint(t)}}. | (33) |
The progress of the population, i. e., the computed candidate optimal trajectory for Scenario 1 is shown in Fig. 3.
The verification as described above yields a perfect match as also indicated by the mean square error of
The cost to the economy turns out to be
7For techniques to deal with infinite horizon optimal control problems we refer to [8], [12], [11], and [26].
Using the optimal control approximately
This scenario has actually happened in
W(T)<WM:=2500. |
The optimal control law stays the same as in the unlimited case, but of course different state and costate values have to be inserted.
The most common solution may be: vaccination stays at the maximum limit until all vaccines are exhausted. This optimal control strategy produces costs to the economy amounting to
Here, the optimal control is "almost bang-bang" i. e. has a sharp decline from the upper to the lower boundary of the admissible set; see Fig. 5. Note that due to the
Bang-bang and even singular subarcs can only appear if an
In summary, it is ---not surprisingly ---cheaper to vaccinate the population than not immunizing them. In fact, a limit of
In order to limit the amount of vaccines at each instant of time we introduce a mixed control-state constraint,
0≤u(t)⋅S(t)≤V0 a.e., | (34) |
where
0≤u(t)≤V0S(t) a.e. | (35) |
We have chosen
Remark 4. It is known that constraints of type (34) can be replaced by an additional term in the scalarized Pareto functional (42) as mentioned in Remark 2. Herewith the multiplier
Due to the mixed control-state constraints we have to augment the Hamiltonian as follows. Since the system is overdetermined, as mentioned above, it is sufficient to consider only four of the five ODE constraints (11) --(14) in addition to (7). Now, we choose (7), (11) --(13) besides the inequality constraint (34) for the optimal control problem describing senario 3,
H(S,E,I,N,u,λ,μ)=A1I(t)+A2u(t)2+λS˙S(t)++λE˙E(t)+λI˙I(t)+λN˙N(t)+μ[V0−S(t)u(t)]. |
Here, a new multiplier
ubound(t)=V0/S(t)>0. | (36) |
Due to the box constraint (18), the complete control law for scenario 3 can therefore be summarized by
u∗(t)=min{umax;ubound;max{0;uint}}. | (37) |
In Figs. 7 the switching time between the control according to (32) and (36) is marked, showing that control values in the interior of the admissible set are optimal, iff these values are less than those obtained by the competing values according to (36).
Overall, there is quite a low final amount of infected people,
In order to calm down an epidemic or pandemia it may be advantageous to limit the number of susceptibles or infected individuals by isolation independent of costs or the influence on the other compartments.
A limitation of the number of susceptibles is acchieved by the following pointwise state constraint,
S(t)≤Smax | (38) |
with an appropriately chosen parameter
Following the guidlines in [18] we firstly examine the order of the state constraint and derive the candidate optimal boundary control on boundary arcs. If the state constraint (38) is active on a non-vanishing time intervall, i. e., if there holds the identity in time,
S(t)≡Smax, | (39) |
a further differentiation of (39) with respect to time, while substituting the right hand sides of the ODE system (11)--(14), reveals the boundary control and therefore that the state constraint is of first order,
ubound1(t)=rN(t)Smax−rKN(t)+cI(t). | (40) |
From optimal control theory, we know that the associated costate
Moreover, since the state constraint is of first order, effective touch points cannot occur; see [9] and [6].
Following the indirect adjoining approach of Bryson, Denham and Dreyfus [3] ---see also Ref. [18] ---we augment the Hamiltonian similar to the case of mixed control-state constraints by a term
8Here we would like to point out that, besides the core of the minimum principle, i. e. Eq. 37, the other necessary conditions associated with the minimum principle can be checked, too, such as the sign of the multipliers associated with path constraints and the signs of the jumps occuring at the junction points of boundary arcs in case of pure state constraints; see [7], Theorem 4.1 and 4.2. See also [10].
Recapitulating, we obtain the same optimal control law (37) as for the mixed control-state constraint; only replace
9Here we have rounded the output along state-constrained boundary arcs by maximal
Numerically we have found that below a value of
Increasing the fineness of the discretization we see that an extremely precise resolution of the switching structure can be obtained even by a method of type First discretize then optimize as used here.
Remark 5. This accuracy could be even increased when we pass on to a switching point optimization technique; see [13]. For this postprocessing step one has to introduce the switching points as new optimization variables. These can be guessed ---in our example surely beyond reasonable doubt ---from the First-discretize-then-optimize approximations. For this, the control variable needs only to be discretized on subintervals where the control law depends on adjoint variables. With other words, in many subinterval the control values can be prescribed either by constants, i. e. the maximal, resp. minimal values allowed by the admissble control set
Herewith, an accuracy can be obtained which is surely beyond the accuracy of the model, but gives almost as much inside as using a First-optimize-then-discretize method based on the full minimum principle and using a multipoint boundary value solver for the resulting multi-point boundary value problem with jump conditions.10
10This approach pays off only if an extremely accurate solution is desired enabling the check of additional necessary conditions. However, there is a tremendous price to be paid, since the user must have sufficient experience to implement all the necessary conditions and to handle the usually sensitive boundary-value-problem solvers such as, for example, multiple shooting codes; see [18].
We will not pursue these ideas furtheron. They would be inappropriate for the more qualitative models discussed here.
The costs for scenario 4 add up to
For our last scenario we impose both constraints (35) and (38),
S(t)≤Smaxand0≤S(t)u(t)≤V0 a.e. | (41) |
After some numerical experiments we have chosen the parameters
V0=400andSmax=1700. |
This choice results in a small time interval, approximately
The numerical results for the susceptibles and infected are shown in Fig. 11. The mixed control-state constraint (35) is active over almost the entire time interval, only interrupted by a tiny state-constrained boundary arc associated with (38).
The overall costs amount to
In economical applications optimal control problems often pertain to functionals that are of discounted type,
T∫t0exp(−rt)(A1⋅I(t)+A2u(t)2)dt, | (42) |
with
Remark 6. The valuation method of a discounted cash flow is used to estimate the attractiveness of an investment. Discounted cash flow analysis takes into account future cash flow projections and discounts them to a value estimate at present time. This is used to evaluate the potential for an investment. If the value obtained through discounted cash flow analysis is higher than the current cost of the investment, the investment is considered to be more advantageous.
Concerning realistic models for the spread of vaccinable diseases discounted functionals may be more appropriate. Usually functionals of type (42) arise from infinite horizon optimal control problems which are naively approximated by truncation of the infinite horizon to a finite time interval. Lykina [11] and Lykina, Pickenhain and Wagner[12] have shown that this truncation is an improper modelling since one often cannot assure the existence of solutions. This is caused by the fact that improper spaces are chosen in which the solutions are assumed to exist. To overcome this hurdle Pickenhain [19] has suggested to use a weighted Sobolev space for the state and a weighted Lebesgue space for the control variables. Moreover, these authors have developed various techniques for solving such problems correctly; see particularly [11]. Unfortunately, there is no single method of choice; it depends on the particular problem.
Let us firstly consider the approximate solution for the discounted functional with time horizon
There one can hardly see any difference to Figs. 3 and 4, even not for higher (more or less realistic) values of
In the present paper a modified SEIR model for infectious diseases is developed which includes logistic growth. It is more appropriate for developed societies. The results of our paper complement the results of the twin paper by Biswas, Paiva and de Pinho [2], where exponential growth is taken into account. In particular several inequality constraints have been included, among others mixed control-state and pure state constraints. Although the numerical results have been obtained only by a first-discretize, then-optimize method ---however on the basis of an existence result ---, optimality is validated by use of Pontryagin's minimum principle. This verification technique shows that the results obtained are close approximations of at least candidate optimal solutions.
As conclusion concerning the numerical method used here, the investigations show that the easy-to-use direct method, particularly the combination of the modelling language AMPL providing the user with exact gradients via automatic differentiation, and the high performance large-scale NLP solver IPOPT, yields numerical results which even allow the determination of the switching structure of the problems if the switching points are not too dense. However, even in these cases a switching point optimization, if meaningful in view of the model accuracy, may yield further improvements; see [13]. This approach may also be used to improve the validation technique towards approximate sufficiency conditions for local optimality; see [13], too.
Concerning the interpretation of the results in view of the application we have seen that there is a correlation between low costs to the economy and a high vaccination rate, which is, of course, to be expected by common sense. What contradicts common sense is not to vaccine at maximal rate in the case of unlimited serums, but the optimal results are quite close to those of common practise with a
Finally it should be mentioned that results have been obtained also for functionals of
Further investigations should focus on the mentioned alternative contact term for time varying populations. However, then the incidence rate
Finally, infinite horizont models could also be worth of future research, since our numerical results, particularly for the
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Parameters | Definitions | Units | Values |
natural birth rate | unit of time | 0.525 | |
| natural death rate | unit of time | 0.5 |
| incidence coefficient | | 0.001 |
| exposed to infectious rate | unit of time | 0.5 |
| natural recovery rate | unit of time | 0.1 |
| disease induced death rate | unit of time | 0.1 |
| maximum vaccination rate | unit of time | 1 |
| maximum available vaccines | unit of capita | various |
| upper bound in Eq. 34 | | various |
| upper bound in Eq. 38 | unit of capita | various |
| weight parameter | | 0.1 |
| weight parameter | unit of money | 1 |
| initial time | unit of time (years) | 0 |
| final time | unit of time (years) | 20 |
| initial susceptible population | unit of capita | 1000 |
| initial exposed population | unit of capita | 100 |
| initial infected population | unit of capita | 50 |
| initial recovered population | unit of capita | 15 |
| initial total population | unit of capita | 1165 |
| initial vaccinated population | unit of capita | 0 |
Parameters | Definitions | Units | Values |
natural birth rate | unit of time | 0.525 | |
| natural death rate | unit of time | 0.5 |
| incidence coefficient | | 0.001 |
| exposed to infectious rate | unit of time | 0.5 |
| natural recovery rate | unit of time | 0.1 |
| disease induced death rate | unit of time | 0.1 |
| maximum vaccination rate | unit of time | 1 |
| maximum available vaccines | unit of capita | various |
| upper bound in Eq. 34 | | various |
| upper bound in Eq. 38 | unit of capita | various |
| weight parameter | | 0.1 |
| weight parameter | unit of money | 1 |
| initial time | unit of time (years) | 0 |
| final time | unit of time (years) | 20 |
| initial susceptible population | unit of capita | 1000 |
| initial exposed population | unit of capita | 100 |
| initial infected population | unit of capita | 50 |
| initial recovered population | unit of capita | 15 |
| initial total population | unit of capita | 1165 |
| initial vaccinated population | unit of capita | 0 |