A pivotal restructuring of modelling the control of COVID-19 during and after massive vaccination for the next few years

This paper presents a new mathematical feedback model to demonstrate how direct observations of the epidemiological compartments of population could be mapped to inputs, such that the social spread of the disease is asymptotically subdued. Details of the stabilization and robustness are included. This is a pivotal restructuring of modelling the control of corona virus from the current models in use world-wide which do not utilize feedback of functions of epidemiological compartments of population to construct the inputs. Although several vaccines have received Emergency Use Authorization (EUA) massive vaccination would take several years to reach herd immunity in most countries. Furthermore, the period of efficacy of the vaccination may be approximately one year only resulting in an unending vaccination. Even during the vaccination, there would be an urgent need to control the spread of the virus. When herd immunity is reached without feedback control and vaccination is discontinued, there could be new surges of the disease. These surges of disease could be prevented in appropriately designed stable feedback models. Moreover, extensive testing, contact tracing, and medical treatment of those found infected, must be maintained.

political jurisdiction will decide on prioritization and optimization. For the rest of the global population, some pharmaceutical companies have announced that more vaccines will be available by the middle of 2021. These companies are still investigating the duration of efficacy of the vaccines. For the near term, there is a need to continue the interventions, or innovative ways to modify the interventions so they will become more effective. It is expected that massive vaccination will be available later this year and in the next two (2) years.
Is there further need for modelling the trend of COVID-19? Yes.
(a) If 100,000 persons can be vaccinated everyday, it will take two (2) years to vaccinate 80% of the Philippine population, assuming vaccinations that require only one shot.
(b) The duration of efficacy of the various vaccines is not known at this time. If the efficacy duration is one (1) year, massive vaccination is needed for the foreseeable future with no end in sight.
(c) During the vaccination period of several years, can the current models used worldwide be used to mitigate the growth of disease spread? After more than a year of COVID-19 pandemic, the Philippines and the entire world are still suffering from the raging disease. A new type of intervention is needed, a pivotal restructuring of modelling the control of corona virus.

Review of science models of disease spread
The first mathematical model to describe the spread of a disease was published in 1927 (Kermack and McKendrick 1927). The basic idea in the model is to divide the population into epidemiological compartments. In the Kermack and McKendrick model, the compartments of population are: Susceptible ( ), Infected ( ), and Removed ( ). The rates of change with respect to time of and , are modeled as functions of and , respectively. This model has been adopted for modeling the spread of COVID-19, with several layers of subsocieties, and each layer with more compartments beyond and , such as Exposed ( ), Hospitalized ( ), and further subdivision of into Symptomatic and Asymptomatic (Chen et al. 2020). Others have further analyzed the Wuhan outbreak (Kucharski et al. 2020, Ndairou et al. 2020, Saad-Roy et al. 2020, Britton et al. 2020. SARS-CoV-2 is known to mutate, and mutation of viruses during propagation has been studied (Eletreby et al. 2019). There are updates available for the progression of COVID-19 (World Health Organization 2020c) including a science model with the updates (Dong et al. 2020). The effect of virus mutation, on therapeutics for COVID-19, has been studied also (Hou et al. 2020), and it appears that the mutation D614G propagates faster than the original SARS-CoV-2. They state that "current vaccine approaches directed against the WT (wild-type) spike should be effective against the D614G strains." (Hou et al. 2020).
In the only paradigm in use today, interventions are deployed to change the environment, resulting in changing direction of observations, and subsequent revised mathematical models.
When the interventions are relaxed or removed altogether, subsequent observations will show that the disease spread will increase exponentially and the process repeats.
The main purpose of this paper is to provide a new engineered closed loop model for stabilizing systems such as a model for the social spread of disease. As in existing science models, the engineered closed loop model in this paper addresses the macro-scale level of modeling the spread of a disease, not necessarily COVID-19. The simplest science model is considered to modify into an engineered closed loop feedback system to focus on the advantages of closed loop models.

Review of SIR science model
The starting point in the development of an engineered closed loop model with feedback of observation in Section 3 is the Kermack and McKendrick model (1927). Figure 1 where is the infectivity rate and is the recovery rate.
The rate of change of with respect to time follows from the constraint, (4)

Review of the SIFD science model
In the SIR model, the compartment is divided into two sub-compartments, and , where is the number of persons that are Free of Virus but possessing antibodies, and is the number of persons that are Deceased. As in the SIR model, and , respectively, are normalized by dividing each by , yielding (5) Figure 1 still represents the science model. The rate equations for and , respectively, are the proposed behavioral postulates as in (Kermack and McKendrick 1927) (6) .
The rate equation for is obtained from the constraint: yielding Equation (4).
Observations are used to calibrate the parameters and . Analysis of the mathematical models leads to obtaining conditions that cause exponential growth, suggesting how instability can be avoided. A typical intervention in science models is to change the environment, such as modification in the use of (a) quarantine or lock down, (b) face masks, constructed. Thus by increasing interventions the resulting system could become stable, but removing the intervention would eventually make the system unstable.

Pivoting from science model to Batangas State University engineered feedback model
There are two steps in the transition. The first step is to create explicit inputs to the science model. The second step is to construct a mapping from the outputs to the inputs of the science model.

Creating the Batangas State University input-output model
The first step in pivoting from a science model to an engineered feedback model is to explicitly create inputs to the science model as indicated in Figure 2. In the science model, the environment could be regarded as input but it is not treated as an external signal. There are intrinsic benefits to creating models with inputs and outputs (Tan et al. 2018). In this paper we create two inputs to the SIFD science model.
As in the science model without inputs, the rate equations, for and , are postulated. The rate equation for is derived from the constraint in Equation (9) where the quantities and are inputs. (10) and Equation (8). Equation (10) is a modification of the Equation (6) and it is one of the equations for the inputoutput model. Equation (12) is a modification of Equation (7) and it is a second equation for the inputoutput model. Equation (8) of the SIFD model is retained as a third equation for the inputoutput model. The fourth equation for the model, Equation (11), is obtained from Equation (9), and Equations (8), (10) and (12). Thus, Equations (8), (10) and (12) are modelling statements similar to the science model, Equations (6), (7) and (8), except that in the Batangas State University model, inputs z1 and z2 have been added.
The input , when scaled up to *N, is the number of persons per day that we plan to change the decrease in the susceptible rate per day by. It has a simultaneous partial effect of opposite change in the infected per day.
The input when scaled up to *N is the additional increase in the number of persons per day that are free of virus and have antibodies in them, per day. It has a simultaneous partial effect of reducing the infected per day.
The Batangas State University input-output model is dynamic and nonlinear. Figure 3 shows the diagram of the closed loop system. One of the key advantages of adding feedback to an inputoutput system is the capability to stabilize the entire system (Åström and Murray 2020, Albertos and Mareels 2010, Cruz 1971, Dorf and Bishop 2004and Kuo and Golnaraghi 2002. The design of the two inputs of the Batangas State University inputoutput model in two-stages. The input is divided into two parts:

Creating the Batangas State University pivotal engineered feedback model
where The value of in the science model is usually uncertain or variable but Ba is chosen to match a specific value, . The value of Ba may be chosen adaptively based on a sequence of estimates of β (Kanellakopoulos et al. 1991) The quantities and are represented by , and The quantities , and are desired constant asymptotes where is always set equal to The first three linear differential equations can be written in matrix form as It can be readily verified that any value of the state is reachable and is controllable (Åström and Murray 2020). Denoting the vector input by (20) where .
The three eigenvalues of can be specified arbitrarily. In particular, the eigenvalues can be set to be real and negative.
It is assumed that β remains in a certain range that includes β * . Summarizing, the Batangas State University feedback control is (27) Where Ba = β * and K is the matrix defined in Eq 21 under the condition that β = β * . β * is a design parameter chosen as one of the possible values of β such as the minimum value of β.

Setting the target asymptotic values, system parameters and initial conditions
Using
In Figures 4 and 6, the plots for S and F , respectively, it is not possible to visually distinguish the curves for four different values of , displaying not only stability, but also robustness as well. In Figure 5

Relationship of controls to interventions
Vaccination intervention is related to control z2. The number of vaccinations need to be counted and related to the desired increase provided by z2. If z2 is greater than the maximum capacity for vaccinations the maximum capacity would be used otherwise the eigenvalues need to be adjusted. The control z1 is related to the intervention of wearing mask, social distancing and quarantine. If the control z1 is greater than the maximum requirements of wearing mask, social distancing and quarantine, the maximum could be utilized or the eigenvalues could be adjusted. In an actual application, iterative adjustments may be needed.

3.5Stabilization and robustness properties of Batangas State University engineered closed loop model
The principal advantage of utilizing a closed loop model that includes a feedback mechanism for mapping the output to the input of the systems is the possibility of stabilization (Cruz 1971, Åström and Murray 2020, Åström and Kumar 2014, and increasing the stability margin. The Batangas State University engineered closed loop model is designed to be stable.
There is extensive literature on stabilization available (Albertos and Mareels 2010, Dorf and Bishop 2004and Kuo and Golnaraghi 2002, for example. Simulations show that the model remains stable for a range of values of . Furthermore, the simulations show that the outputs are robust against variations in the value of (Cruz and Perkins 1964, Cruz et al. 1981a, Cruz et al. 1981b, Freudenberg et al. 1982.

Science models
The prevailing method for the study of disease propagation, such as the spread of COVID-19 is through the construction of mathematical models. Typically, these models consist of sets of differential equations with parameters that are chosen so that computer simulations using the models would show results that closely approximate of real observations. If the observations indicate that the disease keeps growing, the model can be helpful in identifying what environmental conditions influence the growth. For example, the mathematical model could determine that some transmissivity parameters might have values that are too high, and interventions such as increased isolation of infected persons or increased use of face masks are warranted. With the interventions, as suggested by the model, new observations might indicate that growth has stopped or even reversed, and revised models would show that the new situations are stable. If the interventions are removed, there might be resurgence of growth in the spread of the disease. If the fixed intervention is not removed, it would be maintained at the latest level until a resurgence occurs for whatever reason. Note that the latest model is obtained after the latest change in the environment. However, experimental assessment of the effectiveness of the latest model is conducted after it is constructed.

Engineered closed loop models
The principal reason for using an engineered closed loop mathematical model, that adds a feedback mechanism of measured epidemiological compartments of population, to construct the inputs of the input-output model, is to provide a capability to stabilize the composite model Murray 2020, Cruz 1971  of new real observations. One major capability of using a closed loop model is that even if the input-output model is unstable, potentially, the entire system with feedback can be stable.
When the closed loop model is designed to be stable, then the epidemiological compartments of population would tend to the designed asymptotic values, but the closed loop structure needs to be in place indefinitely. Testing will remain and if there is occasional infection, the infected persons will receive medical treatment, as appropriate, and contact tracing will be conducted. A second advantage of using a closed loop model is that it can be robust against variations in the parameters, such as the infectivity rate, β (Cruz and Perkins 1964, Cruz et al. 1981a, Cruz et al. 1981b, Freudenberg et al. 1982.

Other closed loop models
The only closed loop model in the literature was proposed by a team from the University of Montreal (Stewart et al. 2020). They used an input-output model proposed by the University of Notre Dame (Kantor 2020).

Notre Dame University input-output model
The University of Notre Dame model (Kantor 2020) added an input to the Kermack and McKendrick SIR Science Model (Kermack and McKendrick 1927). A control input is introduced by multiplying β by (1-u) in the science model together with Equation (2).
In Kantor (2020), simulations were performed for various fixed values of , where 0 < < 1, to demonstrate mitigation of the propagation of COVID-19. increasing the infected compartment. This paper is the first to incorporate the effect of vaccination on input-output modeling of the spread of a disease. Furthermore, this paper is first to consider the use of a vaccine in a closed loop model incorporating a feedback mechanism to relate observations of epidemiological compartments of population, to the two modulated inputs or interventions.

Concluding remarks
The major contribution of this paper was the design of the Batangas State University engineered feedback model that utilizes the outputs in tempering the inputs to control the spread of COVID-19. The model was designed to be stable and robust against variations in the values of transmissivity, . The Batangas State University engineered closed loop model introduced an input ( ) utilizing feedback, that led to a substantial increase in the number of epidemiological compartment of population that is Free of virus (F), without significant increase in the number of Infected (I). Even when the spread is reduced to almost zero, testing, contact tracing, and medical treatment of those tested positive, must continue as part of the closed loop process, to avoid new surges of infection, and further reduce mortality. The specific design using specific eigenvalues is for illustrative purpose. Before applying to an actual situation, various capacities need to be determined and the eigenvalues need to be chosen so that the values of the controls do not exceed the capacities.

Acknowledgments
Jose B Cruz Jr. acknowledges support from the National Academy of Science and Technology for a Research Fellowship. The authors would also like to acknowledge Batangas State University for all the support and opportunity to complete this project.