\section{Introduction}
The balance between measures to reduce the spread of a virus and the desire to safeguard social and economic activity is quite delicate, as evidenced by the events since the beginning of the pandemic.  The tradeoffs were  assessed in numerous publications using optimal control methods.  Among the first were \cite{NBERw26981}, where a lock-down intervention is incorporated in a SIR model, including the probabilistic occurrence of a vaccine and \cite{Cowles2229} where properties of the optimal social distancing policy are proven. In \cite{Charpentier2020} the same methods are employed to assess the combination of several intervention modes. Our previous work \cite{magirou2020optimal} was in the middle ground between economics and health policy, stressed the seasonality character of the pandemic and came up with policies that a posteriori proved on the right track as they anticipated the need for timely resumption of social distancing by early autumn 2020 and their unabated continuation throughout the winter months\footnote{Seasonality has been included in several models in the literature because the specific disease is airborne and thus more easily transmitted indoors. As the weather improves during the summer, people become more mobile and are less likely to gather indoors.}.  

The arrival of effective vaccination occurred earlier than was expected in the beginning of 2020, but capacity limitations and vaccine avoidance still pose a serious obstacle to the final relief from the pandemic. The modalities of vaccination and its coordination with social distancing were analyzed using optimal control in \cite{Grundel2020.12.22.20248707} which is based on the authors' earlier compartmental model \cite{grundel2020testing}, in \cite{chhetri2021optimal} where existence of optimal controls was examined, in \cite{libotte2020determination} where a multi objective formulation was presented, while political decision making parameters were considered in \cite{PortugalPandemic}. In \cite{Kontoyiannis} the results of a simulation-based evaluation of several policies for vaccine roll out are reported using a detailed epidemiological model.  We will extend these and other similar work by incorporating seasonality in  disease spreading parameters, taking into account vaccination refusal, modeling fatigue from mandatory social distancing and examining the dependence of optimal policy on health system capacity and category mobility. The population fraction unwilling to be vaccinated is an important parameter, and in health systems with limited intensive care facilities it is unavoidable to continue social distancing even after the reduction of susceptibles below the average level required for spontaneous disease extinction.  Incorporating weariness results in reducing the intensity of measures mainly in the middle of the horizon, but as expected has little effect in low capacity health systems.  

An important finding is that vaccination priority depends mainly on the ICU demanded by each demographic group, to a lesser extent on its transmission characteristics and to an even smaller degree on its economic and disease related costs.  It is of interest to see whether the policies derived in our simple model persist in more complex deterministic or stochastic models.  Needless to say  dynamic parameter estimation and  state identification is of paramount importance for the implementation of optimization policies.  In the same spirit, principles of Model Predictive Control are employed in \cite{MPC} and \cite{WatkinsNP20}.

We present the formulation and solution methods for the model in Section \ref{Model}. Computational results for the Cases examined are in Section \ref{Cases}, and a short Conclusion follows.  Appendix \ref{SecParEst} presents the selection of disease, cost and model parameters while \ref{GradAlgor}  gives an informal description of the algorithm used.

\section{Model Formulation and Solution Methods}\label{Model}
\subsection{Model Formulation}\label{modelform}
We consider the standard  SIR epidemiological model of W. O. Kermack and A. G. McKendrick \cite{Kermack},  \cite{HethcoteSIAM} including  population related, demographic categories and endowed with the potential for controlling the contact parameter, a partially controlled vaccination policy and a modeling of fatigue from non pharmaceutical interventions. Population categories are denoted by the indices $j,k$. Thus let $S_j(t),I_j(t),R_j(t)$ be the number of individuals in a population of size $N(t)$ that are in compartments referred to as \textit{Susceptible, Infectives and Removed} at time $t$ in each population category $j$.  We consider a short horizon, and hence assume a constant category size $N_j(t)=N_j$  the total population being $\sum_j N_j=N$.  We work with the corresponding fractions $s_j(t)=S_j(t)/N$, $v_j(t)=I_j(t)/N,\;n_j=N_j/N$, the removed fraction being determined by  $s,v$ as $n_j-s_j(t)-v_j(t)$.  For convenience we denoted the fraction of infectives by \textbf{$v$} instead of \textbf{$i$}.  

The dynamics of an epidemic  with homogeneous mixing of the population are determined using the following standard assumptions: 
\begin{enumerate}
	\item The rate of newly infected in category $j$ by those in $k$ is $\lambda_{jk}(t)s_j(t)v_k(t)$ 
	\item The rate of removal from the infectives in category $j$ is $\gamma_j v_j(t)$, $\gamma_j$ constants.
	\item The rate of immunity loss of those removed is  $\delta_j (n_j-s_j(t)-v_j(t))$ who then revert to the susceptibles in the same category.
\end{enumerate}

In the above equations $\lambda_{jk}(t)$ is the contact rate, which is the average number of adequate contacts in category $j$ per infective in $k$ and unit time at instance $t$ \cite{HethcoteSIAM}, $\gamma_j$ is the removal (recovery plus death) rate, a constant. The possibility of reinfection i.e. immunity loss at rate $\delta_j$ is included in the model so we are dealing with a controlled SIRS model, although in our calculations we do not consider reinfections setting $\delta$ to zero. Vaccination at a rate $w_j(t)$ is applied to the susceptibles of category $j$ and immediately shifts them to the Removed compartment. Consequently, the  epidemic dynamics are 
\begin{equation}\label{SIR}
\begin{array}{l}
\frac{ds_j}{dt}=-\sum_{k}\lambda_{jk}(t)s_j(t)v_k(t)-w_j(t)+\delta_j(n_j-s_j(t)-v_j(t))\\
\frac{dv_j}{dt}=\sum_{k}\lambda_{jk}(t)s_j(t)v_k(t)-\gamma_j v_j(t).
\end{array}
\end{equation}
More detailed models in the literature  include latent infectives (Exposed) that accelerate the contagion rate (SEIRS). We assume that this effect can be incorporated by the proper choice of contact rates.  We allow rates  to depend on time, modeling thus seasonal variations in virus spread.  The vaccination  is modeled in a rudimentary way as transferring susceptibles to the removed at a rate $w_j$.  In order to model the situation of a population fraction refusing to be vaccinated, we introduce categories of susceptibles $s^R_j(t)$ that are not to be vaccinated and that once infected revert to those accepting vaccination.  These categories follow the equations
\begin{equation*}
\frac{ds^R_j}{dt}=-\sum_{k}\lambda_{jk}(t)s^R_j(t)v_k(t).
\end{equation*}
\noindent It is easily shown that the total infectives follow \eqref{SIR}.  We use the detailed equations in our calculations but for ease of exposition make no reference to those refusing vaccination when examining the optimization model.

We assume that we can select the vaccination intensity in each category subject to an exogenous upper bound on the vaccinations, i.e. $\sum_j w_j(t)\le W_{max}(t)$.  More accurate formulations would include a \textit{vaccinated compartment} that will switch to the removed one at a certain rate, in the meantime acting as susceptibles with better disease response as in \cite{Grundel2020.12.22.20248707}.  A separate category that has received the first dose of the vaccine can be introduced to assess a policy of exclusive first dose vaccination versus a two dose one \cite{Kontoyiannis}.  In the literature, for instance in \cite{Grundel2020.12.22.20248707},  vaccination is expressed as a fraction of the corresponding category, which is convenient since the infected population can never go negative. In the same reference a constraint is placed on total vaccines available at all times, and there is no bound on the vaccination rate. By contrast we insist on always observing the appropriate bound, which in turn causes numerical difficulties.

 We assume that the mitigation, suppression and any other policies mentioned say in \cite{Imperial} can be represented by a single, scalar control variable $u(t)$ with values in $[0,1]$, as in \cite{NBERw26981}. This is an oversimplification: one could study modes of intervention $k=1,..,n$ and consider a vector control $[u_1,..,u_n] $ of the corresponding instrument's intensities \cite{Charpentier2020}.  However all proposed interventions consist of reducing the contact rate so we expect to get some insights from the scalar control case.   We thus model the effect of a level $u$ on the contact rate by the expression
$$ \lambda_{jk}(t)=\lambda^o_{jk}(t)(1-u(t)).$$

The contact rate before a social distancing intervention $\lambda_{jk}(t)$ has a seasonality which (as in \cite{magirou2020optimal}) we represent it by  $\lambda_{jk}(t)=\lambda^o_{jk}(1+k_{seas}sin(2\pi t)),\, \lambda^o_{jk},k_{seas}$ constants.  The intervention effectiveness was  assumed linear multiplicative, and we thus finally express the controlled contact as
\begin{equation}
	\lambda_{jk}(t)=\lambda^o_{jk} (1+k_{Seas}sin(2\pi t) )(1-u(t)).  
	\end{equation}
The inter-category contact coefficients $\lambda^o_{jk}$ determine to a great extent the model's behavior, and by a proper selection one can describe "super-spreader" categories, which is crucial in determining vaccination priority. Data on contacts among categories are extensive in the literature \cite{SocContactPatterns} and we present in  Appendix \ref{contactparms} how to parametrize them in accordance with the assumed contact characteristics.

At every distinct time interval $[t,t+\Delta t]$ we assume a cost proportional to the fraction of infectives.  The cost will consist of the reduced output of those infected that exhibit symptoms, the cost of medical service required (perhaps in addition a discomfort cost) and finally a cost for fatalities.  In \cite{NBERw26981} these costs are in economic terms, and we will follow this approach in our parameter selection.   We thus consider a cost element  $m_j(t)v_j(t)\Delta t$ for each category $j$, the time dependence in the cost coefficient $m_j(t)$ reflecting changes in treatment effectiveness and cost. 

It is important to implement the limited capacity of the health system.  We assume that of those infected $v_j$ in category $j$ a fraction $a_j$ exhibits severe symptoms and requires the use of scarce health care facilities and thus their sum should not exceed an exogenous capacity $V_{max}(t)$,  which can be time dependent to reflect capacity changes.  We must adjust the social distancing parameter $u$ and the vaccination schedule so that the infected $v_j(t)$ always satisfy the constraint $\sum_j a_j v_j(t)\le V_{max}(t)$ and the cost (to be specified next) is minimized.  Alternatively as in \cite{magirou2020optimal}, we  append to the cost integral the penalty term $D\sum_ja_jv_j(t) \exp\left(M(\sum_j a_jv_j(t)-V_{max}(t) )\right)$, where the parameters $M,D$ are to be selected so that the cost is small if the infectives do not stress the system but rises steeply if they do. Each alternative has computational difficulties, so we implemented both, obtaining almost identical results.

The intervention intensity $u$ is assumed to impose a cost which is convex since simple mitigation policies have sub linear costs, while suppression type measures have costs that increase more than proportionally.  Thus we consider a control cost of the form $Au(t)^n \Delta t$, usually quadratic.  

We incorporate a cost term reflecting weariness from lengthy interventions which is important in policy making and thus introduce a fatigue index $z(t)$ which accrues while measures are in effect but decreases to some extent when they are relaxed, exhibiting memory loss.  The fatigue index has the dynamics
\begin{equation}
	\frac{dz}{dt}=b_fu(t)-d_fz(t)
\end{equation}
with parameters $d_f,b_f$.  A penalty term $Bz(t)^m$ is included in the cost  function.   Fatigue increases for measures with intensity in excess of $d_f\frac{z(t)}{b_f}$  and conversely. We expected that this could lead to periods of intensive measures followed by relaxations as practiced by several countries, but we failed to observe such an oscillatory effect, see the computations in Section \ref{Cases}.

We consider the model for a fixed horizon $[0,T_H]$, with an instantaneous cost as described in the previous paragraphs and in addition a terminal cost depending on the final state which is important because just minimizing the period cost would lead to a postponement of infections to the horizon's end.  Assuming that costs are additive in time and are to be discounted at a rate $\rho$ the total  cost is given by the expression:
\begin{small}
\begin{equation}\label{totcost}
	\begin{split}
		\int_0^{T_H}\exp({-\rho \tau}) \left\lbrace \sum_{j}v_j(\tau)m_j(\tau)+ D\sum_j (a_j v_j) \exp (M(\sum_j a_jv_j-V_{max})) +Au^n(\tau)+Bz^m \right\rbrace d\tau \\ +\sum_j m^{term}_j v_j(T_H)\exp(-\rho T_H).
	\end{split}
\end{equation}
\end{small}

 For each category $j$ we denote by  $m_jv_j(t)$ its overall cost per unit time and by $m^{term}_jv_j(T_{H})$ the terminal cost, the parameter assessment to be presented in Appendix \ref{SecParEst}.  We are interested in computing a policy consisting of  intervention and vaccination programs $u(t),w_m(t),w_y(t)$, $t\in [0,T_H]$ to minimize \eqref{totcost}. If policy measures are to be selected to satisfy capacity constraints, the penalty coefficient $D$ will be set to zero.  We only consider open loop policies which however is a step towards feedback, stochastic - adaptive control policies.  
 
 \subsection{Solution Methods}
  Since our model is time dependent we use optimal control algorithms \cite{Bryson} instead of the dynamic programming ones used for instance in \cite{NBERw26981}.  For ease of exposition we present only the penalty function formulation.  We write the Hamiltonian including dual variables $\phi^s_j,\phi^ v_j,\phi^ z$ correspond to states $s_j,v_j,z$ respectively:
\begin{small}
\begin{equation}\label{Hamiltonian}
	\begin{split}
		H=\exp({-\rho t}) \left\lbrace \sum_{j}m_jv_j+ D\left(\sum_j a_j v_j\right) \exp\left(M\left(\sum_j a_jv_j-V_{max}\right)\right) +Au^n+Bz^{m} \right\rbrace\\ + \sum_j \left(\phi^s_j \frac{ds_j}{dt}+\phi^v_j \frac{dv_j}{dt}\right)+\phi^z(b_fu-d_fz)
	\end{split}
\end{equation}
\end{small}
The  dual variable dynamics are
\begin{equation}\label{Costateequ}
	\begin{array}{l}
		\frac{d\phi^s_j}{dt}=-\frac{\partial H}{\partial s_j}= (\phi^s_j-\phi^v_j)(1-u)\sum_{k}\lambda_{jk}(t)v_k+\delta_j \phi^s_j
		\\ \frac{d\phi^v_j}{dt}=-\frac{\partial H}{\partial v_j}=(1-u)\sum_{k}\lambda_{kj}(\phi^v_k-\phi^s_k)s_k(t)+ \phi^s_j\delta_j+ \phi^s_j\gamma_j -\\ \quad\quad
		-exp(-\rho t)(m_j+Da_j(1+\sum_{k}a_k v_k M)exp(M(\sum_{k}a_k v_k-V_{max(t)})))
        \\  \frac{d\phi^z}{dt}=-\frac{\partial H}{\partial  z}=-mBz^{m-1}exp(-\rho t)+d_f\phi^z.
	\end{array}
\end{equation}
The partial derivatives of the Hamiltonian with respect to the controls are
\begin{equation}\label{HamPartials}
	\begin{array}{l}
		\frac{\partial H}{\partial u}= \exp(-\rho t)nAu^{n-1}+\sum_{j}\left[(\phi^s_j-\phi^v_j)\sum_{k} \lambda_{jk}(t)s_j(t)v_k(t)\right]+\phi^z b_z
    	\\ \frac{\partial H}{\partial w_j}=-\phi^s_j
	\end{array}
\end{equation}
 The optimal  policy is determined by choosing the values of $u,\,w_j$ minimizing the Hamiltonian \eqref{Hamiltonian}, which is convex in $u$ so we must have $\frac{\partial H}{\partial u}=0$ for an unconstrained control, and hence the optimal social distancing level $u^*$  is given by truncation in $[0,1]$ of the expression 
\begin{equation}\label{OptContr1}
u^*(t)=\left[ \frac{\exp({\rho t})\sum_{j}(\phi^v_j-\phi^s_j)\sum_{k} \lambda_{jk}(t)s_j(t)v_k(t)-b_z\phi^z}{nA}\right] ^{1/(n-1)}.
\end{equation}
The optimal social distancing expression above shows from a policy perspective why it is important to have a good estimates of the product of susceptibles and infectives. 

Vaccination intensities enter linearly  in the Hamiltonian multiplied by the duals of the corresponding susceptibles.  Thus it is almost always optimal to carry out vaccination  exclusively in the category with larger dual variable and at the maximum intensity - a bang/bang control policy.  If the duals are equal for a non zero interval it might be optimal to vaccinate all categories at nonzero levels.  We did not analyze the possibility of such optimal (singular) policies and they did not arise in the calculations. 

The optimal policy is determined in principle by solving a \textit{two point boundary value problem} consisting of the equations \eqref{SIR} and \eqref{Costateequ} with social distancing specified by \eqref{OptContr1}, and the above bang/bang vaccination policy.  The boundary conditions are: at the initial time $t_o$ we are given the values of the state variables $s_j(t_o)=s_{j,o},\, v_j(t_o)=v_{j,o}$ and at the final time $T_{H}$ the dual variables that must attain the boundary values $\phi^s_j(T_H)=0,\,\phi^z(T_H)=0,\,\phi^v_j(T_H)=m^{term}_j\exp(-\rho T_H)$, the terminal cost derivative.  Such problems are difficult to solve numerically (a comprehensive description is in Ch. 7 of \cite{Bryson}) since the dual variables increase backward in time while the state ones decrease. The possibility of using singular paths presents additional complications. We solved the cases presented in Section \ref{Cases} by a  first order gradient search for optimal control problems (Section 7.4 in  \cite{Bryson}) suitably modified to satisfy the state and control constraints.  An informal description of the algorithm is given in Appendix \ref{GradAlgor}, its details are available on demand.

\section{Computational Results}\label{Cases}
We present several calculations that are relevant to Covid 19 policy making.  An analytic derivation of optimality properties is not straightforward and we restrict ourselves to calculations with parameter sets referred to as Cases, selected to reflect policy questions.  A methodology for parameter assessment (as applied to a particular country, Greece) is presented in  Appendix \ref{SecParEst}.

  We will use \textbf{two population categories}, indexed by $y,m$ that correspond to the age groups $[0,49] ,[50+]$. We will use Population Data for Greece in 2020 as compiled by the Hellenic Statistical Service, included in the disease data in Table \ref{ImperData} in the Appendix. The two categories are of almost the same size, with population shares $n_m=42,77\%$ and $n_y=57.23\%$.  
  
  Category dependent economic and related disease parameters that are the same in all Cases are summarized in Table \ref{Parameters}, while those varying among Cases in Table \ref{CaseParameters}.  The model-related parameters (common in all Cases and categories) are
  
\begin{itemize}
	\item  Control Coefficient $A = 1$.
	\item  Discount rate 3\% yearly.
	\item  Seasonality $k_{seas}=80\%$.
	\item  Vaccination capacity: 70\% of the population per year
	\item  Penalty Exponential Coefficient $M=2\cdot10^5$.	
	\item  Penalty Coefficient $D=0.2$.	
	\end{itemize}

The horizon in all Cases lasts for two years unless otherwise mentioned. Time zero is set near the Vernal Equinox, say April 1st, just before massive vaccination starts. The overall contact parameter $\lambda$ takes the value of $120$ in all cases - see Section \ref*{contactparms} for a justification - except in the last where a milder situation with $\lambda=70$ is examined.

\begin{table}	
	\centering
	\caption{Nominal Parameters}  
	\label{Parameters}   
	\begin{tabular}{|l|c|c|c|c|c|c|c|}
		\hline
		Class&Infected&Infected&Vaccine&ICU&Share&Susceptible&Infectives\\
		&Cost&Cost-High&Accept.\%&Demand \%&\%&Initially \%&Initially \%\\
		\hline
		$m$&16.340&24.871&90&2.70&42.77&38.0&0.4\\ \hline
		$y$&1.390&2.353&70&0.05&57.23&52.0&0.5\\
		\hline 
	\end{tabular}
\end{table}
\begin{table}	
	\centering
	\caption{Case Parameters}  
	\label{CaseParameters}   
	\begin{tabular}{|l|c|c|c|c|c|}
		\hline 
		\hspace*{15pt}Case Number&1&2&3&4&5\\
		Parameter Name&  &  & &  &  \\	\hline
		ICU units per $10^4$&3&10&10&1.5&3\\
		\hline
		Fatigue Factor $B$&0&0&2&0&0\\
		\hline
		$\lambda$&120&120&120&120&70 \\
		\hline 
	\end{tabular}
\end{table}

It is interesting that for reasonable parameter variations vaccination policies are dictated by the requirement to satisfy the health capacity constraint rather than by an effort to reduce costs\footnote{A proper analysis of this vaccination robustness property requires studying the dual variable dynamics \eqref{Costateequ} in order to establish inequalities on the susceptibles' duals.}. In particular, for all Cases it is category $m$ - those over 50 - that has vaccination priority. This changes when ICU demand (column 5 in Table \ref{Parameters}) reverses without changing the other table entries.  Then it is the $y$ category - those under 50 - that is vaccinated first in spite of its very low cost per infection (Column 2 of the same Table).  

In order to examine the relation between vaccination and category contact rates, we use a parameter $\zeta$ which characterizes the excess contribution of category $y$ - those under 50 - to transmission speed, see Section \ref*{contactparms} in the Appendix. We take 20\% as the nominal value of $\zeta$.  Then category $m$ is to be exhaustively vaccinated before starting on the other category. Priority switches  only for large  $\zeta$ values of the order of 100\%, a substantial multiple of its nominal value.   An interesting phenomenon is that for some ranges of $\zeta$'s we observe \textit{priority switching}: One must start with category $m$, and then switch to vaccinating $y$ before exhausting $m$, to which one we return next.  We will present several such switching phenomena in the Cases that follow. 
\subsection*{Case 1}
In \textbf{Case 1} a moderate ICU capacity of 3 units per 10.000 individuals was examined, the remaining parameters being at their nominal values.  Figure \ref{Case_1_overall} shows the "best" social distancing policy (i.e. $u(t)$) and the ICU demanded divided with the ICU capacity (Relative ICU Demand). The social distancing measures follow the seasonal contact variation for a year, starting at a level of about $68\%$, decreasing to $40\%$ in the summer period.  They repeat at a lower level up to about $56\%$ in the winter and then stay at about $35\%$ until the horizon's end, with reduced seasonal variation. Continuation of social distancing was expected since we assumed that about $20\%$ of the population refuses vaccination, which is higher than the level required for spontaneous disease extinction (about $10\%$ in the winter).  We see no obvious explanation for the eventual decrease of seasonal variation. Relative ICU demand peaked at $70\%$ and stayed relatively low throughout the horizon. 
\begin{figure}[ht]
	\centering
	\includegraphics[width=7.5cm, height=6cm]{Case1Overall.jpg}
	\caption{\small{\textbf{Case 1: Intensity of Measures, ICU Demand}} }
	\label{Case_1_overall}
\end{figure}
Lengthening the  horizon did not have significant effect on the policies followed, nor changing the cost factors of the infected to the higher values of $m_m=24.871$ and $m_y=2.353$ corresponding to a higher fatality cost ($L=20$).  

Vaccination policies are shown in Figures \ref{Case_1_O} and \ref{Case_1_Y} which show the susceptibles, the susceptibles positive to vaccination and the vaccination intensity in each category.  Vaccination is carried out on category $m$ until the exhaustion of those willing and at the exogenous maximum vaccination rate before starting on category $y$.  Non vaccinated susceptibles, consisting of those refusing vaccination, are considerably reduced but are still sufficient to tax the health system in case measures are lifted. The category contact coefficients are initially at $\zeta=20\%$, and the same vaccination priority persists for value of  $\zeta$ up to about $80\%$.  Then at $\zeta=100\%$ we observe a \textit{switching} policy, starting with $m$, then working on $y$, back to $m$ and then finishing up $y$. When $\zeta$ reaches $200\%$ the policy reverses and category $y$ is vaccinated exhaustively first. In all cases, changing cost coefficients did not affect priorities.

\begin{figure}[ht]
	\centering
	\includegraphics[width=7.5cm, height=6cm]{Case1Mat.jpg}
	\caption{\small{\textbf{Case 1: Vaccination, category $m$}} }
	\label{Case_1_O}
\end{figure}
\begin{figure}[ht]
	\centering
	\includegraphics[width=7.5cm, height=6cm]{Case1Yng.jpg}
	\caption{\small{\textbf{Case 1: Vaccination, category $y$}} }
	\label{Case_1_Y}
\end{figure}
\subsection*{Case 2}
A health system with a high ICU capacity of 10 ICU units for every 10.000 individuals is shown as \textbf{Case 2}, the other parameters as before.  Social distancing  starts almost at the same level as in Case 1, about $63\%$ but decreases drastically in the summer at  $2\%$, then increases mildly to a maximum of $5\%$ next winter and levels at $1.6\%$ for the length of the horizon - see Figure \ref{Case_2_overall}.  Although vaccination is not sufficient for spontaneous disease extinction, very light measures are required to keep the level of infected within the ICU system capacity.
\begin{figure}[ht]
	\centering
	\includegraphics[width=7.5cm, height=6cm]{Case2Overall.jpg}
	\caption{\small{\textbf{Case 2: Intensity of Measures, ICU demand}} }
	\label{Case_2_overall}
\end{figure}

Concerning vaccination policies, again category $m$  is totally vaccinated before proceeding with $y$.  This priority persists even when reversing costs.  As for the dependence on category mobility, the vaccination priority of $m$ is more robust reversing only for $\zeta$ over $1000\%$ with switching among categories for $\zeta$ about $750\%$.  

\subsection*{Case 3}
A situation with the high capacity health care system of the previous Case coupled with fatigue that does not subside if measures are relaxed (fatigue parameters $B=2\;b_c=1,\,d_f=0$) is shown as \textbf{Case 3}. The social distancing measures in Figure \ref{Case_3_overall} are slightly lower than those in Case 2, but those infected increase considerably almost exceeding the health system's capacity.  Social distancing decreases are only significant in relative terms ranging from $3\%$ of the non-fatigue $u(t)$ for high social distancing values to $100\%$ in the summer months, see Figure \ref{Case_3_relative}, but they are quite important in view of the considerable ICU demand dependence on social distancing.  Policy does not change appreciably for higher fatigue ($b_f=5$) or when there is memory loss ($B=2\;b_c=1,\,d_f=1$).  Applying the same weariness parameters to the moderate capacity health system of Case 1 had an appreciably smaller effect, decreasing measures by less than $0.2\%$, which is reasonable given the lower ICU capacity.  Vaccination policy has similar characteristics as in the previous Cases.
\begin{figure}[hbt!]
	\centering
	\includegraphics[width=7.5cm, height=6cm]{Case3boveralB2b1d0.jpg}

	\caption{\small{\textbf{Case 3: Intensity of Measures and Infectives}} }
	\label{Case_3_overall}
\end{figure}
\begin{figure}[hbt!]
	\centering
	\includegraphics[width=7.5cm,height=6cm]{Case3relative.png}
	\caption{\small{\textbf{Case 3: Relative Decrease in Social Distancing}} }
	\label{Case_3_relative}
\end{figure}
\subsection*{Case 4}
This case considers a health system of low ICU capacity at $1.5$  units per 10.000 individuals and the results are presented in Figure \ref{Case_4_overall}.  Social distancing measures are more pronounced starting at $78.3\%$, dropping to $54\%$ in the summer, rising next to $58\%$ and remaining close to $55\%$ until the end, showing minimal seasonality. Capacity is almost exceeded initially but subsides soon.
\begin{figure}[hbt!]
	\centering
	\includegraphics[width=7.5cm, height=6cm]{Case4Overall.jpg}
	\caption{\small{\textbf{Case 4: Very Low Health Care Capacity}} }
	\label{Case_4_overall}
\end{figure}

Vaccination policy is more sensitive to category transmission characteristics.   Thus priority is given to $y$ for $\zeta$ as low as $80\%$ with partial vaccination of categories for a $\zeta$ about $50\%$.  This is significantly lower than in the moderate capacity Case 1, where the corresponding values are  at $80\%$ and over $100\%$ respectively, as well as those of the high capacity Case 2 with $\zeta's$ at $750\%$ and greater than $1000\%$.
\subsection*{Case 5}
In a scenario with a lower contact rate, $\lambda=70$,  \textbf{Case 5}, an average susceptibles level lower than $28.6\%$ seems sufficient for spontaneous extinction, a level that is achieved with the vaccination refusal rates used (10\% for category $m$, 30\% for $y$). However with a seasonality of $80\%$ the susceptibles should be below $28.6/1.8=15.9\%$ at the peak transmission time, and this is not valid with this parameter set.  The results in Figure \ref{Case_5_overall} indeed show that a low health capacity system will impose lower social distancing initially at $38\%$, dropping to $16\%$ in te summer to increase at the level of $25\%$ next winter and then a level of $15\%$ being kept until the horizon's end with minimum seasonal variation. An average contact rate value $L_o\le 55$ guarantees that spontaneous extinction holds for all time after vaccination, and indeed the optimal policy in such a case calls for end of social distancing in a year's time.  
\begin{figure}[ht]
	\centering
	\includegraphics[width=7.5cm, height=6cm]{Case5Overall.jpg}
	\caption{\small{\textbf{Case 5: Intensity of Measures and ICU Demand, Low contact rate}} }
	\label{Case_5_overall}
\end{figure}
Since the measures are mild, calculations with fatigue parameters did not show significant differences. Thus the relative decrease in social distancing for high fatigue ($B=b_f=2.0$) was less than $4.5\%$ with an average of $2.2\%$.  

Category $m$ is vaccinated exhaustively before $y$ for $\zeta$ up $80\%$ when switching occurs, while for $\zeta$ over $100\%$ it is $y$ being exhaustively vaccinated first.  \textbf{Figures \ref{Case_5ksi80Old} and  \ref{Case_5ksi80Yng}} dramatically exhibit the switching effect for $\zeta=80\%$. Thus vaccination switching occurs for the same $\zeta$ as in Case 1, even though the overall contact $\lambda$ parameter there was almost double.  
\begin{figure}[ht]
	\centering
	\includegraphics[width=7.5cm, height=6cm]{Case5_80Old.jpg}
	\caption{\small{\textbf{Case 5: Category $m$ Vaccination, $\zeta=80\%$}} }
	\label{Case_5ksi80Old}
\end{figure}
\begin{figure}[ht]
	\centering
	\includegraphics[width=7.5cm, height=6cm]{Case5_80Yng.jpg}
	\caption{\small{\textbf{Category $y$ Vaccination, $\zeta=80\%$}} }
	\label{Case_5ksi80Yng}
\end{figure}

\section{Conclusions}
 Overall, the findings of this study are quite relevant to policy. While some countries have already vaccinated a large part of their population, most have yet to reach herd immunity, so choosing what groups to focus on to actively promote vaccinations remains relevant. Planning booster jabs, as may be required in the near future, further highlight the importance of such studies. At the same time, vaccination priorities are of great importance for the large number of countries globally that have vaccinated a very small part of their population and need to prioritize based on the small volume of vaccines available. Finally, such modeling applies not only to Covid-19, but also to any future pandemic that requires mass vaccinations. 
 
  It should also be noted that our results are in agreement with the conclusions in \cite{Kontoyiannis} where the optimal strategy is found to be one based on fully vaccinating the elderly/at risk as quickly as possible.  In our computations vaccination priority depends weakly on the infectivity characteristics (contact rates) of the demographic categories. In that sense vaccination decisions take into account a combination of susceptibility and infectivity, in agreement with \cite{Kontoyiannis}, but susceptibility is the dominant factor, while cost differences are not important.  Vaccination policy is "on-off", operating on a single category at every particular time interval.  In most of our computations a category was totally vaccinated before proceeding to the next, but alternating among categories can be optimal although for a limited parameter range.
 
 Our calculations indicate that policies followed by most national health authorities  are sound, although presented from a different perspective.  If one accepts the requirement of staying within the health system's capacity, vaccination priority is necessarily given to those expected to require more resources, while ethical or cost considerations play no role.  The calculations also stress that average herd immunity achievement is not sufficient to stop interventions and that real time state and parameter estimation is of paramount importance.
\clearpage
