---
# System prepended metadata

title: 'PROBLEM 5 : V-KEMS'

---

# PROBLEM 5 : V-KEMS

### Group Members

Dario Domingo: dario.domingo@durham.ac.uk

Paul Heslop: paul.heslop@durham.ac.uk

Gabriele Dian: gabriele.dian@durham.ac.uk

Raymond Pang: r.pang@lse.ac.uk

Stanislaw Biber (in \Latex please write my name as Stanis{\l}aw): s.biber@bristol.ac.uk 

Yue Liu: yue.liu@maths.ox.ac.uk

Ingrid Holm: iholm@ipht.fr

Kays Haddad: kays.haddad@nbi.ku.dk

Patrick Dorey: p.e.dorey@durham.ac.uk

Alan Champneys: a.r.champneys@bristol.ac.uk

Guven Demirel: g.demirel@qmul.ac.uk

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# Initial literature 

1. Duijzer, L. E., van Jaarsveld, W., and Dekker, R. (2018). Literature review: The vaccine supply chain. European Journal of Operational Research, 268(1):174–192. 
2. 
https://www.sciencedirect.com/science/article/pii/S0360835217305272
3. 
https://doi.org/10.1287/mnsc.1120.1661

# Sources of data 

- Johns hopkins - https://coronavirus.jhu.edu/vaccines
- BBC article - https://www.bbc.co.uk/news/world-56698854
- Kaggle - https://www.kaggle.com/datasets?search=covid
- One world - https://ourworldindata.org/coronavirus

# Questions 
- steady, resillient transnational supply networks of  critical products
- model a single vaccine from production to supply (feedforward network)
- ecosystem of different supply chains (more like a food chain/food web)
- inventory control policies
- detailed single firm, or country level stocks (higher level model)
- what decision level do we want to model?
- strategic or operational?
- competition between states and provinces (e.g. in the US or Canada)

# mathematical depiction of the problem
- network theory, (multilayer networks)
- optimisation theory to optimise an inventory level based on an objective function that can have profit or reliability or (at a national government level) health and wellbeing. 
- model flow of stock - orders, delays
- 3 types of agents include manufacturer, supplier, final market (government in the case of a vaccine)
- also logistic network (typically don't make strategic decisions)
- upstream or downstream supply? better to model the upstream network rather than supply to hospital etc.
- model actions of political actors - (perhaps ignore how people react to government decisions because that is the downstream bit)
- game theoretic approach with governments as the agents, 
- how to build the graph? how detailed?
- what type of action can the agents do on the graph?
- do we want the companies to be agents too?
- probably they respond in a certain way and do not have a ulitity that is being optimised.
- two types of flow: 
  final product flow and ingredients flow
  
## what kinds of disruptions?
- can happen anywhere in the supply chain
- problem at manufacturing plant/elec grid
- production stops or flows get interrupted due to virus
- nationalistic restrictions (borders closed)
- demand responses (e.g. people's willingless to take vaccine)

## outputs
- ammount of vaccine
- spread of the disease
- death rates over time
- reputation of different nations as future trading partners 

## data science approach

- Find data and look for intersting supply chain features



---


# Papers reading group
1. Literature review: The vaccine supply chain. European Journal of Operational Research Gabriele, Ingrid, Juliet

2. Supply Chain Resilience: Definitions and quantitative modelling approaches
– A literature review Dario, Kays, Raymond, 

3. Game Theory Yue, Paul, Stanislaw


### Supply Chain Resilience: Definitions and quantitative modelling approaches

 - Types of supply chains:
 Reverse SC: returning goods from customer to vendor.
 Forward SC: manufacturer -> retailer -> consumer.
 Closed loop SC: the union of reverse + forward SC.
 - Definitions of resilience:
 Involve speed of the recovery, magnitude of recovery, adaptability of the supply chain to disturbances, ability to anticipate disturbances, how much the disturbances spread.
 - Quantifying resilience:
 Most studies don't use one single index to quantify resilience, rather considering several resilience factors (availability, connectivity, accessibility, unfulfilled demand after disaster, ...).
 Some studies define a single index. One of these studies assesses the resilience of each node and defines the network resilience as the weighed sum of the resilience of each node.
 Distinction between supply (and non-supply) nodes: how many of the latter have access to supply nodes? What is the maximum and the average supply-path length?
 - Modelling
 Optimizing and re-optimizing the network with each node removed to simulate disruption.
 - Research gaps/directions
 No consensus on the definition of resilience. Authors propose a definition that incorporates focus event (?), performance level, speed, adaptive framing.
 Combination of risk management and resilience awareness as the former typically focuses on frequent but small disruptions while the latter is also concerned with bigger, less probable disruptions.
 Models are often oversimplistic, since real SCs are very complex.

### Literature review: The vaccine supply chain. European Journal of Operational Research

Mainly a review of existing literature, but also compares vaccine supply chain to other supply chains. The parts that add to the supply system efficiency are: 

- Product - What kind of vaccine should be used?
This is mostly about identifying or developing a relevant vaccine. Relevant parts may be choosing which vaccine to use, e.g. Astra-Zeneca scary side effects, and packaging. 
- Production - How many doses should be produced and when?
Trying to make the vaccine market efficient, subject to uncertainties in yield and production lead times. 
- Allocation - Who should be vaccinated?
Identifying high- and low-risk individuals, etc. Decision makers need to carefully analyze the order in which to give vaccines to the population.
- Distribution - How should the vaccines be distributed?
Distributing vaccines from the manufacturer to the end-users. This is unique to vaccines because there is an ethical component.

From their review, they decide that 'Production' and 'Distribution' are comparable to other supply chains, 'Allocation' is unique to the vaccine supply chain, and 'Product' is somewhat in between.  Maybe we can use one of these other supply chains to model vaccination, and modify 'Allocation'?

 **More on Product**
  The vaccines supply chain papers are concerned with: (1) diseases with unknown characteristics that are certain to break out in the near future (seasonal influenza), (2) diseases with unknown characteristics that could suddenly break out (e.g., pandemic influenza). Note that there is also a third category, namely diseases with known characteristics, which has different supply chain structure and is not considered.
  
  There are several paper on how to optimize this part of the supply chain for type (1). The model is usually the following. Decision makers have to chose what vaccine to finance and ultimately use on the population. Every year they know in advance a the set of influence virus that affect the population. Roughly, one virus will be dominant and only one vaccine can be developed at the time. The timing of the composition decision is crucial as it has a direct effect on the production time of the vaccine and therefore on its availability. Post-poning the decision reduces uncertainty and could lead to better decisions about which strains to include in the vaccine but reduces the available time for production of the vaccine, potentially leading to higher production costs. The optimization problem reduces to find the best time to start developing the vaccine.
  
** More on Production **
Vaccines producting is characterized by a high degree of uncertenty due to different factors.

Countries can invest to reduce the production time uncertenty.

Here two papers ( one author in common with the game theory paper) on modelling production:
   - https://pubsonline.informs.org/doi/pdf/10.1287/opre.2016.1552
   - https://pubsonline.informs.org/doi/pdf/10.1287/opre.1080.0527

The first one seems more relevant to our purpose.

### Game theory paper:

- minimisation of cost of the pandemic due to vaccination (cost of vaccinating and treatment)
- Proposition of Global Contract -- coordination of vaccination; main goal -- saving money globaly + minimising total number of infections 
- benefits: provides a starting point; it is mathematical!
- cons: situation in the paper is simplistic, a lot of modification needed for current situation
- analytic vs. numeric?

### Plan for Tuesday
From 9am meet on Zoom (main ESGI Zoom, breakout rooms) to discuss the papers in more details

Meet @10am in the main room **Decide on the next steps!**
Main goal -- identify what our network is (think: what are the nodes, what are the edges)

Zoom: https://durhamuniversity.zoom.us/j/91781364277?pwd=ZnVST2pSZFUvSUMyMTkvTVo4S2R6QT09

# Discussion (Tuesday) -- starting points
1. Add producers to the current Game Theory model: add nodes to the network that are not countries
2. Extend the network to be a larger graph / more countries (easier to test resiliance)


# Matlab code on Github
https://github.com/liuyue002/ESGI165_vaccinesupply

# Initial plan for Wednesday
1. Meet in the main room by 9am
2. Discussion and breakout rooms from c.9:30am
3. REMEMBER: presentations at 3pm


# Wednesday

After lunch (2pm):
Seek approval from Guven and then...
Team 1. -- Keep on Matlabing
Team 2. (multiple) -- find D matrices
Team 3. (one person?) @3pm -- plan the presentation


$$
\frac{d X_1}{dt}= \gamma_1X_1(t)^{-\beta_1}I_1(t) 
$$
$X_1(t)$: vaccine production rate at time t
$I_1(t)$: investment rate at time t
$\gamma_1$: inverse cost of technology

# Gain function social + vaccines cost

The single country gain function $GF_i$ is defined as
$$GF_i= b_i \int g(I(t))dt+ v_i Vn_i,$$
where $I(t)$ is the number of infected, g(I) is a non-linear function representing the cost of the hospital strain, Vn_i is the number of vaccines bought by county $i$ and $v_i$ is the effective cost of vaccine.  In our first model we will assume that all the vaccines bought by country $i$ will be used by country $i$. 
The number of vaccines in this case will be given by 
$$Vn_i= n_d \xi_i$$,
where $n_d$ is the number of days the country will keep vaccinating.
In our second model we allow for countries to donate their vaccines to other country in order to further minimize thier gain function.
![](https://i.imgur.com/yPtbz27.jpg)



## The model

Matrix $D$ represents importation (sum over elements in row $i$) and donation (off-diagonal) of vaccines. The model attempts to minimize a cost function, which represents the cost of having infections plus the cost of vaccines. The cost of vaccination to a country is proportional to the sum of the country's corresponding row in the $D$ matrix.

**The cost function to minimize:**

The optimization problem finds the entries of $D$ to minimize the cost function

$$C=\sum_i\left( \int I(t)\, dt+k_{i}\sum_j D_{ij}\right)$$

The wealth of a country can be encoded in the model by changing the constant representing the cost of the vaccine. For poorer countries, the vaccine is more expensive as a fraction of its total budget, meaning $k_\text{poor}>k_\text{rich}$. More concretely,

$$k_{i}=\frac{\text{cost}}{\text{GDP}\cdot k_{I}}$$

We assume that the social cost of an infection is the same regardless of the wealth of the country. It can therefore be factored out of the cost function and is absorbed into the costs of vaccination.

**Realistic values for the elements of $D$:**

**Realistic values for the $k_{i}$:**

## 2 nodes

A rich country and a poor country.

### asymmetric beta

### asymmetric budged/effective vaccine cost

### egoistic vs collaborative solution

## 3 nodes




# Game theory perspective

 In the game theory approach we are still interested in discussing how to minimize the functions $GF_i$ but from in the form of a competition between countries. Each country will have to chose the best rate of vaccines to minimaize its costs without  knowing what the other countries will do. The strategy chosen by a country influence the infecton rate of the other countries and therefore their optimal vaccination rate creating a competition.  
 A solution to this problem is given by the Nash equilibrium. The Nash equilibrum or equilibria will represent the best strategies the countries can adopt, assuming the other countries will make the same resonment. In our case the Nash equilibria for the first model will be given by any point $\xi^*$ solution  to the equations
  $$\text{min}_{\xi_i}\left(GF_i(\xi_1^*,\cdots,\xi_i,\cdots,\xi_M^*)\right)=GF_i(\xi^*).$$
  In the game theory language $\xi_i$ is called the strategy of the agent $i$. In the second model the strategy of country $i$ will be represented by the row $D_i=(D_{i1},\cdots,D_{iM})$.  The Nash equilibrium will be given by the point $D^*$ solution  to the equations
 $$min_{D_i}(GF_i(D_1\cdots,D_i,\cdots,D_M))=GF_i(D^*).$$
 
 
 The Nash equilibrium is in general different from the minimum for the world gain function $WGF$ which in the first model is given by 
$$min_\xi\left(WGF(\xi)\right)= min_\xi\left(\sum_i GF_i(\xi)\right),$$
and in the second model is given by
 $$min_D\left(WGF(D)\right)= min_D\left(\sum_i GF_i(D)\right).$$
 
The global minimum of $WGF$ can be interpred as the maximal cooperative solution, since correspond to the game scenarion in which all countries care only for the global welfare.

Note that the existence or the uniqueness Nash equilibrium is not garanteed. In the case  more then one or no Nash equilibra exist we have that there is not a best strategy and and uniqueness is obtained by introducing some degree of comunication between the agents.  

 

 
 
 
 

### NOTATION (MIC)
$D^*=(D^*_{-i},D^*_i)$
example: $x=(a,b,c), x_{-1}=(b,c)$ while $x_1=(a)$.
You have a matrix but the notation is the same:
$\bar{\bar{D}}=(\bar{D}_1,\bar{D}_2,\ldots)^T$.


