COVID-19 forecasting in the era of vaccines and variants

# COVID-19 forecasting in the era of vaccines and variants
### Irena Papst, MSc PhD Candidate Cornell University

---

These slides are available at

[papsti.github.io/talks/2021-08-10_phd-defence.html](https:/papsti.github.io/talks/2021-08-10_phd-defence.html)

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- in case you want to **follow along** 
- there are **hyperlinks** that may be of interest sprinkled throughout the talk
- in case i go though too quickly and you want to **flip back** to a previously shown slide

---

I'm giving this talk from Mississauga, Ontario, Canada, and so I would like to begin by acknowledging that the land on which I'm speaking today is part of the **Treaty Lands and Territory of the Mississaugas of the Credit**.

For thousands of years, Indigenous peoples inhabited and cared for this land, and continue to do so today. In particular I acknowledge the territory of the **Anishinabek**, **Huron-Wendat**, **Haudenosaunee** and **Ojibway/Chippewa** peoples; the land that is home to the **Metis**; and most recently, the territory of the **Mississaugas of the Credit First Nation** who are direct descendants of the Mississaugas of the Credit.

I am grateful to have the opportunity to live and work on this land, and by doing so, give my respect to its first inhabitants.

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- Indigenous land acknowledgment statement adapted from the [Region of Peel's statement](https://www.peelregion.ca/council/indigenous.asp)

---

# Mathematical modelling of infectious disease dynamics

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- for this talk, i'd like to start by giving a **brief overview of the other work** i've done for my dissertation, and then spend the majority of the time on my most recent project concerning COVID-19 modelling and forecasting
- the **central theme** of my dissertation is mathematical modelling of infectious disease dynamics

## from recurrence to emergence

???
-----
- and the work i've done for my thesis can be **split into two parts**: 
  - the first concerning recurrent epidemics 
  - and the second concerning an emergent pathogen, which you've probably already guessed is SARS-CoV-2

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## I. Recurrence

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- with respect to **recurrence**, the first project i did was

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- this paper here, published in **2019**

**Context:** School terms **amplify spread** of childhood infectious diseases (such as measles, whooping cough, and rubella).

???
-----
- the context of this work is ...
- because schools **increase contact rates** among children
- school terms act as a **source of periodic forcing**, which, **historically** has contributed to **recurrent** epidemics of these diseases, especially in the era **before mass vaccination**

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## I. Recurrence

**Context:** School terms **amplify spread** of childhood infectious diseases (such as measles, whooping cough, and rubella).

**Question:** Can we model **messy school-term forcing** of childhood infectious diseases using an **idealized sinusoid**?

???

- the question we were investigating is whether we can use a **mathematically convenient sinusoid** to model messy school-term forcing
- **up here**, have a model for school term forcing if transmission is **constant and high** when school in session, and low when schools closed for **holidays**, representative of the school terms as they actually happen
- contrasted with a **nice sinusoid**, which makes the system easier to analyse mathematically
- can we **interchange** these safely when modelling these childhood diseases?
- **not really obvious** that we could given how different the shapes of these two functions are

**Conclusion:** Yes, provided the forcing **amplitude is adjusted**, and the **invariance** in long-term model behavior is surprising.

???
-----
- **based on our work**, we concluded that yes, we **absolutely can** use a sinusoid instead of a messier piecewise forcing function
- though the **forcing amplitude** must be adjusted, and we describe how to do so in the paper
- we also demonstrate a **surprising invariance** in long-term model behavior between models whose forcing functions have very different shapes
- we also show that this invariance isn't limited to the epidemiological model we were originally studying, but it also appears in a **qualitatively different predator-prey model**

---

## I. Recurrence

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- my second project about recurrent epidemics is currently available **as a preprint**, though it's been submitted to a journal and we've received relatively **minor revisions** which i'm currently working on
- here we were interested in seasonal influenza, and specifically about **vaccination decisions**

**Context:** Every year, many individuals choose whether or not to receive a **flu shot**, and the way in which they make their choice isn't entirely **"rational"**.

???
----
- the context is that ...
- partially because people often don't have **all of the information** they would need to make a perfectly rational choice (e.g. probability of infection, risk of flu complications/infection someone immunocompromised, likelihood of adverse reaction)

**Questions:** How do flu model predictions change if **vaccination decision-making** is more **realistic**? Can the population ever **self-organize** into herd immunity?

???
-----
- how do...
- if we incorporated a more **realistic decision-making framework**
- can the population...
- given **voluntary vaccination**

**Conclusion:** Model results are more **nuanced** with more realistic vaccination decisions. Self-organized herd immunity is attainable if the vaccine is **perceived as costless**.

???
-----
- model results...
- compared to previous work relying on **the rationality assumption**
- SOHI...
- i'll just mention here since it's probably on your minds: this work **pre-dates** COVID-19, and so we didn't consider what seem to be **drivers** of COVID-19 vaccine hesitancy, like **misinformation**, but that's certainly something that could be added to the vaccine decision-making framework we set up with this work

---

## II. Emergence

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- the second part of my dissertation concerns the **emergence** of a new infectious disease: **COVID-19**

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- the first project in this part was published in **2021**

**Context:** COVID-19 severity varies somewhat mysteriously, but seems to **scale with age**. Understanding this relationship is important for **COVID-19 models and forecasts**.

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-----
- in the summer of 2020, **when this project started**, when we were looking at the COVID-19 data for Ontario, we realized that there was a **gap in the literature**
- the context is ...
- important when considering whether outbreaks from, for instance, reopening, are likely to **challenge healthcare capacity** depending on population demographics

**Question:** **Age distributions** for cases and deaths were readily available, but how did those compare to distributions of **severe outcomes**?

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-----
- we knew a lot about the ages of people **getting infected** and people **dying** from COVID-19, but we didn't know a lot about the ages of those **battling severe cases** of COVID-19

---

## II. Emergence

**Context:** COVID-19 severity varies mysteriously, but seems to **scale with age**. Understanding this relationship is important for **COVID-19 models and forecasts**.

**Question:** **Age distributions** for cases and deaths were readily available, but how did those compare to distributions of **severe outcomes**?

???
- so we looked at the ages of  **hospitalizations, ICU admissions, intubations, ventilations** in Ontario, Canada 
- hospitalizations in blue, top panel, other outcomes nested within, and age groups on the x-axis

**Conclusion:** In Ontario, severe outcomes represented a **much broader age-range** than deaths, so **deaths could increase in younger ages** if healthcare were overwhelmed.

???
-----
- these severe outcomes represent a much **broader age range** than deaths, bottom panel 
- so the **need** for serious healthcare interventions exists in **younger ages than** those represented by deaths alone
- so we concluded that if the healthcare system **were to be overwhelmed** by a wave of infection (has been a consistent fear throughout this pandemic)
- and if the need for care **was not met** in younger individuals, deaths could **start occurring more often in younger individuals** than had been previously observed
-----
- the fourth and final project in my dissertation is my **most recent project**: the one i'll **focus on** for the rest of this talk

---

## II. Emergence

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- this work has been drafted and will be submitted to a journal at some point in the next few months

**Context:** The COVID-19 pandemic **continues to evolve** and short-term forecasting is still needed to inform public health planning.

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- this project was **borne out of the need** to continue to provide useful short-term forecasts to help inform public health planning during the **continually evolving COVID-19 pandemic**

**Question:** How do we ensure COVID-19 models continue to provide **useful and informative forecasts**?

???
- so the question i'll try to answer today is ...
- the way we've approached this problem is by

---

# Extended model

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- extending our model **as the pandemic has evolved**
- **two key extensions** that i'll focus on today are

## Vaccination

## Variants of Concern

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-----
- but before we talk about the extended model, we need a bit of

---

# Background

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## Forecasting is an important tool

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- short-term forecasting of COVID-19 has proven itself to be an **enormously useful** tool for **public health planning** throughout this pandemic

Source: [Public Health Agency of Canada](https://www.canada.ca/content/dam/phac-aspc/documents/services/diseases-maladies/coronavirus-disease-covid-19/epidemiological-economic-research-data/update-covid-19-canada-epidemiology-modelling-20210730-en.pdf)

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-----
- here's a **recent example** from a public briefing by **Dr. Theresa Tam, Canada's Chief  Public Health Officer**, showing forecasts for the **late summer** and **early fall** in Canada
- looking at the **number of reported cases we can expect** in Canada in the short-term, depending on whether we **increase our contacts with other people**, or **maintain the current level of interaction**
- this forecast was prepared by one of my collaborators, Mike Li, at the Public Health Agency of Canada, using the **model that i've been working on developing further** for this final project in my dissertation

---

[https://github.com/mac-theobio/McMasterPandemic](https://github.com/mac-theobio/McMasterPandemic)

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- i'll also mention that the model and forecasting software are implemented as a **publicly-available R package**, which you can find on **Github**

---

## Base model

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- let's start with the **base model** that we've been using to make COVID-19 forecasts

We use a **compartmental** epidemiological model (SEIR framework).

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- SEIR refers to the main compartments in this model: susceptible, exposed, infected, recovered
- the population is **partitioned** into one of these states
- **everyone starts in the susceptible class **and then individuals can get **exposed** to (and therefore infected by) SARS-CoV-2
- but they aren't infectious until they enter **one of the four infectious classes** representing different infection types important for COVID-19
- the first two are:
1. **asymptomatic**, where you don't show symptoms but can still infected others (albeit in a reduced capacity)
2. **pre-symptomatic**, where you will eventually develop symptoms
- asymptomatic infections simply recover
- while presymptomatic infections turn into either mild or severe infections 
- **mild** infections recover and never result in hospitalization or death
- while **severe** infections always result in either hospitalization 
or death before hospitalization
- and then we have some downstream outcomes important to modelling the healthcare system, which we won't focus on today
-----
- each of these arrows represents a **flow rate** specified with various parameters

Model is encoded in a system of **ordinary differential equations**.

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- and the model... which I have as a **supplementary slide** if anyone wants to see them in the question period
- i will also mention that, while the underlying model is deterministic, we add stochastic elements to the simulation software to be able to estimate confidence intervals around our forecasts
- so that's the abstract model, but how do we actually use to it forecast?

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## Calibrating transmission

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- the key is to understand what's **happening with transmission** by calibrating, **or fitting**, a transmission rate

We both fit to and forecast COVID-19 infection **reports**:

- **underestimate** of actual infections
  - **delay** of 2-3 weeks
  - depend on **testing availability** and **behavior**
  
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- both **fit** and **forecast**

**Question:** How many **new infections** will we *observe* each day?

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-----
- **question**

**Answer:** It depends on **transmission**, which converts susceptibles `$S$` to exposeds `$E$`:

$$
\quad  S(t)  \xrightarrow{(\color{#B31B1B}{\text{force of infection}}) \times S(t)} E(t)
$$

???
-----
- the **per capita rate** with which this **conversion** happens is called the **force of infection**

$$
\color{#B31B1B}{\text{force of infection} = \beta(t) \times \frac{I(t)}{N}}
$$
???
-----
- the **basic structure** of the force of infection in our model

- `$\color{#B31B1B}{\beta(t)}$`: time-dependent transmission rate (# of **transmission opportunities** per individual per day, depends on the average **contact rate** between individuals)

???
-----
- **transmission opportunities**

- `$\color{#B31B1B}{\frac{I(t)}{N}}$`: probability of **encountering an infective** during a transmission opportunity at time `$\color{#B31B1B}{t}$`

???
-----
- probability of encountering an infective (**encounters are random**)
- this is a **simplified version** of the force of infection used in the base model
- we also **facet by infection type** (asymptomatic, presymptomatic...)
- i'm just **bundling** all infection types into this `$I(t)$` term - total infectives at time `$t$` - for simplicity right now

---

## Calibrating transmission

We **fit `$\beta(t)$`** as a **piecewise-constant** function to available infection report data.

???
- we fit...
- to make transmission **time-varying**, we

We re-estimate `$\color{#B31B1B}{\beta}$` when there has been a **significant change** in transmission-related behavior (e.g. restrictions loosening, stay-at-home order):

--
 
<img src="data:image/png;base64,#figs_files/figure-html/betademo-1.png" width="1344" style="display: block; margin: auto;" />

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-----
- this is a **demo** of what beta might look like
  
---

## Forecasting transmission

We make **assumptions** on `$\color{#B31B1B}{\beta(t)}$` in the forecast period.
  
???
- we make...

---

## Forecasting transmission

We make **assumptions** on `$\color{#B31B1B}{\beta(t)}$` in the forecast period.

Start with **latest** `$\color{#B31B1B}{\beta}$` (from last estimation window).

???
- start with...

---

## Forecasting transmission

We make **assumptions** on `$\color{#B31B1B}{\beta(t)}$` in the forecast period.

Start with **latest** `$\color{#B31B1B}{\beta}$` (from last estimation window).

In **forecast period**, either:

???
- in forecast period...

---

## Forecasting transmission

We make **assumptions** on `$\color{#B31B1B}{\beta(t)}$` in the forecast period.

Start with **latest** `$\color{#B31B1B}{\beta}$` (from last estimation window).

In **forecast period**, either:

1. use latest `$\color{#B31B1B}{\beta}$` (**status quo** forecast)

???
- use latest...

---
  
## Forecasting transmission

We make **assumptions** on `$\color{#B31B1B}{\beta(t)}$` in the forecast period.

Start with **latest** `$\color{#B31B1B}{\beta}$` (from last estimation window).

In **forecast period**, either:

1. use latest `$\color{#B31B1B}{\beta}$` (**status quo** forecast)

2. increase (or decrease) transmission by **x%** relative to latest `$\color{#B31B1B}{\beta}$`
  
???
- increase (or decrease)...
- that's the general procedure we employ

---

# Results

???
- so that was the **background**, and **instead of** showing you the extended model and methods first, allow me to **whet your appetite** with the results, which are

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# Forecast from 21 Feb 2021 in Ontario

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- examples of some forecasts that we've been making for ontario
- this way, i can **highlight the value** of the extended model **before delving** into the details
- the first forecast is from **late february**
- **what was going on** in Ontario at the time?

---

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- in the **lead up** to the forecast date, marked here with a **dashed line**, we saw a slow and steady increase in **infection reports** 
- and while **2000/3000 infections** reported in a province of almost **15 million** may not seem like much, the **strain** on the healthcare system was beginning to show

---

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-----
- you can see **ICU occupancy** had also been rising with infection reports

---

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- and we had actually surpassed the **pre-COVID ICU capacity** threshold for the entire province
- while we did expand ICU capacity in response to the pandemic, it was achieved through **hugely disruptive measures**, including **cancellations of "elective" surgeries** that sound optional, but may actually be lifesaving down the line
- this was the beginning of a **potential crisis** in our hospitals, and we had to act

---

???
- in response to rising infection reports and ICU occupancy, the government implemented **several interventions**, namely
- lockdowns in Toronto and Peel in late november, the **two most populous health regions in the province** (representing about 30% of the provincial population)
- followed by a **province-wide shutdown** on 26 december 
- and then ultimately a **province-wide stay-at-home order** in mid-January

---

???
- the combination of these interventions finally began **decreasing** infection reports and **pretty sharply** compared to the original incline, though **decline in ICU occupancy** was **slower**

---

???
- the stay-at-home order was **set to expire** in early march, and since things were looking good, it was looking **likely** that the province **wouldn't extend it**
- so the **question** at the time (as it usually is in these forecasts) was

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# Is it safe to loosen restrictions?

---

???
- **just** looking at the trends in infection reports and ICU occupancy, you may think *"yes, it should be safe to reopen!"*
- ICU occupancy was **still high** but **trending downward**, and we know the ICU trend is **lagged** from infection reports---people enter the ICU some time after testing positive and then they may stay in the ICU for a while
- but there's **another piece of the puzzle** that isn't clear from infection reports or ICU occupancy

---

???
- **under the hood**, the Alpha varaint (aka strain **B.1.1.7**) had made its way into the province
- this variant is about **50% more transmissible** compared to resident strains at the time
- it was first confirmed in the province **coincidentally** on the same day as the province-wide shutdown, **26 december**
- by the forecast date, it represented about **20%** of infection reports
- we were particularly concerned about Alpha because **despite** the fact that our public health interventions were **successfully driving infection reports down pretty rapidly**, Alpha was finding a way to **increase its frequency** over this period... _scary!_

Trends in infection reports and ICU occupancy favored reopening, while Alpha threatened it.

---

???
- so we fit transmission to the infection report data (the **black curve**) and then
- so we made **two forecasts** at the time:
  1. one where we **explicitly modelled the increasing frequency** of the Alpha variant through both the fit and the forecast period (**extended model**)
  2. another where we detected **whatever effect** the variant was having on infection reports by fitting the transmission rate as usual, then we just **propagated that effect forward**, with no change to Alpha frequency going forward (**base model**)
- the curve is the median forecast in ecah case and the band around each curve represents 95% confidence intervals
- and here is **what happened** actually happened

---

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- these **hollow points** are still infection reports, just those observed after the forecast date
- you can see that the forecast using the **extended model**, with the effect of the variant, **caught the third wave takeoff**
- while the forecast using the base model showed some take-off, but **grossly underestimated the magnitude of the uptick** resulting from the measures relaxing

Infection reports *alone* could not signal the start of the third wave.

???
- infection reports alone...
- and that **including the variant** in our model was **crucial** for this forecast

---

# Forecast from 5 Jun 2021 in Ontario

---

???
- so **fast-forward** now to another forecast in **june**
- for reference, here is the **previous forecast date**
- this was the start of the **third *wave** of infection in the province, **driven by** the Alpha variant
- by the beginning of april, we had almost hit the **previous peak** of infection reports from January, which was around 4000
- but even worse, **ICU**s didn't get chance to empty out from the second wave and the numbers just kept climbing

---

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- so the province instituted another stay-at-home order in the **beginning of april**

---

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- which again succeeded in bringing infection reports down, though the **ICU debt** was much more severe
- we ended up at **triple the pre-COVID capacity**, and the situation was very dire
- doctors were **on the verge** of having to decide **who would get care and who wouldn't**
- patients were being **flown out of province** for critical care

---

???
- by the forecast date, we were again in a similar position where **the stay-at-home order was expiring**, and the province was planning to enter **step 1** of its latest reopening plan, so we were once again thinking about whether

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# Is it safe to loosen restrictions?

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- it would be safe to loosen restrictions

---

???
- although ICU occupancy was **still high**, we were **less worried** about a hospital crisis at this point because

---

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- we had **vaccines**!
- the provinces' mass vaccination effort took off in **early 2021**
- we actually **prioritized** getting **first doses** in as many arms as possible, initially planning to delay second doses by **up to 16 weeks**
- in practice, the delay ended up being about **8-12 weeks** for most people, which turns out to be very protective, possibly more protective than the minimum 3 week interval
- though this the delay posed a risk because...

---

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- the Delta variant had **arrived** in the province

---

???
- Alpha had become **dominant** in mid-march
- and while surveillence for **Delta** variant was a bit patchier, we believe Delta began to make up a majority of the share of "non-Alpha" strains in early May
- so although we had **mass vaccination taking off**, we had a new variant that was 50% more transmissible than _Alpha_, so posed to **take it over** as the most dominant strain 
- but most problematically,

First-dose vaccine-efficacy against Delta: 30%

???
-----
- Delta seemed to decrease vaccine efficiacy, **especially first dose efficacy** to about 30% against symptomatic disease
- with such a low first-dose efficacy, it wasn't clear whether the the **first-dose strategy** would be **undermined** by a strain that could still be **spread by partially vaccinated people**
- one dose of the vaccine would certainly reduce the amount of severe disease to an extent, but if it wasn't **blocking transmission** sufficiently, the virus could potentially **find holes** in the population and take off again

---

Trends in infection reports, ICU occupancy, and vaccination favored reopening, while Delta threatened it.

???
- so trends in...

---

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- here is the **set of forecasts** we produced in early june, with the extended model featuring two-dose vaccination and increasing frequency of the delta variant
- we had to consider scenarios for the effect of **step 1 reopening** on **transmission** (colour of each line)
- but we also had to consider scenarios for the **vaccination effort**, as it seemed poised to **continue expanding** based on announced supply increases at the time (line type)
- for reference, the **180k doses/day** scenario is most representative of what actually happened
- we saw that although step 1 reopening would be relatively safe if we kept transmission from increasing too much, **we could still see a fourth wave**, even with significantly **more vaccination**, which was a problem, because we were still coming down from that **massive ICU wave**, and we didn't want to challenge our hospital system any more, or risk the possiblity of **another stay-at-home order**

---

???
- and here's what happened
- this is not too surprising because step 1 was centred on **outdoor activities**, so although we felt that the low transmission scenarios were the most likely, we wanted to show that we **weren't impervious to a Delta wave** if we pursued a more aggressive reopening, as that's what we **saw in the UK** a few weeks before we made this forecast

Safe reopening was **possible** but not **guaranteed**.

???
- so the message was that...
- to **emphasize the effect** of including both vaccination and the Delta variant in our model, we can consider **counterfactual forecasts** where we omit one or both of these mechanisms, as we did with the february forecast, where we omited the explicit variant effect
- i'll just show **one counterfactural** forecast now in the interest of time

---

???
- here is the exact same forecast **without vaccination** but **with the Delta variant**

The **stay-at-home order** followed by **expedited vaccination** stifled Delta, even with the **first-dose strategy**.

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- vaccination helped... **even with Delta in the picture**

---

# Extended model

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- so i hope that i've **whetted your appetite** with those forecasts
- and now we can dig into the **extended model and methods**

---

# Vaccination

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- let's start with the vaccination model

---

background-image: url("data:image/png;base64,#figs/flowchart1.png")
background-position: center
background-size: contain

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- **recall** the base model 
- this is the **epidemiology** we're working with
- for vaccination, we're going to add a **layer for each vaccine state**

---

background-image: url("data:image/png;base64,#figs/flowchart2.png")
background-position: center
background-size: contain

???
- each vaccine stratum has **its own copy** of the epidemiological compartments
- there are **five vaccine strata**...
- when forecasting infection reports, the effect of vaccination will be most important in **transmission**  
- though vaccines will also reduce disease severity in our model)
- but implementing vaccination in disease transmission is a **little trickier** but for that reason it's also more interesting, so lets look at that now 
- vaccination will **modify the flow from susceptible to exposed**, or if you'll remember from earlier the...

---

## Force of infection

**Base model**:

`\begin{align}
\text{force of infection} &=  \beta \left( C_{\mathrm{a}} I_{\mathrm{a}} + C_{\mathrm{p}} I_{\mathrm{p}} + C_{\mathrm{m}} I_{\mathrm{m}} + C_{\mathrm{s}} I_{\mathrm{s}} \right)/N 
\end{align}`

<center>
a = asymptomatic, p = presymptomatic, m = mild, s = severe
</center>

???
- in the **base model**, the force of infection expands to the following
- this time, i've actually included the different infection types (**asymptomatic**, **presymptomatic**, **mild**, **severe**)
- the **C's** are factors that adjust for the relative contribution to infection from each infection type (**e.g. severe infections** contribute relatively less because they're likely isolating due to their symptoms)

**Factor** this as a matrix multiplication:

`\begin{align}
\text{force of infection} = \beta/N \begin{bmatrix} C_{\mathrm{a}} & C_{\mathrm{p}} & C_{\mathrm{m}} & C_{\mathrm{s}} \end{bmatrix} \begin{bmatrix} I_{\mathrm{a}} & I_{\mathrm{p}}  & I_{\mathrm{m}}  & I_{\mathrm{s}} \end{bmatrix}^{T}.
\end{align}`

???
- which will be **convenient** in a second

Expand transmission coefficients using **Kronecker product**.

???
- to include vaccination, we will expand these transmission coefficients using the **Kronecker product**

---

## Kronecker product expansion

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- here is the kronecker product
- basically taking the matrix `$B$` and distributing through `$A$` as a block being scalar multiplied by the entries of `$A$`

---

## Kronecker product expansion

Let `$B$` contain the **epidemiological** coefficients:

Let `$A$` contain coefficients for the expansion **subcategories** (*e.g.,* vaccination):

???
- this is a simplified version of the vaccination model, we're **omitting the layers where people are dosed but not protected** (that would just add two more rows to the matrix A)

Expanded coefficients:

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- this is a simplified example, but we use this trick to efficiently expand the model with vaccination and/or with age

---

## Expanded force of infection

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- instead of just `$B$` times the vector of infectives, we now have `$A$` kronecker `$B$` times the vector of infectives for the force of infection

---

# Variants of Concern

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- so that's the vaccination model
- the extended model also includes the...

---

# Variant model

???
- so which model did we use?

Let `$p(t)$` be the proportion of infection reports caused by the variant at time `$t$`. Then

$$
p(t) = \frac{p(0)}{p(0) + (1-p(0))e^{-\Delta r t}}
$$
???
- then we take p(t) to be **modelled by this function**, which depends on

- `$p(0)$` is the variant proportion **upon introduction** (at `$t=0$`)

- `$\Delta r$` is the **selective advantage** of the variant against the dominant strain

???
- the selective advantage, from evolutionary theory
- but as mathematicians, we can see that this is just a logistic curve and the parameter `$\Delta r$` dictates the **steepness of the curve** in this middle section

???
- as an example, here is the variant model we used for the february forecast
- now we didn't know, and don't often know, the initial variant prevalence, `$p(0)$`, with a lot of certainty
- instead, we usually **peg this curve** to a point when we believe the estimate of `$p(t)$` is particularly reliable
- so for instance, here, we pegged the variant model to a **point prevalence study** conducted by Public Health Ontario in late january, where they checked every single sample positive for SARS-CoV-2 from a single day for Alpha

---

# Variant model

Let `$p(t)$` be the proportion of infection reports caused by the variant at time `$t$`. Then

$$
p(t) = \frac{p(0)}{p(0) + (1-p(0))e^{-\Delta r t}}
$$

- `$p(0)$` is the variant proportion **upon introduction** (at `$t=0$`)

- `$\Delta r$` is the **selective advantage** of the variant against the dominant strain

Modify **force of infection** with the following factor:

- `$\theta$`: **transmission advantage factor** of variant relative to resident
 (e.g `$\theta = 1.5$` for **50%** more transmissible variant)

- `$\rm{VE}$`: **vaccine efficacy**

???
- once we have the variant proportion model, we can go ahead and modify the **force of infection** based on variant proportion and transmissibility advantage
- so we take a weighted average relative to the variant proportion of the **change to transmission** depending on the variant's **transmission advantage** over the resident strain as well as the **vaccine's efficacy** against the variant or the resident
- so **if the variant is 50% more transmissible** than the resident strain, the transmission advantage, `$\theta$` is 1.5
- we make this adjustment for each vaccination stratum, so when there is **no vaccination** (as in the february forecast), it's as if we only have the unvaccinated layer, where the vaccine efficacies are **both 0** and we just have...

---

# Looking ahead

---

# Age structure

???
- the last extension that's **likely to come into play** soon is age structure

---

.pull-left[
<img src="data:image/png;base64,#figs_files/figure-html/vax-by-age-1.png" width="576" style="display: block; margin: auto;" />
]

???
- here we have **vaccine coverage over time by age**, split into at least one dose and received two doses
- for the lines, **darker green** corresponds to **older ages**
- you can now also see that not only did we pursue a **first-dose strategy**, but our vaccine distribution primarily prioritized people by age, from **oldest to youngest**

.pull-right[
 
 
 
 
 
 
 
Vaccination is still restricted to those **12+** in Canada (18+ in other regions).

]

Vaccination is **saturating**, with **lower** coverage in **younger** age groups.

???
-----
- and at this point it's not about **supply** or the **prioritization by age**, because these curves are all saturating, it's about **demand**, or lack thereof

**Schools** set to reopen in the fall, increasing contacts among children.

### How do we **anticipate** issues with schools reopening?

---

## Limitations

???
- so what are some limitations of our work?

1. Infection reports cannot be used to **anticipate** changes in transmission.

--
  
  - **delayed** by 2-3 weeks from transmission
  
  - impose **scenarios** for transmission changes
  
???
- this is a bit of an art
- we've gotten **lucky** that the scenarios we've considered have ended up being representative of what happened

- use **mobility** data

???
- one idea is to use mobility data as a **proxy** for transmission increases (not delayed, like infection reports)
- this is something that my collaborators have worked on

1. **Population-level** models breaks down as population becomes more **heterogeneous**.

- **vaccine** coverage may be patchy

???
- we already know there's heterogeneity in vaccine coverage **by age** but it may also vary **by region**, depending on demographics and difference attitudes toward the pandemic and vaccine
- big problem in the US right now

- outbreaks may become **more localized**

???
- so models may not perform well over large regions (e.g. an entire state) and should instead be run on smaller regions (e.g. at the county level)

1. **Testing behavior** varies over time.

- integrate with existing **testing model**

???
- testing **demand** changes over time
- testing has been **reactive**, prompted by symptoms (and in fact, you must have symptoms to be tested in general in Ontario, ta least), the likelihood of which which varies with vaccines and variants
- my collaborators have worked on a testing layer to the model, but we have yet to integrate it with the other extensions i've discussed today

---

## Summary

**Context:** The COVID-19 pandemic **continues to evolve** and short-term forecasting is still needed to inform public health planning.

**Question:** How do we ensure COVID-19 models continue to provide **useful and informative forecasts**?

???
- so to recap this work

---

## Summary

**Context:** The COVID-19 pandemic **continues to evolve** and short-term forecasting isstill needed to inform public health planning.

**Question:** How do we ensure COVID-19 models continue to provide **useful and informative forecasts**?

**Conclusion:** Look **beyond infection reports**, monitor the **global situation**, and update models according to **changes in the epidemiological landscape**.

---

# Thank you for listening!

---

# Acknowledgements

???
- doing a PhD is somehow simultaneously both very isolating and utterly impossible to do without the support of others
- i'd like to take a few minutes now to acknowledge some people that have been very important to me over the course of doctoral journey

---

???
- starting with the people i've been working with closely over the last year or so, primarily on the COVID-related work you saw today

.pull-left[
*McMasterPandemic Team:* Ben Bolker, Jonathan Dushoff, David Earn, Morgan Kain, Zachary Levine, Michael Li, Matthew So, Steve Walker

{{content}}
]

???
- led by ben, jonathan, and david

*Public Health Agency of Canada:* Michael Li, David Champredon, Aamir Fazil

???
-----
- PHAC, for helping guide the work that i presented today, as well as for funding it
- especially Mike, with whom i've worked closely on the forecasts you saw today

*McMaster Theoretical Biology Lab*

???
- for welcoming me back with open arms

----

Steve Strogatz

???
- my advisor, for all of his guidance during my time at cornell
- i'll forever cherish our fun and winding coversations during our meetings
--

Chris Myers, Richard Rand

???
- for serving on my special committee engaging with my dissertation work

Steve Ellner

???
- for working really hard to convincing me that cornell was the place to be

Erika Fowler-Decatur, David Bindel, Alex Vladimirsky, Karen Benavidez, Gennady Samorodnitsky

???
- who kept CAM running smoothly throughout my years in the program
- now on to other people who have made my time at cornell special

{{content}}
]

???
- my first real friend in ithaca, and such a pleasure to collaborate with

John Chavis

???
- one of the strongest people i've ever met and quite possibly the best hype man ever

*CAM students*, including Kath Landgren, Elizabeth Wesson, Aditya Vaidyanathan, Isabel Kloumann, Danielle Toupo, Sumedh Joshi, Evan Randles

???
- one of the main reasons i chose cornell was the student community in CAM that was obvious to me from just a few days spent visiting
- so i'd like to thank all of my fellow grad students in the program that made it such a wonderful place to spend the last few years, but i'd especially like to acknowledge...

David Mimno

???
- was a pleasure to teach with, even through a global pandemic, and taught me that we're all humans first, teachers/students/academics second
- and then there are a number of people who helped make ithaca home for the last few years

-----

Devon Lang, Jena Andres, Felice Doynov

???
- never failed to brighten my week when we'd get together

Marie MacDonald

???
- my weekend baking partner-in-crime

Andy Ruina, the Sewing Sunday crowd

???
- andy transformed his home into a sewing mecca every sunday, which is very quintessentially ithaca, and i miss it so much

David Shuman, Debbie Benzer

???
- the kindest and fairest landlords that i was lucky to have stumbled upon in one of my first visits to the city
- and then there are some people that have been important to my time at Cornell and beyond

---

### Lindsay Mercer

???
- hands down one of the best humans that i know

### Kate Sinclair

???
- with whom i can't wait to share a plate of nach again soon

### Jérémie Bérubé

???
- qui est une des personnes les plus tendres que j'aie jamais connues, et avec qui je me sense toujours à l'aise

### Madeleine Baker

???
- who is someone i'm proud to unironically call a bestie

### Spencer Hunt

???
- who became the sibling i never had

### Natalie Jaworski, Dimitri Podubni

???
- who i've now been friends with for nearly two decades, so they've seen me at my best *and* my most awkward

### David Earn

???
- who i have no better way to describe than my academic dad, and without whom i never would have ended up with a PhD from Cornell of all places, because i never would have aimed that high on my own

### Sanja, Mate, Marko Marković

???
- who i've been so lucky to consider family for a very long time

### Danica, Mirko, Gordana Aničić

???
- koji su me oduvijek tješili i podržavali

---

## Predrag & Ljiljana Papst

???
- finally, i absolutely would not be here without my parents, who left basically everyone and everything they had thousands of kilometers away to escape a civil war so that i might one day have the chance to graduate from a place like Cornell
- i'll never be able to thank them enough for all of the opportunities they've given me 
- volim vas do neba i natrag
- i dedicate this talk to you

---

# Thank you!

---

# Questions?

---

# Extras

---

## Base model equations

---

background-image: url("data:image/png;base64,#figs/fig4-3.png")
background-position: center
background-size: contain

### Reopening would have been *very* safe thanks to vaccination without the introduction of the Delta variant.

???
- here is another counterfactual forecast, this time where we omit the Delta variant but include vaccination

---

background-image: url("data:image/png;base64,#figs/fig4-4.png")
background-position: center
background-size: contain

### Even in the absence of the Delta variant, safe reopening would have been difficult without vaccination.

???
- and finally here is the counterfactual forecast with the **base model** (so no vaccination or variant)

---

## Age-structured force of infection

- `$i$`: age group of **susceptible**

- `$j$`: age group of **infective**

- `$x$`: **infection type** (asymptomatic, presymptomatic, mild, severe)

- `$p_{ij}$`: **probability** that a contact for age group `$i$` is with age group `$j$`

---

background-image: url("data:image/png;base64,#figs/fig5-1.png")
background-position: center
background-size: contain

## Age-structured projections

???
- example age structured projection
- forecast date is also 5 june (as with the second set of forecasts we looked at before)
- forecast is the "worst case" previously considered (highest transmission scenario upon step 1 reopening, lowest vax rate considered)
- since we have age-structure, we can break down the projection by age group
- just considered three
- note these are not raw infection reports anymore but we're normalizing by the size of each sub-population so we can compare panels head to head 
- see reasonably good agreement between the projection and the actual for ages over 12, but under 11s we have an over-prediction
- there may be more underreporting in younger age groups (actually supported by the work in chapter 3 of my dissertation)
- note too that there is an overprediction for the april wave in seniors---this model does not distribute vaccines from oldest to youngest, as the province did, vax distribution is random, so the model will have less vax coverage in seniors at this point than we actually did

---

Source: [Ontario Science Advisory Table](https://covid19-sciencetable.ca/ontario-dashboard/)

---

<img src="data:image/png;base64,#figs/vaxmap.png" width="600" style="display: block; margin: auto;" />
 
Source: [Public Health Ontario](https://www.publichealthontario.ca/en/data-and-analysis/infectious-disease/covid-19-data-surveillance/covid-19-data-tool?tab=maps)

---

<img src="data:image/png;base64,#figs/casemap.png" width="600" style="display: block; margin: auto;" />
 
Source: [Public Health Ontario](https://www.publichealthontario.ca/en/data-and-analysis/infectious-disease/covid-19-data-surveillance/covid-19-data-tool?tab=maps)

---

## Testing model