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# System prepended metadata

title: A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2
tags: [articles]

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# A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2
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[link](https://science-sciencemag-org.ezproxy.lib.nctu.edu.tw/content/369/6505/846.abstract)

###### tags: `articles`

## Introduction

1. Introduce a heterogeneous model
2. The result of the heterogeneous model shows herd immunity is less than homogeneous model
3. How preventing measures affects the epidemic

## Homogeneous Model

$SI, SIR, SEIR...$ we've seen before

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| Symbol | Term                          | Definition                                                                                                                               |
|:------:|:----------------------------- |:---------------------------------------------------------------------------------------------------------------------------------------- |
| $R_0$  | Basic Reproduction number           | Number of secondary cases generated by a typical infectious individual when the rest of the population is susceptible                    |
| $v_c$  | Critical vaccination coverage | Proportion of the population that must be vaccinated to achieve herd immunity threshold, assuming that vaccination takes place at random |
|  $E$   | Vaccine efficacy         | The percentage reduction of disease in a vaccinated group of people compared to an unvaccinated group                   |


|     | w/o vaccine | w/ vaccine       |
| --- | ----------- | ---------------- |
| $R$ | $R_0$       | $(1-v)R_0$ |
| Immunity |    $h_c = 1 - {1 \over R_0}$         |$v_c = E^{-1}(1 - {1 \over R_0})$                  |

## Population Heterogeneity Model

On the basis of **SEIR model**, we add a feature.

`Partition the population by age and activity level`

### The Model

:::info
**Notations.**

* Label the types (combination of age and activty level) from $1$ to $m$
* For all $j \in \{1, ... , m \}$, the population consists of $n_j$ people of type $j$
* Denotes
    *  $\pi_j = {n_j / n}$ as proportion of type $j$
    * $a_{jk}$ as how much an $j$ individual has contact with a specific $k$ individual
    * $\alpha$ as scaling parameter, quantify the impact of control measures
    * $\mu$ as recovery rate of infected individuals
    * $\sigma$ as rate of individuals become infectious
* Set
    * $s_j(t) = S_j(t)/n_j$
    * $e_j(t) = E_j(t)/n_j$
    * $i_j(t) = I_j(t)/n_j$
    * $r_j(t) = R_j(t)/n_j$
:::

* For $j,k \in \{1,...,m\}$, every given person of type $j$ makes infectious contacts with every given person of type $k$ at rate $\alpha a_{jk}/n$

* The expected number of people of type $k$ infected by person of type $j$
is $n_k \times (\alpha a_{jk} / n) \times (1/ \mu) = \alpha \pi_k a_{jk} / \mu$, <br> where $1/ \mu$ is the expected duration of an infectious period

* The $m \times m$ next-generation matrix $M$ has $\alpha \pi_k a_{jk} / \mu$ as element in $j$-th row and $k$-th column

* The basic reproduction number $R_0$ is given by the largest eigenvalue of $M$ [[1]](https://ieeexplore.ieee.org/document/8191446)

#### The system of differential equations

![](https://i.imgur.com/5pzEhA7.png)

* The final fractions of the different groups in the population becoming infected are obtained by solving the equations above.

## Results
![](https://i.imgur.com/ybeWmP1.png)
- This was obtained by assuming that
    - Preventive measures having factor $\alpha \lt 1$ are implemented at the start of an epidemic and then exposing the population to a second epidemic with $\alpha=1$. 
    - We obtain $\alpha_*$ , the greatest value of $\alpha$ such that a second epidemic cannot occur. 
    - $h_D$ is given by the fraction of the population that is infected by the first epidemic. 

![](https://i.imgur.com/JUl07yv.png)
![](https://i.imgur.com/zbwNTHG.png)
![](https://i.imgur.com/PdAdy6e.png)
### Gradual lifting of restrictions
- Consider: restrictions are relaxed gradually linearly between June 1 (day 105) and August 31 (day 195).
![](https://i.imgur.com/LEFwGEx.png)
![](https://i.imgur.com/QAXUVVh.png)
- Figure S3 shows the eﬀective $R_0$ as a function of time $t$
![](https://i.imgur.com/HGxhucr.png)

## Discussion
- Our simple model shows how the $h_D$ may be substantially lower than the $h_C$  .
- It seems reasonable to assume that additional heterogeneities will have the effect of lowering the $h_D$ even further
- One assumption of our model is that preventive measures act proportionally on all contact.
- It is not obvious what effect school closure and WFH would have on the herd immunity level.
- The impact of allowing people to change their their activity levels over time is still unknown.
- In model, we assume that infection with and subsequent clearance of the virus leads to immunity. 
- Vaccination policies selecting [example](https://www.sciencedirect.com/science/article/abs/pii/S002555640400104X)

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## Further Reading
trc@iis.sinica.edu.tw

Shen:
1.[COVID-19 herd immunity: where are we?](https://www-nature-com.ezproxy.lib.nctu.edu.tw/articles/s41577-020-00451-5?fbclid=IwAR1XjD0YOvTNAXN5Lw4VureGOzzxGfvXRwx4lpoxUaMysAq_4b_D51l0gUw)
2.[Challenges in creating herd immunity to SARS-CoV-2 infection by mass vaccination](https://www.thelancet.com/journals/lancet/article/PIIS0140-6736(20)32318-7/fulltext?utm_campaign=cover20&utm_content=146747890&utm_medium=social&utm_source=twitter&hss_channel=tw-27013292)
3. [Model-informed COVID-19 vaccine prioritization strategies by age and serostatus](https://science-sciencemag-org.ezproxy.lib.nctu.edu.tw/content/371/6532/916.full?_ga=2.147303287.152673584.1628583565-2085592763.1625074585) (非引用 另外找的)

Ting-En Liao:
1. [Uniqueness of Nash equilibrium in vaccination games](https://www.tandfonline.com/doi/full/10.1080/17513758.2016.1213319)
2. [Vaccine escape in a heterogeneous population: insights for SARS-CoV-2 from a simple model](https://royalsocietypublishing.org/doi/full/10.1098/rsos.210530)

我:
*[L. Zdeborová, P. Zhang, H.J. Zhou  "Fast and simple decycling and dismantling of networks" Scientific Reports 6, 37954 (2016)](https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=Fast+and+simple+decycling+and+dismantling+of+networks&btnG=)*

Chih-Cheng Liao:
1. [Epidemiological and evolutionary considerations of SARS-CoV-2 vaccine dosing regimes](https://science.sciencemag.org/content/372/6540/363/tab-pdf)
2. [Trajectory of individual immunity and vaccination required for SARS-CoV-2 community immunity: a conceptual investigation](https://royalsocietypublishing.org/doi/full/10.1098/rsif.2020.0683)

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