---
# System prepended metadata

title: A Further Improved Algorithm
tags: [others]

---

# A Further Improved Algorithm
###### tags: `others`

[TOC]

## The Original

![](https://i.imgur.com/ebEETz9.png)
![](https://i.imgur.com/knfeBaL.png)


![](https://i.imgur.com/gDAc8MF.png)
- black - one contact group
- red - susceptible members
- orange - RBT for each age

## The Improved

- compute $A_g=[n_{g,1},\ldots,n_{g,N_{age}}]$ for each contact group $g$ with nonzero $A_g$
    - $N_{age}=|Age|$ is the number of age groups
    - $n_{g,i}=|\{u\mid u\in I\cap g,age(u)=i\}|$, that is, infected individual $u$ is in the contact group $g$ in this period(so it is attempting to infect susceptible individuals in contact group $g$), and grouped by age


### Algorithm

Given a contact group $g$ with computed $A_g$ and $C=\{c_{i,j}\mid i,j\in Age\}$ be contact matrix(prob from $i$ to $j$)

- let $p_i$ be $1-\prod\limits_{j\in Age}(1-c_{j,i})^{n_{g,j}}$ for $i\in Age$
- randomly choose $B(||)$ members among $g$ of age $i$

### Improvement
e.g.

- only one age, $N_{age}=1$, $c_{i,j}=p$
- $n$ infectors
- then org we have to
    - $n$ infection with $K=p|g|$
    - cost $np|g|$
- and improved
    - one infection with $K=(1-(1-p)^n)|g|\leq (1-(1-np))|g|=np|g|$
    - cost less than $np|g|$