Covid 수리모델링

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# Covid 수리모델링
## 서울시 확진자데이터 이용
### 김진섭; <a href="https://github.com/jinseob2kim"><span><i class="fab  fa-github faa-float animated "></i>jinseob2kim</span></a>
### 차라투(주)
### 2021-02-08

---

<div class="my-footer"><span><a href="https://www.zarathu.com">Zarathu Co.,Ltd</a>   
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;
<a href="https://github.com/jinseob2kim">김진섭</a></span></div>

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# Executive summary

- 확진자수 등 **실제 데이터를 활용**, parameter 들을 추정.

- SEIR 에 서울시 확진자수를 적용, **시간에 따라 변화하는 감염률**을 계산함.

- Parameter와 그 파생지표의 범위를 제한하고 신뢰구간을 계산하기 위해, **베이지안통계** 이용 예정. 
]

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class: center, middle

# 감염병 수리모형 소개

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# SIR model

---
# `$\beta$`, `$\gamma$`

`$\gamma$`: 회복률

`$$\frac{1}{\gamma}: \text{평균회복시간(일)}$$`

감염재생산수 `$R_0 = \frac{\beta}{\gamma}$`
- `$> 1$` 유행, `$< 1$` 소멸
]

---
# [SEIR model](https://covid19.uclaml.org/model.html)

$$
`\begin{align}
      \frac{d S}{d t}&= -\frac{\beta I S}{N}\\
      \frac{d E}{d t}&= \frac{\beta I S}{N} -\theta E\\
      \frac{d I}{d t}&= \theta E - \gamma I\\
      \frac{d R}{d t}&=\gamma I
\end{align}`
$$
]

---
# Simulation
.large[
https://benbennben.github.io/covid-19/

- 미분방정식(사실은 점화식) 은 초기값( `$S, I, R$` ), parameter( `$\beta$`, `$\gamma$` )가 정해지면 단순계산으로 구할 수 있다.

- 그럼 끝?

]

- 서울시분석: 회복기간( `$\frac{1}{\gamma}$` ) 6.8일, 잠재기( `$\frac{1}{\theta}$` ) 3.5 일 가정, `$\beta$` 는 추정.
]

---
# 보도자료

https://www.nrc.re.kr/board.es?mid=a10301000000&bid=0008&list_no=171543&act=view&nPage=1&otp_id=

---
# 추정 과정

2. SEIR 모델: `$\beta$` 만 추정( `$\gamma$`, `$\theta$` 고정)

3. `$\beta$` 시간에 따라 변한다고 가정. 
- B-Spline Basis For Polynomial Splines 이용: 여러 3차함수 조합
- Basis 함수 갯수만큼 parameter 존재 
 
4. 최소제곱 or 포아송분포: 실제 확진자수를 가장 잘 맞추는 `$\beta$` 선택 
]

---
# [Data](http://data.seoul.go.kr/dataList/OA-20279/S/1/datasetView.do)

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# 확진자 수

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앞 데이터 이용
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# 인구수
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---
# SEIR model

```r
incidata <- inci[지역 == "합계"]
times   <- seq(0, nrow(incidata) - 1, by = 1)
tt <- seq(min(times), max(times), len = max(times) +1)
*B  <- splines::bs(tt, knots = seq(min(times), max(times), len = ceiling(nrow(incidata)/7)), Boundary.knots = c(min(times), max(times)+1), degree = 3)
  
  
covODE <- function(time, state, modelparams) {
    with(as.list(c(state, modelparams)), {
      ## modelparams 1, 2: theta, gamma
*     beta <- exp(predict(B, time) %*% modelparams[-c(1, 2)])
      N <- S + E + I + R
      dS <- -beta*S*I/N 
      dE <- beta*S*I/N - theta*E
      dI <- theta*E - gamma*I
      dR <- gamma*I
      return(list(c(dS, dE, dI, dR)))
    })
  }
```

---

<div class="figure" style="text-align: center">
<img src="index_files/figure-html/unnamed-chunk-5-1.png" alt="B-splines of degree 3"  />
<p class="caption">B-splines of degree 3</p>
</div>

---
# 오차함수

```r
## Least square: x- basis parameter
ftemp <- function(x) {
  modelparams <- c(c(theta = 1/3.5, gamma = 1/6.8), x)
  tempoutput <- as.data.frame(ode(y = c(S = 9953009 - 1 - 1, E = 1, I = 1, R = 0), times = times, func = covODE, parms = modelparams))
  return(abs(tempoutput$R - cumsum(incidata$N)))
  }

## Poisson likelihood
ftemp2 <- function(x) {
  modelparams <- c(c(theta = 1/3.5, gamma = 1/6.8), x)
  tempoutput <- as.data.frame(ode(y = c(S = 9953009 - 1 - 1, E = 1, I = 1, R = 0), times = times, func = covODE, parms = modelparams))
  llik <- -sum(dpois(incidata, lambda = c(tempoutput$R[1], diff(tempoutput$R)), log = TRUE)) 
  return(llik)
}
```

---
# Estimation

```r
# Estimation
initbeta <- rep(0, ncol(B))                                 ## initial parameter
fitresult <- pracma::lsqnonlin(ftemp, initbeta)             ## nonlinear least square fit
#fitresult <- optim(initbeta, ftemp2, method = "L-BFGS-B")  ## poisson fit
betas <- fitresult[[1]]     
```

---
# 확진자수 예측

```r
params = c(c(theta = 1/3.5, gamma = 1/6.8), betas)
output <- ode(y = c(S = 9953009 - 1 - 1, E = 1, I = 1, R = 0), times = times, func = covODE, parms = params)
```

코드는 https://github.com/jinseob2kim/covidmodel-seoul/tree/main/docs/modelling/sources 에
- 11월부터, 시군구별 분석 수행
]

---
# `$\beta$` 추정

![](beta.jpg)

---
# 누적 확진수 예측
![](cum.jpg)

---
# 보완점
.large[
- `$\beta$`, 예측값의 신뢰구간 구하기가 어려움. 0보다 크다는 제한 두기 위해 `$\exp$` 이용.

-  여러 지표들의 제한범위를 설정하고, 신뢰구간을 자유자재로 구하기 위해 MCMC 등을 이용한 베이지안통계 필요. R에서 [stan](https://mc-stan.org/users/documentation/case-studies/boarding_school_case_study.html) 이용가능.

- 일부 데이터만 이용시(예: 11월 이후) 초기조건 정하기 어려움.
]

---
# ShinyApps

[![](shiny.png)](http://app.zarathu.com/konkukmath/covid19/)

---

# Executive summary

- 확진자수 등 **실제 데이터를 활용**, parameter 들을 추정.

- SEIR 에 서울시 확진자수를 적용, **시간에 따라 변화하는 김염률**을 계산함.

- Parameter와 그 파생지표의 범위를 제한하고 신뢰구간을 계산하기 위해, **베이지안통계** 이용 예정. 
]

---

# END