Disease modeling for public health
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다니엘 베르누이의 천연두 연구: 최초의 감염병 수리모형

감염병
수리모형
역학
역사
18세기 유럽은 천연두(smallpox)의 그림자 아래 살고 있었다. 이 무서운 질병은 매년 수십만 명의 목숨을 앗아갔으며, 생존자들에게는 평생 지울 수 없는 흉터를 남겼다 (Razzell, 1977). 이런 상황에서 한 수학자가 수학적 사고를…
May 6, 2025
6 min

왜 수학으로 감염병을 설명하는가?

수학은 세상을 설명하는 언어다. [전통적인 text. 혹은 유클리드, 아르케메데스, ] 바이러스는 숫자를 모른다. 바이러스 어떤 수식을 계산해서 다음 날 환자수를 결정하지는 않을 것이다. 그런데 우리는 바이러스의 전파 과정을 숫자로 설명한다. 왜 그럴까?
Apr 18, 2025
4 min

The Ross-Macdonald Framework: Foundational Mathematical Models in Vector-Borne Disease Epidemiology

The mathematical modeling of infectious diseases has become an indispensable tool in modern epidemiology, guiding intervention strategies and policy decisions. While…
Apr 17, 2025
10 min

The Birth of Epidemic Modeling: Daniel Bernoulli and the Smallpox Inoculation Controversy

Mathematical modeling of infectious diseases is now a cornerstone of public health decision-making. During COVID-19, we all became familiar with concepts like “flattening…
Apr 14, 2025
7 min

Regularization in statistical models

Regularization is a fundamental technique in statistical modeling that helps prevent overfitting and improves model generalization. This blog post explores different…
Feb 2, 2025
4 min

Understanding correlation, covariation, and least squares regression

Understanding relationships between variables is a fundamental aspect of statistical analysis. Three key concepts in this domain are correlation, covariation, and least…
Jan 30, 2025
3 min

Distribution Matching Methods: From Theory to Practice

When working with data from different sources or distributions, we often need to…
Oct 31, 2024
7 min

Bayesian workflow: fake social contact data example

Bayesian workflow
negative binomial
social contact
This post is my attempt to follow along a Bayesian Workflow by Gelman et al and create examples for better understanding. While my ultimate goal is to study infectious disease transmission using a dynamic model following…
Jul 21, 2024
12 min

Modeling human behavior: Infectious disease transmission modeling perspective

Bayesian workflow
negative binomial
social contact
감염병의 전파 양상을 연구하는 감염병 역학…
Jul 21, 2024
1 min

Kalman filter to estimate R using the FKF package

kalman filter
Arroyo-Marioli et al used a Kalman filter approach to estimate. I tried to reproduce in R. Let’s use a SIR model as was used in my previous post to…
Jul 11, 2024
5 min

Learning ChatGPT 4: Universal Approximation Theorem

Universal Approximation Theorem
Neural network
Arbitrary-Width
Arbitrary-Depth
Neural networks can approximate any function according to the Universal Approximation Theorem. A more precise statement of the theorem is that neural networks with a single hidden layer…
Jun 24, 2024
3 min

Parallel simulation in R

parallelism
foreach
doParallel
I find that parallel, doParallel and foreach packages provide the easiest approach for parallel computing in R. The doParallel vignette provides a great overview. librar…
Jun 14, 2024
2 min

Basic reproduction number: An algorithmic approach

Basic reproduction number
Mathematica
algorithmic approach
next generation matrix
A recent article published in Mathematics discusses an approach to calculating \(\mathcal{R}_0\). Since I have previously written a post about calculating \(\mathcal{R}_0\) using Sympy, I wanted to explore a new approach proposed by the…
Jun 2, 2024
5 min

Claude 3

LLM
Claude
Testing Claude 3 with R
May 30, 2024
2 min

Learning ChatGPT 2: Approximating a tower function using a neural net

Neural network
Adam
matrix
The blog post by Stephen Wolfram discusses a neural network for approximating some sort of a step function. This post is my attempt to reproduce it. A YouTube video and a book…
May 28, 2024
3 min

A very basic implementation of a neural network

GPT-2
ChatGPT
Wolfram
I am documenting my learning of a neural network. The contents are mostly based on the e-book.
May 27, 2024
3 min

Variables selection in statistical models

R
variable selection
generalized linear model
When building statistical models, one of the most critical steps is variable selection. This process involves choosing the appropriate predictors or features that will be…
May 18, 2024
5 min

Learning ChatGPT 1: Probabilities for the next word

GPT-2
ChatGPT
Wolfram
Inspired by the blog post by Stephen Wolfram about the workings of the GPT-2 system, I decided to learn a bit about ChatGPT myself. Luckily, GPT-2 is now available for R. My first task is simply to learn to run the model and generate the probability table for the words that can follow the text, “The best thing about…
May 1, 2024
3 min

Logistic function in R

Logistic function
indirect vaccine effectiveness
oral cholera vaccine
The logistic function, represented as: \[ f(x) = \frac{L}{1+e^{-k(x-x_0)}} \] , where \(x_{0}, L\), and \(k\) represent tht value of the function’s midpoint, the supremum of the values of the function…
Apr 19, 2024
6 min

Waning vaccine efficacy on susceptibility

Vaccine
waning
SEIR
Erlang distribution
In this article, I examined the process of modifying the disease transmission model (for instance, the \(SEIR\) model) to include vaccination and the waning of vaccine-induced immunity as it might happen in a clinical trial. A straightforward method to represent this…
Apr 8, 2024
11 min

SEIR model with varying number of compartments

SEIR
Erlang distribution
The \(SEIR\) …
Apr 5, 2024
6 min

Waning of vaccine effectiveness

vaccine efficacy
clinical trial
SEIR
The protection derived from vaccination often wanes over time and require the second or the third doses (so-called booster doses). For example, the study showed the efficacy of cholera vaccines over five years. The vaccine efficacy (VE) over the period seems to indicate that the VE wanes over time. In this…
Apr 4, 2024
7 min

Solutions for the steady states of the SEIR model

SEIR model
steady states
ODE
deSolve
Mathematica
In this post, I compared the numerical solutions obtained using deSolve::ode with the algebraic solutions derived from Mathematica for the steady states of a SEIR…
Mar 28, 2024
10 min

Vaccine effectiveness

SEIR
vaccine efficacy
direct
indirect
total
overall
Vaccine efficacy and effectiveness (VE) is generally estimated as one minus some measure of relative risk (RR) in the vaccinated group compared to the unvaccinated group Hal…
Mar 26, 2024
5 min

Counterintuitive effects in disease transmission dynamics

Infectious diseases
nonlinearity
transmission dynamics
An article by Heesterbeek et al. provides a few examples on the counterintuitive behavior of a dynamical system of…
Mar 20, 2024
1 min

Prevalence vs. incidence for the SIR model

R
prevalence
incidence
deSolve
In the following, I implemented a simple model of disease incidence and recovery. Susceptibles are infected at a constant rate, go through the natural history of infection…
Mar 12, 2024
3 min

감염병의 대유행 가능성

probability of a large outbreak
reproduction number
감염병 인류 라는 책을 재미있게 읽는 중이다. 136페이지에는 기초감염재생산지수와 대유행의 가능성에 대한 간단한 수식이 나온다. \[\text{대유행의 가능성} = 1 - \frac{1}{R_0}\]
Mar 4, 2024
4 min

Modeling the waning of the vaccine-derived immunity in the ODE model

vaccine-derived immunity
waning
cholera
ODE
exponential
Gamma
The use of ordinary differential equation (ODE) models to simulate disease spread and…
Feb 27, 2024
10 min

Incremental Cost-Effectiveness Ratio (ICER)

cholera
sub-Saharan Africa
The Incremental Cost-Effectiveness Ratio (ICER) is a crucial metric in health economics, offering insights into the value of medical interventions by comparing their costs…
Feb 25, 2024
5 min

Implementing incubation period of cholera in the ODE model

incubation period
ODE
cholera
exponential
Erlang
In the realm of infectious disease modeling, accurately simulating the incubation period–the time between exposure to a pathogen and the onset of symptoms–is crucial for…
Feb 22, 2024
3 min

Mass-action assumption: density- vs. frequency-dependent transmission

mass action
frequency-dependent
density-dependent
In models of transmission of directly transmitted pathogens, e.g., COVID-19, the transmission is assumed to occur via so-called mass action principle. It means the rate of newly infected people per unit area, per unit of time is proportional to the product between the numbers (or densities) of susceptible and…
Feb 19, 2024
3 min

LabelledArrays and NamedTupleTools make it easy to use the ODE model in Julia

julia
ODE
LabelledArrays
NamedTupleTools
SEIR
The LabelledArrays pa…
Jan 26, 2024
2 min

SIR model benchmarks: deSolve, odin, and diffeqr

ODE
R
deSolve
odin
diffeqr
C++
Julia
Euler method was implemented
Jan 19, 2024
5 min

diffeqr: R interface to the Julia’s DifferentialEquations.jl

differential equation
julia
DifferentialEquations.jl
diffeqr
Julia DifferentialEquations.jl provides an impressive collection of differential equation solvers. The DE…
Jan 15, 2024
4 min

Universal differential equation using Julia

universal differential equation
julia
sub-exponential growth
The UDE refers to an approach to embed the machine learning into differential equations. The resulting UDE has some parts of the equation replaced by universal approximators i.e., neural network (NN). The UDE model approach allows us to approximate a wide, if not infinite, variety of functional relationships. As an example, I will test how well…
Jan 12, 2024
5 min

Fitting a straight line in Julia: Flux machine learning

julia
Flux
linear model
This post is my attempt to learn machine learning in Julia. The contents of this page came from the Flux. Flux is a machine learning package written in Julia.
Jan 4, 2024
1 min

Critical vaccination threshold

vaccine
population immunity
critical vaccination threshold
The following article by Fine provides a great introduction to the critical vaccination threshold.
Dec 14, 2023
4 min

Generation interval

generation interval
reproduction number
Although not published, I wrote a correspondence to Lancet to commenting the article. In the article, the authors stated that the generation interval is the sum of the incubation period and the infectious period. I argued that this statement holds only for…
Dec 7, 2023
8 min

Generation interval, growth rate, reproduction number

generation interval
growth rate
reproduction number
Wallinga and Lipsitch wrote a highly cited paper about the reproduction number. It discusses how to derive reproduction number, \(R\), given the growth rate, \(r\), and the…
Dec 6, 2023
3 min

Convolution

particle filter
The following content was adapted from Grant Sanderson’s YouTube video and the Wikipedia article.
Dec 5, 2023
2 min

Idiosyncrasies and generalities

ecology
idiosyncransy
generality
COVID-19
Debates in the population ecology Bjørnstad and Grenfell.
Dec 4, 2023
2 min

SIR model in Stan: Euler method

R
Stan
Euler method
SIR model
I developed a SIR model and solved…
Nov 28, 2023
6 min

Estimating a time-to-event distribution in Stan

news
code
analysis
As in the previous post, let’s create a sample through a non-homogeneous process for the infection events and a Gamma distribution for the serial (or generation) interval.
Nov 24, 2023
6 min

Estimating a time-to-event distribution from right-truncated data

right truncation
exponential growth
Poisson process
Seamen writes: Data on time to an event are said to be right truncated if they come from a set of individuals who have been…
Nov 23, 2023
5 min

Estimating serial interval: doubly interval-censored data

R
serial interval
interval censoring
We start simple. Our task is to estimate parameters of a probability density function used to model the serial interval. Suppose dates of onsets of infectors, \(A\)…
Nov 17, 2023
3 min

Estimating serial interval for a growing epidemic

R
serial interval
interval censoring
In this case, the above likelihood function may be modified as follows:
Nov 17, 2023
2 min

Estimating serial interval: interval cenoring

R
serial interval
interval censoring
MLE
Suppose dates of onsets of infectors, \(t^{A}\), and infectees, \(t^{B}\), are given…
Nov 15, 2023
2 min

Branching process model 2

R
branching process
final epidemic size
In the branching process model, the number of secondary infections is realized as a random number (e.g., Poission or Negative binomial).
Nov 14, 2023
2 min

Final epidemic size: uniroot vs. optimize

epidemic
size
R
uniroot
optimize
Miller 2012 shows that the…
Nov 10, 2023
2 min

SEIR model

SEIR
deterministic
stochastic
Gillespie's algorithm
SEIR 모형은 잠복기가 어느 정도 긴 감염병 (예를 들어 코로나19)의 전파를 모형하는 데 사용한다. 이번 포스트에서는 SEIR 모형을 만드는 방법을 알아본다. 결정론적 (deterministic) 그리고 확률론적 (stochastic) 방법으로 SEIR 모형을 R언어로 만들어 본다.
`Nov 9, 2023`{=html}
8 min

Branching process model

R
branching process
final epidemic size
In the branching process model, the number of secondary infections is realized as a random number (e.g., Poission or Negative binomial).
Nov 8, 2023
3 min

Negative binomial regression with censored data: POLYMOD data

R
regression
contact
censor
This post describes my attempt to reproduce Table 1 of the paper, Social Contacts and Mixing Patterns Relevant to the Spread of Infectious Diseases. Data were downloaded from Social Contact…
Nov 2, 2023
9 min

Apartment transactions in Korea via API provided by the Ministry of Land, Infrastructure, and Transport

R
API
apartment
I will plot the price
Nov 1, 2023
2 min

Confidence interval using profile likelihood

SEIR
profile likelihood
likelihood ratio
수리 모형을 이용하여 연구를 하게되면 관찰값을 이용하여 모형의 모수를 보정하는 과정을 거치게 된다. 이 과정을 소위 결과 (관찰값)로 부터 원인 (모형)을 알아내는 과정이라 하여 inverse problem 이라 부르기도 한다. 이 글에서는 \(SEIR\) 모형과 중국 우한 에서의 초기 코로나-19 발열자 자료를 이용하여 모형의 모수 (기초재감염지수)와 신뢰구간을 구해본다. 모수는 푸아송 (Poisson) 분포를 이용한 최대 우도 (maximum likelihood) 방법으로 그리고…
Oct 19, 2023
4 min

SIR model using SymPy

SIR
SymPy
I attempted to replicate some of the simple analytical results presented in the book, Mathematical Epidemiology by Brauer et al.
Oct 16, 2023
1 min

Regression with censored data: AER::tobit and optim

R
regression
censor
tobit
The following example was adapted from the Tobit model in Model Estimation by Example. The dataset contains 200 observations. The academic aptitude variable is apt, the reading and math…
Oct 15, 2023
4 min

Mulitple regression: POLYMOD data

R
regression
contact
This post describes my attempt to reproduce Table 1 of the paper, Social Contacts and Mixing Patterns Relevant to the Spread of Infectious Diseases. Data were downloaded from Social Contact…
Oct 12, 2023
3 min

Regression using optim

R
optim
regression
I will use the cars data with give the speed of cars and the distances taken to stop.
Oct 11, 2023
2 min

Extract raster based on a polygon

R
raster
shapefile
crop
mask
sf
raster 이미지의 일부를 추출해보자. 특히, shapefile에 담겨져 있는 polygon에 해당하는 raster 를 추출해보자. raster 패키지의 crop 과 mask 함수를 이용할 수 있다.
Sep 23, 2023
1 min

odin package

ODE
R
odin
이번 Vaccine Impact Modeling Consoritum (VIMC) 연례회의에서 odin이라는 패키지에 대해 알게 되었다. deSolve의 업그레이드…
Sep 15, 2023
2 min

PubMed search, ChatGPT summary, and sending an email in Python

ChatGPT
R
xml
httr
Sep 1, 2023
2 min

PubMed search, ChatGPT summary, and sending an email in R

ChatGPT
R
xml
httr
Sep 1, 2023
5 min

Polygon 면적 구하기: sf 와 raster 패키지

R
shapefile
ggplot2
sf
raster
RColorBrewer
shapefile에 담겨져 있는 polygon의 면적을 구해보자 raster 패키지의 area 혹은 sf 패키지의 st_area 함수를 이용할 수 있다.
Aug 30, 2023
1 min

ggplot2로 지도 그리기

R
map
ggplot2
sf
RColorBrewer
ggplot2를 이용하여 지도 그리기를 해보자. 지도는 shapefile에 담겨져 있다고 가정하자. shapefile을 읽는 방법은 여러가지가 있을 수 있는데 sf 패키지의 st_read 혹은 read_sf 함수를 이용한 후 ggplot2의 geom_sf를 이용하여 그리는 것이 가장 쉬운 것 같다.
Aug 30, 2023
2 min

Writing a paper: Start with an outline

writing
paper
연구자의 업무 중에 연구 만큼 중요한 것이 글쓰기, 특히 논문 쓰기이다. 논문으로 쓰여지지 못한 연구는 타인에게는 존재하지 않는 것이나 다름 없는 것이다.. Writing a paper by George M. Whitesides 에 논문 쓰기에 유용한 팁이 있어 여기에 기록으로 남긴다. 한 마디로 요약하면 outline (개요)을 이용하는 것이다. 개요를 연구과제의 초기에 작성하여 연구의 계획표로 활용하며 공저자 (주로 제 1저자와 책임저자) 간에 논문에 대한 의견 교환시 개요를 사용하는…
Aug 30, 2023
1 min

Importance sampling

importance sampling
Importance sampling is a Monte Carlo…
Aug 30, 2023
2 min

Important figures from the book, How to avoid a climate diaster? by Bill Gates

parameter estimation
R
maximum likelihood
profile likelihood
How to Avoid a Climate Disaster: The Solutions We Have and the Breakthroughs We Need by Bill Gates is a comprehensive and accessible guide on how to tackle the urgent issue of climate change. Gates begins by laying out the scope of the problem, explaining…
Aug 28, 2023
2 min

Regression toward the mean

parameter estimation
R
maximum likelihood
profile likelihood
In his lecture Joseph Blitzstein talks about two basic statistical…
Aug 25, 2023
4 min

Survivor bias

parameter estimation
R
maximum likelihood
profile likelihood
In his lecture titled “The Soul…
Aug 25, 2023
3 min

Estimating the instantaneous reproduction number using the particle filter

particle filter
COVID-19
파티클 필터 (particle filter) 를 이용하여 잠재 변수 (latent variable)를 추정하는 과정을 지난 글에서 다루었다. 관찰값들이 코로나 19 일별 감염자일때 감염병 수리 모형을 이용하여 일별 감염재생산지수 (\((R_t\)) 를 추정한다. 아래 글은 2020년 Kucharski et al. 논문에 사용되었던 방법을 차용하였다. 이해를 돕기 위해 모형을 단순화 하였고 가상의 데이타를 만들어 내는 과정을 더하였다. 우선 SEIR 모형을 이용해서 가상의 데이타 (일별 감염자 수)를 만든다. 누적 감염자 (cumulative…
Aug 19, 2023
12 min

Maximum Likelihood and Profile Likelihood for the SEIR model

parameter estimation
R
maximum likelihood
profile likelihood
통계학은 많은 부분 확률모형의 모수를 추정하는 (inferential statistics) 과정이고 모수 추정방법으로 가장 많이 사용되는 방법이 maximum likelihood (ML)이다. 이번 포스트는 2014년 출간된 Cole et al.의 Maximum Likelihood, Profile…
Aug 14, 2023
2 min

Basic reproduction number using SymPy

Basic reproduction number
SymPy
감염병의 전파를 이해하는 데 있어 가장 기본적인 개념이 재감염지수, 특히 기초재감염지수 (\(\mathcal{R}_0\)) 이다. 재감염지수는 한 명의 감염자로부터 생산되는 평균 후속 감염자의 수를 일컫는데…
Aug 11, 2023
3 min

Population Monte Carlo 파퓰레이션 몬테카를로

Monte Carlo
R
parameter estimation
code
analysis
최근에 파티클필터링 (particle filtering; PF) 방법을 이용하여 \(\mathcal{R}_t\) 추정하는 과정에 대한 논문을 썼다. 그런데, 항상 의문이었던 것은 PF를 조금만 변형하면 감염병 모형의 감염속도 \(\beta=\mathcal{R}_0 \gamma\) 와 같은 time-invariant 파라미터를…
Aug 10, 2023
8 min

Simple mathematical models with very complicated dynamics

R
code
analysis
Robert M. May Nature Vol. 261 June 10, 1976
Aug 8, 2023
2 min

Sub-exponential growth

news
code
analysis
대부분의 SIR 모형은 감염병 확산의 메커니즘을 아래와 같은 식으로 표현한다.
Aug 7, 2023
6 min

ODE-based SIR models in Stan

R
Stan
ODE
SIR
Stan은 통계 모형 뿐 아니라 ODE 모형을 시물레이션하고 모수를 추정하는 데에도 유용하다. 이 포스팅에서는 일별 감염자 자료가 주어졌을 경우 Stan을 이용하여 SIR 모형의 두 개의 모수 (\(\beta, \gamma\))를 추정하는 과정을 기술하겠다. 먼저 deSolve 패키지 양식을 따라 SIR 모형을 아래와 같이 구현하고 모형에서…
Aug 6, 2023
5 min

감염재생산지수 계산하기

modeling
reproduction number
코로나19에 효과적으로 대응하고자 방역 당국과 연구자들이 코로나19의 전파 양상을 분석한 결과가 뉴스에 종종 보도 되었는데 그 내용 중에 빠지지 않는 것이 감염재생산지수이다. 영어로는 reproduction number (\(\mathcal{R}\)) 로…
Aug 4, 2023
4 min

Particle filter using R

particle filter
The following example was adapted from the post in RPubs.
Jul 19, 2023
3 min

Multinomial distribution

multinomial
Rcpp
pomp
When implementing a model of stochastic disease transmission, one has to…
Jun 19, 2023
3 min

SEIR model using the Nimble pacakge

nimble
MCMC
posterior predictive check
trace plot
감염병 수리 모형을 개발하는 데 있어 가장 근본적인 질문 중 하나는 주어진 관찰값 (시계열)하에서 어떤 모형을 선택하고 그 모수의 값을 어떻게 결정하는가이다. 모형을 선택하는 과정은 따로 다루기로 하고 여기서는 일반적으로 사용되는 감염병 수리 모형 (i.e., SIR)을 사용할 때 모수를 추정하는 과정에…
Jun 19, 2023
8 min

모델과 현실: 수학으로 감염병 확산을 읽다

수학 모델로 감염병 확산을 예측한다. 현실과 얼마나 맞을까? 코로나19는 이 질문의 정답을 알려줬다.
Apr 18, 2023
14 min

The Model and Reality: Bridging Mathematical Frameworks with Real-World Infectious Disease Dynamics

As infectious disease modelers, we face a fundamental epistemological question: How can abstract mathematical constructs help us…
Apr 18, 2023
6 min

Origins of major human infectious diseases

infectious disease
emergence
density
Major human infecious disease are believed to have arisen after agriculture revolutionOrigins of major human infectious diseases. By the way, this emergence of infectious diseases are one reason that agricultural revolution is callled one of the…
Mar 4, 2023
2 min
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