Week 2: Discussion on disease characteristics and model structure

[Wirichada Pan-ngum]

By the end of the week, I would like you to submit to the discussion board.

-What model structure would the disease you are interested be and please start adjusting the R code to that structure.

– Please state the key characteristics of the disease you are focusing on e.g. transmission, pathogenicity, symptoms, control measures etc. You can write it as text or start using the parameter table in the template provided. Anything you want to let me know about the disease you are going to work on would be useful. (10 points)

Variable name	description	Value or range	Source of information/data

 

[Wirichada Pan-ngum]

I realize it may be a bit difficult to think about the model structure all by yourself. Please look in some literature related to the disease you have chosen and see what sort of model structure they used. Most infectious diseases would have SIRS or SEIRS structure if we know some immunity would be developed after the infection. If not then, SIS or SEIS may be the one. ‘E’ compartment sometimes is ignored if the latent period is not significant or if it lasts very short. For chronic diseases, may be it is not suitable to ignore this stage. This is specially true if you plan to run the model for a long timeframe (e.g. over 5 or 10 years).

[Thundon Ngamprasertchai]

Pneumococcal disease modeling
The model divides the total population(N) into six sub-classes according to their disease status. susceptible S(t), vaccinated V(t), infected I(t), and recovery R(t).
The total population size: N = V+S+I+R
SVIR model
Here is the link for table; https://1drv.ms/w/s!AjwS_uCrdS0jg_sI48FLeMyEWx3YLw?e=BPw4sA

[Wirichada Pan-ngum]

I am not clear about the model structure. I think it would be nice to draw the compartmental model diagram. I am not sure about the treated individuals as there exists no compartment of treated population. Does the vaccine wane over time?

[Kridsada Sirichaisit]

Acinetobacter baumannii is the gram negative bacteria that is the major serious MDR bacterial infection in ICU patients. By the SIR model I find the R0 from one study in Austraria (Doan TN, Kong DCM, Marshall C, Kirkpatrick CMJ, McBryde ES (2015) Characterising the Transmission Dynamics of Acinetobacter baumannii in Intensive Care Units Using Hidden Markov Models. PLOS ONE 10(7): e0132037. https://doi.org/10.1371/journal.pone.0132037) that A. baumannii has R0 equal to 1.5 when transmitted by cross transmission. And the every time of treatment in the overall course of antibiotics is 14 days. then I can created the model from R0 (1.5) , gramma (1/14) , beta (1.5/14) then I create the R code for plot SIR model graph of A. baumannii infection in ICU.

SIR.dyn <- function(t,var,par) {

S <- var[1]
I <- var[2]
R <- var[3]
N <- S+I+R
beta <- par[1]
gamma <- par[2]
dS <- -beta*S*I/N
dI <- beta*S*I/N – gamma*I
dR <- gamma*I
list(c(dS,dI,dR))}

beta <- 1.5/14
gamma <- 1/14
SIR.par <- c(beta,gamma)
SIR.init <- c(14,1,0)
SIR.t <- seq(0,60,by=1)

SIR.sol <- lsoda(SIR.init,SIR.t,SIR.dyn,SIR.par)
TIME <- SIR.sol[,1]
S <- SIR.sol[,2]
I <- SIR.sol[,3]
R <- SIR.sol[,4]
N <- S + I + R

plot(0, 0, type=’n’, xlim=c(0,60),ylim=c(0,20), xlab=’time’, ylab=’infecteds’)
points(TIME,S,type=’l’,col=’blue’,lwd=3)
points(TIME,I,type=’l’,col=’red’,lwd=3)
points(TIME,R,type=’l’,col=’green’,lwd=3)
legend(10,20,legend=c(“Susceptible”,”Infected”,”Recovered”),col=c(‘blue’,’red’,’green’),lty=rep(1,3))

In the study that I citation, this paper use Hidden Markov models (HMMs) that can solve the problem of the undetected colonization patients by the imputation. The nosocomial infection is the complex problem and there are many risk factor the can influenced the infection such as host (underlying disease), LOS, previously antibiotic used, Colonization of A. baumannii in the environment. The result of this study is like to my observation epidemiology of A. baumannii in my hospital. the susceptible cases are dynamic due to the new case that was admitted to the hospital and some infected patient was died due to A. baumannii infection (the mortality rate is 30-60%). The SIR model of R-code may use loop for achieve the true model with the dynamic parameters.

[Wirichada Pan-ngum]

I looked at the reference https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0132037 and pretty sure that they went with SIS model. What is R compartment mean in yours? Please check the structure again. Should it just be uncolonized and colonized?

[Kridsada Sirichaisit]

Thank you very much for your comment when I carefully read this paper again, I agree with your comment that the model of this study is SIS model. The major cause of A. baumannii is the colonization but the colonization divided into in patients and in the environment. In previous surveillance study of the environment in ICU I found that A. baumannii can colonization in bed, floor, and the medical equipment and with the conventional cleansing can not eliminate A baummannii from the environment.

[Wachirawit Supasa]

Mycobacterium tuberculosis is the bacteria that cause Tuberculosis or TB which is easily infected, widely spreading from cough aerosol, and causes pathological disease in the susceptible population especially immunocompromised people. TB is also a concerning disease at a multi-national level as such WHO has planned to eradicated TB in 2030. TB spreading in Thailand varies from geographical settings and the highest endemic area usually consists of borders with neighboring countries such as the northeast and northern region.

Creating a model for TB SIR could help medical professionals determine health control policy to cope with the TB problem.

Variables:
S = 0-100, I = 0-100, R = 0-100, N = S+I+R (Total 100)
t = 100 (because TB has long spreading period)

Defind equation
beta = 0.326666
gamma = 0.123111

Coding in R

library(deSolve)

SIR.dyn <- function(t,var,par) { 
  # Rename the variables and parameters
  S <- var[1]
  I <- var[2]
  R <- var[3]
  N <- S+I+R
  beta <- par[1]
  gamma <- par[2]
  
  # Derivatives
  dS <- -beta*S*I/N
  dI <- beta*S*I/N - gamma*I
  dR <- gamma*I
  
  # Return the 3 values
  list(c(dS,dI,dR))
}

beta <- 0.326666
gamma <- 0.123111
SIR.par <- c(beta,gamma)
SIR.init <- c(100,1,0)
SIR.t <- seq(0,100,by=1)

SIR.sol <- lsoda(SIR.init,SIR.t,SIR.dyn,SIR.par)

TIME <- SIR.sol[,1]
S <- SIR.sol[,2]
I <- SIR.sol[,3]
R <- SIR.sol[,4]
N <- S + I + R

plot(SIR.sol[,2],type='l',main='TB SIR Model',xlab='Time',ylab='Number',col='blue',ylim=c(0,100))
lines(SIR.sol[,3],type='l',col='red')
lines(SIR.sol[,4],type='l',col='green')

legend(60,60,c("Susceptible","Infected","Recovered"),fill=c("blue","red","green"))

Here is the result

Reference: Side, Syafruddin. (2015). A Susceptible-Infected-Recovered Model and Simulation for Transmission of Tuberculosis. Advanced Science Letters. 21. 10.1166/asl.2015.5840. Link

[Wirichada Pan-ngum]

Great start but TB has a long latent period which I think it is hard to ignore. Also what kind of interventions you are planning to explore. I have previously done some TB work, the simplest model we could have when exploring shorter course treatment. https://journals.plos.org/plosone/article/authors?id=10.1371/journal.pone.0248846

[Saravalee Suphakarn]

Rabies is viral zoonotic disease cause by Rabies virus (Rabies lyssavirus, Genus Lyssavirus). All mammal species could be infected with rabies virus including humans, dogs, and domestic cats. The virus is transmitted through direct contact, such as broken skin or mucous membranes in eyes, nose, or mouth, with saliva or nervous system tissue of infected host (cdc, 2019). After entering host, the virus invade to exon of peripheral nervous system (PNS) and it is transported in a retrograde direction to central nervous system (CNS). Rapid viral replication happen within the CNS, the results in pathologic effect of nerve cell and tissue such as encephalitis. The virus also spread from the CNS to the adjacent tissue via peripheral nerve including tissue of salivary gland. Many of virus are shed through saliva from the infected salivary gland at the time of clinical sign onset. Rabies patient present progressive neurologic signs that consistent from encephalitis or myelitis, for example, dysphagia, hydrophobia, and paresis. Once symptoms of the disease develop, rabies is inevitably fatal to both animals and humans (WHO, ). Although rabies hasn’t no effective treatment, it is preventable. For humans, have to avoid contract with risk animals or saliva and nervous tissue, which possible contain the virus. If accidental biting or scratching happens, the patient should receive post-exposure prophylaxis (PEP) as soon as possible. Vaccination is effective preventing measure for both human and animals

Model structure:
For rabies transmission from dog to dog, there is 4 compartment in the model
1. Susceptible population (S)
2. Vaccination population (V)
3. Exposure population (E)
3. Infected population (I)
Which Total population (N) = S + V + E + I

Parameter
– Rate of loss of vaccine efficacy (lambda)
– Rate of vaccination in population (alpha)
– Efficacy of vaccine (nu)
– Mortality rate (mu)
– Transmission rate (beta)
– Rate of progression from exposed to infected state (sigma)
– Probability of exposed developing rabies (kappa)

R-code
library(deSolve)

SVEI.dyn <- function(t,var,par)
{
S <- var[1]
V <- var[2]
E <- var[3]
I <- var[4]
N <- S+V+E+I

lambda <- par[1]
alpha <- par[2]
nu <- par[3]
mu <- par[4]
beta <- par[5]
sigma <- par[6]
kappa <- par[7]

dS <- (V*lambda) – (beta*keppa*I/N) – (nu*alpha*S)
dV <- (nu*alpha*S) + (nu*alpha*E) – (V*lambda)
dE <- (beta*keppa*I/N) – (nu*alpha*E) – (sigma*E)
dI <- sigma*E

# Return the 3 values
list(c(dS,dV,dE, dI))
}

lambda <- 5.5*(10^-3)
alpha <- 2.96*(10^-3)
nu <- 0.94
mu <- 1.23
beta <- 0.0292
sigma <- 0.239
kappa <- 0.49

SVEI.par <- c(lambda,alpha,nu,mu,beta,sigma,keppa)
SVEI.init <- c(100,50,1,0)
SVEI.t <- seq(0,365,by=1)
SVEI.sol <- lsoda(SVEI.init, SVEI.t, SVEI.dyn, SVEI.par)

TIME <- SVEI.sol[,1]
S <- SVEI.sol[,2]
V <- SVEI.sol[,3]
E <- SVEI.sol[,4]
I <- SVEI.sol[,5]
N <- S + V + E + I

plot(0,main=’Rabies in dog population model’, xlab=’Time (day)’, ylab=’Number’, col=’blue’, ylim=c(0,100))

lines(SIR.sol[,2],type=’l’,col=’blue’)
lines(SIR.sol[,3],type=’l’,col=’green’)
lines(SIR.sol[,4],type=’l’,col=’orange’)
lines(SIR.sol[,5],type=’l’,col=’red’)

legend(locator(1),legend=c(“Susceptible”,”Vaccinated”,”Exposed”,”Infected”),col=c(“blue”,”green”,”orange”, “red”))

References
Laager, M. Mathematical modelling of dog rabies transmission in N’Djamena, Chad (Dissertation). Basel: University of Basel; 2018. 106p.

[Wirichada Pan-ngum]

I am following you code and have a question on what happened when exposed dogs (E) are vaccinated? Can they return to susceptible? Really?

[Saravalee Suphakarn]

Thank you for reviewing. I my opinion, exposed dogs have probability return to susceptible dogs after they receive immediately post-exposure vaccination. However, the protocol of post-exposed vaccination in animal depended on previous vaccination status. For unvaccinated dog, there is evidence that the use of post-expose vaccine alone will not reliably prevent the disease in these animals. In case that any dog exposes to a confirmed or suspected rabid animal should be assessed, good would cleaning, and vaccinate (according to protocol) by veterinarian as soon as possible. After that strict quarantine is necessary. If the dog current on rabies vaccination, already get good wound cleaning, and received post-expose vaccine, it possible to undeveloped rabies and return to susceptible. In otherwise, the dog didn’t vaccinate before may develop rabies in quarantine period even through already received immediately post-expose vaccine.

[Wirichada Pan-ngum]

Excellent progress, over expectation on the coding. I will probably need a bit more time to look at the details. Also wondering if most of you sketched the structure diagram before adjusting the R code?

[Wirichada Pan-ngum]

For those who are not yet to confident to adjust the code, don’t worry just keep thinking and searching for more information on the disease, parameters, relevant questions and issues.

[Naphat]

Coronavirus disease 2019 (COVID-19) is an infectious disease emerging in China in December 2019 that has rapidly spread around China and many other countries. On 11 February 2020, the World Health Organization (WHO) renamed the epidemic disease caused by 2019-nCoV as strain severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)

SEIR Model structure;
S is the number of people at risk of being infected. (Susceptible)
E is the number of infected populations in the incubation period (Exposed).
I is the number of infectious agents that can be transmitted (Infectious).
R is the number of groups that have recovered from infection. and there is no chance of recurrence including being unable to infect others (Recovered)
N is the total system population.
𝑁(𝑡) = 𝑆(𝑡) + 𝐸(𝑡) + 𝐼(𝑡) + 𝑅(𝑡)

Parameter;
Beta is transmission rate
Gamma is Recovery rate
Sigma is Incubation period

R Codes;

library(deSolve)

SEIR.dyn <- function(t,var,par) {
S <- var[1]
E <- var[2]
I <- var[3]
R <- var[4]
N <- S+E+I+R

beta <- par[1]
gamma <- par[2]
sigma <- par[3]

#Differential equations
dSdt <- -beta*S*I/N
dEdt <- beta*S*I/N – sigma*E
dIdt <- sigma*E – gamma*I
dRdt <- gamma*I

list(c(dSdt, dEdt, dIdt, dRdt))
}

beta <- 14.866/7
gamma <- 1/10
sigma <- 1/14
#14 days quarantine

SEIR.par <- c(beta,gamma,sigma)
SEIR.init <- c(1000,500,100,0)
SEIR.t <- seq(0,60,by=1)

SEIR.sol <- lsoda(SEIR.init, SEIR.dyn, SEIR.par, SEIR.t)

TIME <- SIR.sol[,1]
S <- SEIR.sol[,2]
E <- SEIR.sol[,3]
I <- SEIR.sol[,4]
R <- SEIR.sol[,5]
N <- S + E + I + R

plot(0,main=”COVID-19 SEIR Simulation”, xlab=”Time”, ylab=”Number of population”, col=”Blue”, xlim = c(0,60), ylim=c(0,1000))

lines(SEIR.sol[,2],type=”l”, col=”Blue”)
lines(SEIR.sol[,3],type=”l”, col=”Red”)
lines(SEIR.sol[,4],type=”l”, col=”Green”)
lines(SEIR.sol[,5],type=”l”, col=”Yellow”)

legend(locator(1),legend=c(Susceptible,Exposed,Infectious,Recovery),col=c(‘Blue’,’Red’,’Green’,’Yellow’),lty=rep(1,4))

[Wirichada Pan-ngum]

Ok, you seem to have a simple COVID-19 model structure there. Next, what are you planning to do with it? Are you planning to add in some interventions? What assumptions have you made when using this simple model? You have mentioned 14 quarantine days, which I think you used it to estimate the rate leaving E to I compartment there, right? Keep going!

[Naphat]

Thank you for your reccomended. I do have some research and see many parametors and interventions for COVID-19 Model. As you said, the real world problems are a lot more complex than the maths world! Thank you krab 🙂

[Wirichada Pan-ngum]

Those who have tried running their model here using parameters specifically for their disease of interest, please start documenting where these values come from and think more about the assumptions you made when using this simple structure of disease transmission dynamic. Remember, the real world problems are a lot more complex than the maths world!

[Mingkhwan Vithayaveroj]

#Infection and death rate of COVID-19 with and without intervention.
#I think I will choose lockdown intervention for this report.

library(deSolve)

SIRD.dyn <- function(t,var,par) {

S <- var[1]
I <- var[2]
R <- var[3]
D <- var[4]

N <- S+I+R+D

beta <- par[1]
gamma <- par[2]
alpha <- par[3]

dS <- -beta*S*I/N
dI <- beta*S*I/N – gamma*I – alpha*I
dR <- gamma*I
dD <- alpha*I

list(c(dS,dI,dR,dD))}

#There will be 2 groups of these below data for with and without intervention.

beta <- 0.2169 #infection rate
gamma <- 0.0267 #recovery rate
alpha <- 0.00065353 #death rate

SIRD.par <- c(beta,gamma,alpha)
SIRD.init <- c(2762, 1, 0,0)
SIRD.t <- seq(0, 500, by = 1)
SIRD.sol <- lsoda(SIRD.init,SIRD.t,SIRD.dyn,SIRD.par)

TIME <- SIRD.sol[,1]

S <- SIRD.sol[,2]
I <- SIRD.sol[,3]
R <- SIRD.sol[,4]
D <- SIRD.sol[,5]

N <- S+I+R+D

library(ggplot2)

SIRD.sol2 <- as.data.frame(SIRD.sol)

ggplot(SIRD.sol2,aes(x = TIME))+
geom_line(aes(y = S,colour=”Susceptible”))+
geom_line(aes(y = I,colour=”Infected”))+
geom_line(aes(y = R,colour=”Recovered”))+
geom_line(aes(y = D,colour=”Death”))+
xlab(label = “Time”)+

theme_classic()+

theme(legend.justification=c(1,0), legend.position = c(1,0.5))+ theme(legend.title = element_blank(), legend.background = element_blank(), legend.text = element_text(size = 10), legend.key = element_rect(colour=”#FFFFFF”, fill = ‘#C2C2C2’, size = 0.25, linetype = “solid”))+

scale_colour_manual(breaks=c(“Susceptible”,”Infected”,”Recovered”,”Death”), values=c(“blue”,”red”,”darkgreen”,”purple”))

[Wirichada Pan-ngum]

Ok! Another COVID-19 example with death compartment added. I think here when you plan to explore the impacts of lockdown, it is fine not to consider E compartment like the previous model because lockdown is applied to all population groups. So lockdown can simply reduce beta then?

[Wirichada Pan-ngum]

See my work on COVID-19 modelling…bearable structure, not extremely complicated. https://www.mdpi.com/2079-7737/10/2/80

[Mingkhwan Vithayaveroj]

thank you so much ka ajarn.

[Mingkhwan Vithayaveroj]

yes, I think lockdown can simply reduce beta, but with many assumptions for a simple model.

here I try to plot Infecteds with the different beta, maybe it can explain the concept.

compare I

I plot with assumptions that lockdown has 20 and 25 per cent efficiency.

[Wirichada Pan-ngum]

Kind of make sense, the plot. At least it goes in the right direction but it has little impact here. Often the output of the model we are most interested is incidence (new case per unit of time). It is not obvious in the model, and it is not I. New cases is calculated as force of infection x susceptible. FOI is explained more in my lecture as well.

[Mingkhwan Vithayaveroj]

Thank you so much for your advice. I have reviewed the lecture again and get a better understanding. 😀

I do some plot again here >> incidence

I think that I will increase more efficiency variations, maybe it can see more about impact.

[Navinee Kruahong]

Ebola also known as Ebola hemorrhagic fever, is a rare and deadly disease caused by an infection of one of the Ebola virus strains. Ebola can cause diseases in humans and nonhuman primates (monkeys, gorillas, and chimpanzees). EVD is mainly transmitted through direct contact with infected bodily fluids and contaminated materials. At the early stage, the disease is characterized by initial-flu symptoms, high fever, severe headache followed by pharyngitis and abdominal pain, whereas the late stage of the disease is marked by vomiting, diarrhea, rash, and internal and external bleeding. The disease was first identified in Sudan and Zaire in 1976 where it infected over 284 people with a mortality rate of 53%, and subsequently, few months later after the first outbreak, another outbreak emerged in Yambuku, Zaire, affecting 315 and killing 254 people.

Model structure:
The SEIR model, which have 4 compartment in the model;
1. the susceptible compartment S,
2. the exposed compartment E,
3. the infected compartment I
4. the removed compartment R (which includes all individuals who either recover from the disease or die).
N is the total system population.
𝑁(𝑡) = 𝑆(𝑡) + 𝐸(𝑡) + 𝐼(𝑡) + 𝑅(𝑡)

Constant parameters and their numerical values appearing in the epidemic model
λ Rate of increase (recruitment rate) of susceptible humans 0.6321
μ Natural mortality rate of susceptible humans 0.9704
β1 Rate of infection from susceptible to exposed humans 0.2877
β2 Rate of infection from exposed to infected humans 0.7613
β3 Rate of infection from infected to recovered humans 0.4389 [
β4 Rate of infection due to wild animals from susceptible to exposed humans 0.1234
β5 Rate of infection due to wild animals from susceptible to infected humans 0.2431
β6 Rate of infection due to domestic animals from susceptible to exposed humans 0.4000
β7 Rate of infection due to domestic animals from susceptible to infected humans 0.3000
μ1 Natural mortality rate of exposed humans 0.0432
μ2 Disease induced mortality rate of exposed humans 0.2006
μ3 Natural mortality rate of infected humans 0.0656
μ4 Disease induced mortality rate of infected humans 0.9764
μ5 Natural mortality rate of recovered humans 0.6704

R-code
library(deSolve)

SEIR.dyn <- function(t,var,par) {
S <- var[1]
E <- var[2]
I <- var[3]
R <- var[4]
N <- S+E+I+R

beta1 <- par[1]
beta2 <- par[2]
beta3 <- par[3]
beta4 <- par[4]
beta5 <- par[5]
beta6 <- par[6]
beta7 <- par[6]
gamma <- par[7]
sigma1 <- par[8]
sigma2 <- par[9]
sigma3 <- par[10]
sigma4 <- par[11]
sigma5 <- par[12]
sigma <- par[13]

#Differential equations
dSdt <- gamma-sigma*S-(beta1+beta4+beta6)*S*E-(beta5+beta7)*S*I
dEdt <- (beta1+beta4+beta6)*S*E- beta2*E*I-(sigma1+sigma2)*E
dIdt <- beta2*E*I+(beta5+beta7)*S*I-(beta3+sigma3+sigma4)*I
dRdt <- beta3*I-sigma5*R
list(c(dSdt, dEdt, dIdt, dRdt))
}

beta1 <- 0.2877
beta2 <- 0.7613
beta3 <- 0.4389
beta4 <- 0.1234
beta5 <- 0.2431
beta6 <- 0.4000
beta7 <- 0.3000
gamma <- 0.6321
sigma1 <- 0.0432
sigma2 <- 0.2006
sigma3 <- 0.0656
sigma4 <- 0.9764
sigma5 <- 0.6704
sigma <- 0.9704

SEIR.par <- c(beta1,beta2,beta3,beta4,beta5,beta6,beta7,gamma,sigma,sigma1,sigma2,sigma3,sigma4,sigma5)
SEIR.init <- c(0.1,0.1,0.1,0.1)
SEIR.t <- seq(0,90,by=1)

SEIR.sol <- lsoda(SEIR.init,SEIR.t,SEIR.dyn,SEIR.par)

TIME <- SEIR.sol[,1]
S <- SEIR.sol[,2]
E <- SEIR.sol[,3]
I <- SEIR.sol[,4]
R <- SEIR.sol[,5]
N <- S + E + I + R

plot(0,main= “EBOLA SEIR Simulation”, xlab= “Time”, ylab= “Number of population”, col= “Blue”, xlim = c(0,90), ylim=c(0,1000))

lines(SEIR.sol[,2],type= “l”, col= “Blue”)
lines(SEIR.sol[,3],type= “l”, col= “Red”)
lines(SEIR.sol[,4],type= “l”, col= “Green”)
lines(SEIR.sol[,5],type= “l”, col= “Yellow”)

legend(locator(1),legend=c(Susceptible,Exposed,Infectious,Recovery),col=c(‘Blue’,’Red’,’Green’,’Yellow’),lty=rep(1,4))

Reference: Rafiq, M., Ahmad, W., Abbas, M. et al. A reliable and competitive mathematical analysis of Ebola epidemic model. Adv Differ Equ 2020, 540 (2020). https://doi.org/10.1186/s13662-020-02994-2

[Wirichada Pan-ngum]

I haven’t done much in Ebola so will need to read the detail about the disease first and would get back. In the mean time, move forward with the final report on this one, how would you like to apply the model to, any intervention?

[Wirichada Pan-ngum]

Although this structure is probably ok for Ebola, the code needed some adjustment. You need to rescale the y-axis to c(0,1) to see what happened. Then you can see the dynamic.

[Weerada Trongtranonth]

Forecasting Trend of spreading of Covid-19
using SIR model

S = Total number
I = cumulative confirmed cases
R = number of cured cases

Parameter :
Beta = transmission rate
Gamma = recovery rate

source: https://dl.acm.org/doi/pdf/10.1145/3429889.3429893

[Anawat ratchatorn]

Leptospirosis has high prevalence in Thailand, in Northeastern (Srisaket) and in Southern (Nakhon Sri Thammarat) are in particular. Leptospirosis caused by spirochete bacteria that transmit to human by accidentally contact with different kind of mammals. Leptospirosis cause around 50000 death each year. Symptoms of Leptospirosis include high fever, chill, muscle aches, red eyes and also jaundice. Most mathematical model use SIR to evaluate epidemiology of Leptospirosis. There are many different transmission rate and rate of recovery used in different studies. This model will define transmission rate = 0.01 [1.] and recovery rate = 1/15 (from 15days taken to recover) [1.]

##SIR Model

library(deSolve)

SIR.dyn <- function(t,var,par) {

S <- var[1]

I <- var[2]

R <- var[3]

N <- S+I+R

beta <- par[1]

gamma <- par[2]

dS <- -beta*S*I/N

dI <- beta*S*I/N – gamma*I

dR <- gamma*I

list(c(dS,dI,dR))}

beta <- 0.01

gamma <- 1/15

SIR.par <- c(beta,gamma)

SIR.init <- c(99,1,0)

SIR.t <- seq(0,1000,by=0.1)

SIR.sol <- lsoda(SIR.init,SIR.t,SIR.dyn,SIR.par)

plot(SIR.sol[,1],SIR.sol[,2],type=’l’,main=’Leptospirosis’, xlab=’t’,ylab=’N’,xlim=c(-0,100),ylim=c(0,100),col=’blue’)
lines(SIR.sol[,1],SIR.sol[,3],col=’red’)
lines(SIR.sol[,1],SIR.sol[,4],col=’green’)

– comments -> All study that I found use transmission rate quite low, I tried to use many value and every value result in chart like this because of transmission rate is much lower than recovery rate.

Ref.
1. Pongsumpun, P. (2012). ‘Mathematical Model for the Transmission of Leptospirosis in Juvennile and Adults Humans’. World Academy of Science, Engineering and Technology, Open Science Index 72, International Journal of Mathematical and Computational Sciences, 6(12), 1639 – 1644.
2. Narkkul, U., Thaipadungpanit, J., Srisawat, N., Rudge, J. W., Thongdee, M., Pawarana, R., & Pan-Ngum, W. (2021). Human, animal, water source interactions and leptospirosis in Thailand. Scientific reports, 11(1), 3215. https://doi.org/10.1038/s41598-021-82290-5
3. Felzemburgh RDM, Ribeiro GS, Costa F, Reis RB, Hagan JE, et al. (2014) Prospective Study of Leptospirosis Transmission in an Urban Slum Community: Role of Poor Environment in Repeated Exposures to the Leptospira Agent. PLOS Neglected Tropical Diseases 8(5): e2927. https://doi.org/10.1371/journal.pntd.0002927
4. Paisanwarakiat, R., & Thamchai, R. (2021). Optimal Control of a Leptospirosis Epidemic Model. Science & Technology Asia, 26(1), 9-17. Retrieved from https://ph02.tci-thaijo.org/index.php/SciTechAsia/article/view/214606

[Anawat ratchatorn]

I try to adjust SIR.init to (80,20,0) and the chart look better.

[Wirichada Pan-ngum]

Tuning the model to get the results that make sense is time consuming. Good start anyway!

[Author]

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