After a brief overview of the epidemiology and clinical picture of Chikungunya, I analyze the evolution of the current epidemic in Paraguay.

Introduction

An increase in the cases of Chikungunya (CHIKV) has been registered in Paraguay since October 2022 (40th epidemiological week), with most of the cases localized in the city of Asunción and in the neighboring department of Central (Ministerio de la Salud, CDC). While in the year 2015, a total of 4297 cases were reported in the departments of Asunción and Central, and in 2016 this number was 1239, as of the 3rd of March of 2023 the number of notified cases in these two departments is already about 9000. The magnitude of the current epidemic episode, in comparison with the previous ones, can be fully appreciated by looking at Figure 1, from the Alerta Epidemiológica N° 3/2023 (here).

Figure 1. Weekly cases of Chikungunya in Paraguay from 2015 to the 7th week of 2023, from the Alerta Epidemiológica N° 3/2023, Ministerio de la Salud, Paraguay (here).

Epidemiology of Chikungunya

CHIKV is a disease due to an RNA virus in the alphavirus genus (of the family Togaviridae), transmitted to humans by mosquitos of the genus Aedes. Homo sapiens is the principal reservoir of the pathogen. The disease is endemic in the tropical areas of Africa and Asia, it is characterized in the acute phase by high fever and polyarthralgia. The name comes from a Tanzanian language and it means “to walk bent over”, “to be contorted”, and it alludes to the abnormal gait of the victims, due to joint and muscle pain. In 2013 the disease appeared for the first time in the Caribbean sea (Saint Martin Island) and in the following years it spread in the Americas, from north to south (Yactayo S. et al. 2016).

Figure 2. Weekly cases of Chikungunya in the Dominican Republic from the 8th week of 2014 to the 39th. From Ministerio de Salud Pública, República Dominicana (Pimentel R et al. 2014).

We know from previous outbreaks outside the Americas that the percentage of the population that develops the disease during an epidemic episode (attack rate) goes from 10 to 75% and that those who become infected but are asymptomatic are between 17.5 and 27.8% (Ayu Mas S. et al. 2010). During the outbreak of 2014 in the Dominican Republic the attack rate in the main cities of the nation ranged from 40% to 81%; the epidemic episode lasted from the eighth epidemiological week to the 39th of 2014 (see Figure 2) (Pimentel R et al. 2014). After an epidemic episode in the Italian region of Emilia Romagna (summer of 2007), the attack rate was only 10.2% while the percentage of asymptomatic cases was 18% (Moro ML et al. 2010).

Clinical aspects of Chikungunya

The incubation period is typically 3–7 days (range 1–12 days). The most common clinical findings are acute onset of fever and polyarthralgia. Joint pains are usually bilateral, symmetric, and often severe and debilitating. Other symptoms can include headache, myalgia, arthritis, conjunctivitis, nausea, vomiting, or maculopapular rash. Clinical laboratory findings can include lymphopenia, thrombocytopenia, and elevated creatinine. Rare complications include uveitis, retinitis, myocarditis, hepatitis, nephritis, bullous skin lesions, hemorrhage, meningoencephalitis, myelitis, Guillain-Barré syndrome, and cranial nerve palsies. People at risk for more severe disease include neonates exposed intrapartum, older adults (e.g. age > 65 years), and people with underlying medical conditions (e.g., hypertension, diabetes, or cardiovascular disease). The whole paragraph has been taken from (CDC).

While some patients with the acute disease recover fully, severe and debilitating polyarthralgia may persist and become chronic (duration above 3 months to years). Chronic symptoms were reported in up to 64% of the individuals after one year from the acute phase (Ayu Mas S. et al. 2010). After 2.5 years after the epidemic episode that stroke the Island of Curaçao, in the Caribbean Sea in July 2014, of those who developed the acute phase of the disease only 43% fully recovered: 35% were still mildly affected and 22% were highly affected. Highly affected patients reported the highest prevalence of ongoing rheumatic and non-rheumatic/psychological symptoms, with an increased prevalence of arthralgia in the lower extremities, fatigue, and pain (Doran C et al. 2022). Six years after the epidemic episode of CHIKV of 2005-2006 in Reunion Island (Indian Ocean), a follow-up on 252 military employees (95% males, mean age 44 years) compared those who had CHIKV (81) to those who didn’t (171). CHIK+ patients complained of more frequent and intense joint pain than CHIK- (38% vs. 17% declared suffering at least once a week; 48% vs. 16% declared moderate to intense pain, p 0.0001). The frequency of nonrheumatic symptoms such as fatigue, headache, and depression, was markedly higher among CHIK+ subjects (Marimoutou C et al. 2015). In another follow-up study 18 months after the 2005-2006 outbreak of Reunion, it was found that among 362 adult subjects who had reported either rheumatic pain or fatigue at the onset of the infection, 23.9% developed a CFS-like illness while only 7.4% among initially asymptomatic peers did (p< 0.01) (Duvignaud A et al. 2018).

Figure 3. Weekly cases of Chikungunya in Paraguay from the 40th week of 2022 to the 8th of 2023. From Ministerio de Salud Pública, Paraguay (here).

Analysis of the current Chikungunya epidemic episode in Paraguay

I have considered the latest resumen epidemiologico semanal de arbovirosis (as of the 7th of March 2023) with the weekly reported cases of Chikungunya in the whole national territory (Figure 3). I have searched for a differential equation that could describe the evolution of the cumulative number of cases (f). I considered an exponential model, a logistic one, and a Gaussian model. The mathematical details of the analysis and the script that performs the calculations are reported in the following paragraph. The result of the analysis is reported in Figure 4. The best model (according to both the $R^2$ and the statistic test specified in Table 1) is the exponential one, followed by the Gaussian one. The Logistic model of growth does not fit well the data. The Gaussian model predicts that the maximum rate of growth will be reached by the mid of April. If we dismiss the first five weeks, focusing on the latest development of the epidemic, the Gaussian model is more optimistic (while the exponential one does not change much, Figure 5) but the statistical parameters indicate a worse fit (Table 2). In this case, the Gaussian model predicts that the maximum rate of growth has been reached by the mid of February.

Figure 4. Cumulative cases of Chikungunya in Paraguay from the 40th week of 2022 to the 8th of 2023. Experimental data from Ministerio de Salud Pública, Paraguay (here). The best model is the exponential one, followed by the Gaussian; the logistic model of growth is clearly not good. The statistical parameters of the regression models are reported in Table 1.

Figure 5. Cumulative cases of Chikungunya in Paraguay from the 45th week of 2022 to the 8th of 2023 (we skip the first five weeks). Experimental data from Ministerio de Salud Pública, Paraguay (here). The statistical parameters of the regression models are reported in Table 2.

Table 1. Statistical parameters for the regression models. The statistical test on the coefficients is the standard test used for the angular coefficient of the linear regression for the logistic and the exponential model (with $H_0:\beta_0=0$), while in the case of the Gaussian model, it is an F-test. We consider here the first available data point as the starting point for the regressions.

Table 2. As in Table 1, but in this case, we start from the fifth week available.

Informe del Ministerio de Salud Publica de Paraguay, 3 marzo 2023 (R).

Noticiero de ADN, 8 marzo 2023 (R).

I will add here the simulations based on the updates by the Ministerio de la Salud of Paraguay. I note that each time there is an update of the page containing the data (this one), not only there is the add of the data of a week, but there is also a substantial change in the cases for all the previous weeks (not sure why this is the case, I assume that notified case must then be verified, so the number will be adjusted over time).

Figure 5.b. Cumulative cases of Chikungunya in Paraguay from the 40th week of 2022 to the 8th of 2023. For the simulation, we use only the last 10 weeks available. Experimental data from Ministerio de Salud Pública, Paraguay (here). F-test: 1.85e-06 (logistic), 1.74e-08 (exponential), 4.312e-07 (gaussian). Update of 10th March 2023.

Figure 5.b.2. Weekly cases of Chikungunya in Paraguay from the 40th week of 2022 to the 8th of 2023. For the simulation, we use only the last 10 weeks available. Experimental data from Ministerio de Salud Pública, Paraguay (here). F-test: 1.85e-06 (logistic), 1.74e-08 (exponential), 4.312e-07 (gaussian). Update of 10th March 2023.

Mathematical notes

The logistic model of growth is described by the differential equation

(1) $\frac{df(t)}{dt}\,=\,f(t)\left(r-\frac{r}{k}f(t)\right)$

Its solution is given by

(2) $f(t)\,=\frac{k}{1+\left(\frac{k}{f_0}-1\right)}e^{-r\left(t-t_0\right)}$

We can transform eq.1 into a linear relationship by writing

(3) $\frac{\frac{df(t)}{dt}}{f(t)}\,=\,r\,-\,\frac{r}{k}f(t)$

We then define the response variable $Y\,=\,\frac{\frac{df(t)}{dt}}{f(t)}$ and the explanatory variable $X\,=\,f(t)$ and we perform a linear regression between them (Figure 6), which gives us the parameters

$\beta_0\,=\,r,\,\beta_1\,=\,\frac{r}{k}$

From these two relations, we can calculate r and k; by substituting them into eq. 2 we get the logistic curve of Figure 4 and Figure 5.

The exponential model of growth is described by the (1) for $k\rightarrow\infty$ and the solution is

(4) $e^{r\left(t\,-\,t_0\right)+\ln(f_0)}$

By taking the linear logarithm of both hands of the equation, we have a linear relationship and we can perform a linear regression.

The gaussian model cannot be reduced to a linear model, in general (only when the derivative is smaller than 1). But it can be reduced to a polynomial model of the second order. We consider the differential equation

(5) $\frac{df(t)}{dt}\,=\,Ke^{-\frac{1}{2}\left(\frac{t-\mu}{\sigma}\right)^2}$

whose solution is

(6) $f(t)\,=\,f(t_0)+\sigma K \sqrt{2\pi}\left[\Phi\left(\frac{t-\mu}{\sigma}\right)-\Phi\left(\frac{t_0-\mu}{\sigma}\right)\right]$

If we now take the natural logarithm of both hands of the equation, we get a linear relationship:

(7) $\ln\left(\frac{df(t)}{dt}\right)\,=\,-\frac{1}{2\sigma^2}t^2+\frac{\mu}{\sigma^2}t-\frac{\mu^2}{2\sigma^2}+\ln (K)$

We can now perform a polynomial regression and calculate the unknowns K, $\mu$, $\sigma$. Once we know them, we put them into eq. 6 and we have the red curves in Figure 4 and Figure 5. The R script used to perform the regressions and to plot all the figures is reported after the figures.

Figure 6. Linear regression for the logistic model. Starting from the first data point available.

Figure 7. Linear regression for the exponential model. Starting from the first data point available.

Figure 8. Polynomial regression for the gaussian model. Starting from the first data point available.

# file name: chikungunya
#
# Asunciòn, 5 marzo 2023
#
#
#
# we define the vector for the cumulative number of cases (updated every week)
#
f<-mydata\$CHIKV_not_PY_cum # cases
len<-length(f)
fc<-8 # weeks of forecast
stl<-1 # starting week for logistic regression
ste<-1 # starting week for exponential regression
stg<-1 # starting week for gaussian regression
#
#---------------------------------------------------------
# Logistic growth
#---------------------------------------------------------
#
# we define X (explanatory) and Y (response)
#
st<-stl # starting week
X<-f[st:(len-1)]
n<-length(X)
Y<-c()
for (h in 1:n) {
i<-st+h-1
Y[h]<-f[i+1]-f[i]
Y[h]<-Y[h]/f[i]
}
#
# linear regression
#
Lin_re<-lm(Y~X)
coefficients<-coef(Lin_re)
B0<-coefficients[[1]]
B1<-coefficients[[2]]
#
# we calculate the residues
#
R<-0
for (i in 1:n) {
R[i]<-Y[i]-B0-B1*X[i]
}
#
# plotting the linear regression and the residues
#
fitted<-predict(Lin_re) # it contains the line of the regression
#
plot(X, Y, xlab = "f", ylab = "f'/f", pch=21,bg="blue")
abline(lm(Y~X),lwd=1.5, col="black")
for (i in 1:n) lines (c(X[i],X[i]),c(Y[i], fitted[i]), col="red", lwd=2.5)
grid (col="black",lty="dashed",lwd=1.0)
#
# We calculate and store the logistic curve
#
t<-c(1:(len+fc)) # weeks
r<-B0
k<--r/B1
f2<-c()
t2<-c()
for (h in 1:(n+fc)) {
i<-st+h-1
den<-1+(k/f[st]-1)*exp(-r*(t[i]-t[st]))
f2[h]<-k/den
t2[h]<-t[i]
}
#
#---------------------------------------------------------
# Exponential growth
#---------------------------------------------------------
#
st<-ste
X<-t[st:(len-1)]
n<-length(X)
Y<-c()
for (h in 1:n) {
i<-st+h-1
Y[h]<-log(f[i])
}
#
# linear regression
#
Exp_re<-lm(Y~X)
coefficients<-coef(Exp_re)
B0<-coefficients[[1]]
B1<-coefficients[[2]]
#
# we calculate the residues
#
R<-0
for (i in 1:n) {
R[i]<-Y[i]-B0-B1*X[i]
}
#
# plotting the linear regression and the residues
#
fitted<-predict(Exp_re) # it contains the line of the regression
#
plot(X, Y, xlab = "log(cases)", ylab = "weeks", pch=21,bg="blue")
abline(lm(Y~X),lwd=2.5, col="black")
for (i in 1:n) lines (c(X[i],X[i]),c(Y[i], fitted[i]), col="red", lwd=2.5)
grid (col="black",lty="dashed", lwd=1.5)
#
# We calculate and store the exponential curve
#
t<-c(1:(len+fc)) # weeks
f3<-c()
t3<-c()
for (h in 1:(n+fc)) {
i<-st+h-1
f3[h]<-exp(B0+B1*t[i])
t3[h]<-t[i]
}
#
#---------------------------------------------------------
# Gaussian growth
#---------------------------------------------------------
#
st<-stg # starting week
X<-t[st:(len-1)]
n<-length(X)
Y<-c()
for (h in 1:n) {
i<-st+h-1
Y[h]<-log(f[i+1]-f[i])
}
#
# polynomial regression of the second order
#
Gau_re<-lm(Y~X+I(X^2))
coefficients_P<-coef(Gau_re)
B0P<-coefficients_P[[1]]
B1P<-coefficients_P[[2]]
B2P<-coefficients_P[[3]]
#
# we calculate the residues
#
R<-0
for (i in 1:n) {
R[i]<-Y[i]-B0-B1*X[i]
}
#
# We plot the regression
#
plot(X,Y,pch=21,col="black",bg="red",xlim=c(X[1],X[n]),ylim=c(Y[1],max(Y)),xlab="weeks",ylab="log(f')")
par(new=T)
plot(X,(X^2)*B2P+X*B1P+B0P,type="l",col="red",lty=6,lwd=2,xlim=c(X[1],X[n]),ylim=c(Y[1],max(Y)),xlab="",ylab="")
#
# We calculate and store the logistic curve
#
s<-sqrt(-1/(2*B2P))
mu<-(s^2)*B1P
lnK<-B0P+0.5*(mu/s)^2
K<-exp(lnK)
t<-c(1:(len+fc)) # weeks
f4<-c()
t4<-c()
for (h in 1:(n+fc)) {
i<-st+h-1
mul<-pnorm((t[i]-mu)/s) - pnorm((t[st]-mu)/s)
f4[h]<-f[st]+s*K*sqrt(2*pi)*mul
t4[h]<-t[i]
}
#
# We plot the actual data and the logistic fit
#
plot(t[1:len],f,xlab="semanas",ylab="notificaciones",pch=21,bg="blue",xlim=c(1,t[len+fc]),
ylim=c(0,2*f[len])) # experimental data
par(new=T)
plot(t2,f2,xlab="",ylab="",xlim=c(1,t[len+fc]),ylim=c(0,2*f[len]),type="l",lty=4,lwd=2,col="black") # fitted log
par(new=T)
plot(t3,f3,xlab="",ylab="",xlim=c(1,t[len+fc]),ylim=c(0,2*f[len]),type="l",lty=2,lwd=2,col="black") # fitted exp
par(new=T)
plot(t4,f4,xlab="",ylab="",xlim=c(1,t[len+fc]),ylim=c(0,2*f[len]),type="l",lty=2,lwd=2,col="red") # fitted exp
grid (col="black",lty=1, lwd=1)
abline(v=c(14,18.5,22.5,27),col="black",lty=4,lwd=1.5)
text(x=c(16,20.5,25),y=c(5000,5000,5000),labels = c("Jan","Feb","Mar"))
legend("topleft",legend=c("logistic","exponential","gaussian"),lty=c(2,2,2),col=c("black","black","red"),lwd=c(4,2,2))
summary(Lin_re)
summary(Exp_re)
summary(Gau_re)