Datos y librerias

library(ggplot2)

d <- data.frame(Fecha = paste0(c("06", "07", "08", "09", "10", "11", "12",
                                 "13", "14", "15", "16", "17", "18", "19",
                                 "20", "21", "22", "23"), "/03/2020"),
                y = c(1, 2, 9, 11, 13, 17, 22, 38, 43, 71, 86, 117, 155,
                      234, 263, 318, 363, 395))
d$Fecha <- as.Date(d$Fecha, "%d/%m/%Y")

Modelo exponencial

temp <- d[1:14, ]
model <- lm(log(y) ~ Fecha, data = temp)
pred <- data.frame(Fecha = seq(as.Date.numeric(0, d$Fecha[1]),
                               as.Date.numeric(17, d$Fecha[1]), 1))
pred.mod <- predict(model, newdata = pred)
pred$y <- exp(pred.mod)

Grafico comparativo

ggplot(d, aes(x = Fecha)) +
  geom_point(aes(y = y, colour = "Observado")) +
  geom_smooth(aes(y = y, colour = "Observado"),
              method = "auto", span = .75) +
  geom_smooth(data = pred, aes(y = y, colour = "Modelo exponencial"),
              method = "auto", span = .75) +
  labs(x = "Fecha", y = "Número de casos identificados", color = "") +
  theme(legend.position = "bottom",
        legend.box = "vertical")
## `geom_smooth()` using method = 'loess' and formula 'y ~ x'
## `geom_smooth()` using method = 'loess' and formula 'y ~ x'