diff --git a/functions/opt_vax_rate.R b/functions/opt_vax_rate.R
new file mode 100644
index 0000000..997d633
--- /dev/null
+++ b/functions/opt_vax_rate.R
@@ -0,0 +1,139 @@
+library(lpSolve)
+
+#' opt_vax_rate.
+#' 
+#' Optimal vaccination rate function, 
+#' its optimaze the problem of rollout vaccine solving a by non-linear programming solver.
+#'
+#' @param VAX.INITIAL.STORAGE.NUM # Number of READY vaccines on day one 'VAX.INITIAL.STORAGE.NUM'.
+#' @param VAX.PRODUCTION.RATE # Number of vaccines produced per day 'VAX.PRODUCTION.RATE'.
+#' @param MAX.VAC.RATE # Max number of vaccine applications per day 'MAX.VAC.RATE'.
+#' @param VAX.WINDOW.DAYS # Time window between first and second vaccines 'VAX.WINDOWS.DAYS'. 
+#' Its varies with the vaccine type.
+#' @param SECOND.VAX.LOSS.FRAC # Fraction of people that take the first but not the second dose 'SECODNE.VAX.LOSS.FRAC'.
+#' @param MAX.TIME.DAYS # time until end of vaccination schedule, in days 'MAX.TIME.DAYS'.
+#' @import lpSolve
+#'
+#' @return
+#' @export 
+#'
+#' @examples
+opt_vax_rate <- function(VAX.INITIAL.STORAGE.NUM, VAX.PRODUCTION.RATE,
+                         MAX.VAC.RATE, VAX.WINDOW.DAYS, SECOND.VAX.LOSS.FRAC,
+                         MAX.TIME.DAYS) {
+    times <- seq(0, MAX.TIME.DAYS)
+
+    alpha = 1 - SECOND.VAX.LOSS.FRAC
+    V.T = max(VAX.WINDOW.DAYS * (alpha * MAX.VAC.RATE - VAX.PRODUCTION.RATE), 0)
+   
+    # boundary-free case
+    if (VAX.INITIAL.STORAGE.NUM + MAX.TIME.DAYS * (VAX.PRODUCTION.RATE - MAX.VAC.RATE) -
+        (MAX.TIME.DAYS - VAX.WINDOW.DAYS) * alpha * MAX.VAC.RATE >= V.T) {
+        return(rep(MAX.VAC.RATE, length(times)))
+    }
+
+    # feasibility is trivial: V(T) > 0 if VAX.RATE == 0
+
+    # solution using LP
+    N <- length(times)
+    M <- matrix(0, N, N)
+    M[row(M)-col(M) >= 0] = 1
+    M.2 <- matrix(0, N, N)
+    M.2[seq(1 + VAX.WINDOW.DAYS, N),] <- M[seq(1, N - VAX.WINDOW.DAYS),]
+
+    M.total <- - M - alpha*M.2
+
+    # objective
+    C.T <- colSums(M.total)
+    ## inequality constraints (Ax <= b)
+    A <- M.total
+    # diagonal matrix offset by window
+    offdiag <- matrix(0, nrow = N, ncol = N)
+    offdiag[row(offdiag) - col(offdiag) == VAX.WINDOW.DAYS] = 1
+    A <- rbind(- M.total,
+               diag(nrow = N, ncol = N) + alpha * offdiag,
+               M[N,]
+               )
+    b <- c(VAX.INITIAL.STORAGE.NUM + VAX.PRODUCTION.RATE * times,
+           rep(MAX.VAC.RATE, N),
+           (VAX.INITIAL.STORAGE.NUM + VAX.PRODUCTION.RATE * (times[N] + VAX.WINDOW.DAYS))/(1+alpha)
+           )
+    xopt <-  lp(direction="min",
+                objective.in = C.T,
+                const.mat = A,
+                const.dir = rep("<=", length(b)),
+                const.rhs = b)
+    #print(paste("Vacinas restantes no tempo final:",
+    #            sum(M.total[nrow(M.total),] * xopt$solution) + b[length(times)]))
+    return(xopt$solution)
+}
+
+#' plot_vac_schedule.
+#' 
+#' A function to plot the rollout vaccine after the optimal solution.
+#' 
+#' @param OPT.VAX.RATE # Optimal vaccine rate from the `opt_vax_rate` 'OPT.VAX.RATE'.
+#' @param VAX.INITIAL.STORAGE.NUM # Number of READY vaccines on day one 'VAX.INITIAL.STORAGE.NUM'.
+#' @param VAX.PRODUCTION.RATE # Number of vaccines produced per day 'VAX.PRODUCTION.RATE'.
+#' @param MAX.VAC.RATE # Max number of vaccine applications per day 'MAX.VAC.RATE'.
+#' @param VAX.WINDOW.DAYS # Time window between first and second vaccines 'VAX.WINDOWS.DAYS'. 
+#' Its varies with the vaccine type.
+#' @param SECOND.VAX.LOSS.FRAC # Fraction of people that take the first but not the second dose 'SECODNE.VAX.LOSS.FRAC'.
+#' @param MAX.TIME.DAYS # time until end of vaccination schedule, in days 'MAX.TIME.DAYS'.
+#' @import lpsolve
+#'
+#' @return
+#' @export
+#'
+#' @examples
+plot_vac_schedule <- function(OPT.VAX.RATE, VAX.INITIAL.STORAGE.NUM,
+                              VAX.PRODUCTION.RATE, MAX.VAC.RATE,
+                              VAX.WINDOW.DAYS, SECOND.VAX.LOSS.FRAC,
+                              MAX.TIME.DAYS) {
+    times <- seq(0, MAX.TIME.DAYS)
+    alpha = 1 - SECOND.VAX.LOSS.FRAC
+    V <- VAX.INITIAL.STORAGE.NUM + VAX.PRODUCTION.RATE * times -
+        cumsum(OPT.VAX.RATE) -
+        alpha * c(rep(0, VAX.WINDOW.DAYS),
+                  cumsum(OPT.VAX.RATE[seq(1, length(times) - VAX.WINDOW.DAYS)]))
+
+    par(mar = c(5, 4, 4, 4) + 0.3)
+    plot(times, OPT.VAX.RATE,
+         xlab="time (days)",
+         ylab="vaccination rate (doses/day)",
+         type='l')
+    lines(times, alpha * c(rep(0, VAX.WINDOW.DAYS),
+                           OPT.VAX.RATE[-seq(length(times)+1-VAX.WINDOW.DAYS, length(times))]),
+          type='l', lty=2)
+    par(new = TRUE)
+    plot(times, V, type = "l", axes = FALSE, bty = "n", lty = 3, xlab = "", ylab = "", col="blue")
+    axis(side=4, at = pretty(round(range(V), -5)))
+    mtext("Vaccine doses stored", side=4, line=3, col="blue")
+}
+
+#' interpolate.VAX.RATE.
+#' 
+#' A function to interpolate vaccine rate rollout give the vaccine rate
+#'
+#' @param OPT.VAX.RATE # Optimal vaccine rate from the `opt_vax_rate` 'OPT.VAX.RATE'.
+#'
+#' @return
+#' @export
+#'
+#' @examples
+interpolate.VAX.RATE <- function(OPT.VAX.RATE) {
+    VAX.RATE <- function(t){
+        if (t <= 0)
+            return(0)
+        tu <- ceiling(t)
+        x <- tu - t
+        s = 0.1
+        if (x > 1-s && tu > 1)
+            return((x)*OPT.VAX.RATE[tu] +
+                   (1-x)*(OPT.VAX.RATE[tu] + OPT.VAX.RATE[tu-1])/2)
+        if (x < s)
+            return((1-x/s)*OPT.VAX.RATE[tu] +
+                   (x/s)*(OPT.VAX.RATE[tu] + OPT.VAX.RATE[tu+1])/2)
+        return(OPT.VAX.RATE[tu])
+    }
+}
\ No newline at end of file
diff --git a/main.R b/main.R
new file mode 100644
index 0000000..9758370
--- /dev/null
+++ b/main.R
@@ -0,0 +1,74 @@
+library(shiny)
+source('functions/opt_vax_rate.R')
+
+# Define UI for app that draws a histogram ----
+ui <- fluidPage(
+
+  # App title ----
+  titlePanel("Otimização de aplicação de duas doses"),
+
+  # Sidebar layout with input and output definitions ----
+  sidebarLayout(
+
+    # Sidebar panel for inputs ----
+    sidebarPanel(
+      numericInput(inputId = "VAX.INITIAL.STORAGE.NUM",
+                   label = "Estoque inicial de doses",
+                   value = 0),
+      numericInput(inputId = "VAX.PRODUCTION.RATE",
+                   label = "Produção de novas doses or dia",
+                   value = 0),
+      numericInput(inputId = "MAX.VAC.RATE",
+                   label = "Capacidade máxima de aplicação por dia",
+                   value = 0),
+      sliderInput(inputId = "VAX.WINDOW.DAYS",
+                  label = "Intervalo entre a primeira e a segunda dose, em semanas",
+                  min = 3,
+                  max = 12,
+                  step = 1,
+                  value = 3),
+      numericInput(inputId = "MAX.TIME.DAYS",
+                   label = "Tempo total da simulação, em dias",
+                   value = 120),
+      numericInput(inputId = "SECOND.VAX.LOSS.FRAC",
+                   label = "Fração de pessoas que não tomam a segunda dose",
+                   value = 0.1)
+    ),
+# VAX.INITIAL.STORAGE.NUM, VAX.PRODUCTION.RATE, MAX.VAC.RATE, VAX.WINDOW.DAYS, SECOND.VAX.LOSS.FRAC, MAX.TIME.DAYS
+
+    # Main panel for displaying outputs ----
+    mainPanel(
+
+      # Output: Histogram ----
+      plotOutput(outputId = "distPlot")
+
+    )
+  )
+)
+
+# Define server logic required to draw a histogram ----
+server <- function(input, output) {
+
+  # Histogram of the Old Faithful Geyser Data ----
+  # with requested number of bins
+  # This expression that generates a histogram is wrapped in a call
+  # to renderPlot to indicate that:
+  #
+  # 1. It is "reactive" and therefore should be automatically
+  #    re-executed when inputs (input$bins) change
+  # 2. Its output type is a plot
+  
+  output$distPlot <- renderPlot(with(reactiveValuesToList(input), {
+    OPT.VAX.RATE <- opt_vax_rate(VAX.INITIAL.STORAGE.NUM, VAX.PRODUCTION.RATE,
+                                 MAX.VAC.RATE, VAX.WINDOW.DAYS,
+                                 SECOND.VAX.LOSS.FRAC, MAX.TIME.DAYS)
+    
+    plot_vac_schedule(OPT.VAX.RATE, VAX.INITIAL.STORAGE.NUM,
+                      VAX.PRODUCTION.RATE, MAX.VAC.RATE, VAX.WINDOW.DAYS,
+                      SECOND.VAX.LOSS.FRAC, MAX.TIME.DAYS) 
+
+    }))
+}
+
+shinyApp(ui, server)
+