SIR model without ode

[Laufrech]

Hello guys,
I have got following code of basic SIR model. I need the differential equation to be solved without ode() function or any other. Does anybody know, how to make it happen?

library(deSolve)
library(shiny)


ui <- fluidPage(
  
  titlePanel("Choose amounts"),
  
  fluidRow(column(width = 4,


  sliderInput(inputId = "time_values", label = "Days", value = 10, min = 1, max = 10),
  sliderInput(inputId = "beta", label ="Disease", value = 0.05, min = 0, max = 1, step = 0.05),
  sliderInput(inputId = "gamma", label ="Cure", value = 0.5, min = 0, max = 1, step = 0.1),
  ),
  column(width = 8,

  (plotOutput("plot"))
    )
  )
)

server <- function(input, output) {
  sir_equations <- function(time, variables, parameters) {
    with(as.list(c(variables, parameters)), {
      dS <- -beta * I * S
      dI <-  beta * I * S - gamma * I
      dR <-  gamma * I
      return(list(c(dS, dI, dR)))
    })
  }

  initial_values <-  c(S = 1000, I = 1, R = 0)

  sir_values_1 <- reactiveValues(val = data.frame())

  observe({
    sir_values_1$val <- as.data.frame(ode(
      y = initial_values,
      times = seq(0, input$time_values),
      func = sir_equations,
      parms = c(beta=input$beta, gamma=input$gamma) 
    ))
  })

  output$plot <- renderPlot({
    with(sir_values_1$val, {
    plot(sir_values_1$val$time, sir_values_1$val$S, type = "l", col = "blue",
         xlab = "Days", ylab = "Number of people")
    lines(sir_values_1$val$time, sir_values_1$val$I, col = "red")
    lines(sir_values_1$val$time, sir_values_1$val$R, col = "green")
    legend("right", c("halthy", "infected", "cured"),
           col = c("blue", "red", "green"), lty = 1, bty = "n")
    })
  })
}

shinyApp(ui = ui, server = server)

Thank you