Discrete version of continuous SIR modelI'm working with a SIR infection model, which is d S d t = − β I S d I d t = β I S − γ I d R d t = γ I in continuous time, where S, I, and R are the proportion of Susceptible, Infected, and Recovered, respectively.However, since I am working with fixed-width discrete-time data, I think it would be more appropriate to modify the equations accordingly. I know this is incorrect (based on getting a negative values for γ ^ and β ^ , neither of which should be negative): S t + 1 = − β I t S t I t + 1 = β I t S t − γ I t R t + 1 = γ I t Ultimately, I would like to estimate β and γ by doing regression on I t + 1 = β ( I t S t ) + γ ( − I t ) + U t , where U t ~ N ( 0 , σ 2 ) or whatever the discrete version is.

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Discrete version of continuous SIR modelI&#039;m working with a SIR infection model, which is                                          d            S                                d            t                                      =                    −        β        I        S                                                  d            I                                d            t                                      =                    β        I        S        −        γ        I                                                  d            R                                d            t                                      =                    γ        I            in continuous time, where S, I, and R are the proportion of Susceptible, Infected, and Recovered, respectively.However, since I am working with fixed-width discrete-time data, I think it would be more appropriate to modify the equations accordingly. I know this is incorrect (based on getting a negative values for             γ      ^       and             β      ^      , neither of which should be negative):                              S                      t            +            1                                      =                    −        β                  I                      t                                    S                      t                                                        I                      t            +            1                                      =                    β                  I                      t                                    S                      t                          −        γ                  I                      t                                                        R                      t            +            1                                      =                    γ                  I                      t                              Ultimately, I would like to estimate   β and   γ by doing regression on      I          t      +      1        =  β  (      I          t            S          t        )  +  γ  (  −      I          t        )  +      U          t        , where       U          t            ~          N      (      0      ,              σ        2            )      or whatever the discrete version is.

Finnegan Stone · Accepted Answer

Step 1You&#039;ve missed some things. Remember a derivative is the small   Δ limit of, for example,   (  S  (  t  +  Δ  )  −  S  (  t  )  )      /    Δ. So, your first differential equation is like            S      (      t      +      Δ      )      −      S      (      t      )        Δ    =  −  β  I  Sor, where   Δ is your time interval width,      S          t      +      1        =      S    t    (  1  −  Δ  β      I    t    )  ..You missed the S(t) term on the right hand side. S should be decreasing with time, and looking at the equation we see that this would be the case if   β is positive.Step 2Also note that   S  +  I  +  R  =  N is constant, so you only need 2 differential equations, you can always find R by doing   n  −  S  −  I.

I'm working with a SIR infection model, which is (dS)/(dt)=-betaIS, (dI)/(dt)=betaIS-gammaI, (dR)/(dt)=gammaI

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