Optimal Control of a Delayed HIV Infection Model with Immune Response Using an Efficient Numerical Method

Algorithm 1

Step 1: for 𝑖=−𝑚,,0,do∶

     𝑥_𝑖=𝑥₀, 𝑦_𝑖=𝑦₀, 𝑣_𝑖=𝑣₀, 𝑧_𝑖=𝑧₀, 𝑢^𝑖₁=0, 𝑢^𝑖₂=0.

   end for

   for 𝑖=𝑛,,𝑛+𝑚,do∶

     𝜆^𝑖₁=0, 𝜆^𝑖₂=0, 𝜆^𝑖₃=0, 𝜆^𝑖₄=0.

   end for

Step 2: for 𝑖=0,,𝑛−1,do∶

     𝑥_𝑖+1=𝑥_𝑖+ℎ[𝑠−𝑑𝑥_𝑖−(1−𝑢^𝑖₁)𝑘𝑣_𝑖𝑥_𝑖],

     𝑦_𝑖+1=𝑦_𝑖+ℎ[𝑘(1−𝑢^𝑖₁)𝑒^−𝛿𝜏𝑣_𝑖−𝑚𝑥_𝑖−𝑚−𝛿𝑦_𝑖−𝑝𝑦_𝑖𝑧_𝑖],

     𝑣_𝑖+1=𝑣_𝑖+ℎ[(1−𝑢^𝑖₂)𝑁𝛿𝑦_𝑖−𝜇𝑣_𝑖],

     𝑧_𝑖+1=𝑧_𝑖+ℎ(𝑐𝑦_𝑖𝑧_𝑖−𝑏𝑧_𝑖),

     𝜆₁^{𝑛−𝑖−1}=𝜆₁^𝑛−𝑖−ℎ{1+𝜆₁^𝑛−𝑖[𝑑−(1−𝑢^𝑖₁)𝑘𝑣_𝑖+1]

             +𝜒_{[0,𝑡𝑓−𝜏]}(𝑡_𝑛−𝑖)𝜆₂^{𝑛−𝑖+𝑚}(𝑢₁^𝑖+𝑚−1)𝑘𝑒^−𝛿𝜏𝑣_𝑖+1},

     𝜆₂^{𝑛−𝑖−1}=𝜆₂^𝑛−𝑖−ℎ[𝜆₂^𝑛−𝑖𝛿−𝜆₃^𝑛−𝑖(1−𝑢^𝑖₂)𝑁𝛿−𝑐𝑧_𝑖+1𝜆₄^𝑛−𝑖+𝑝𝑧_𝑖+1𝜆₂^𝑛−𝑖],

     𝜆₃^{𝑛−𝑖−1}=𝜆₃^𝑛−𝑖−ℎ[𝜆₁^𝑛−𝑖(1−𝑢^𝑖₁)𝑘𝑥_𝑖+1+𝜇𝜆₃^𝑛−𝑖

             +𝜒_{[0,𝑡𝑓−𝜏]}(𝑡_𝑛−𝑖)𝜆₂^{𝑛+𝑚−𝑖}(𝑢₁^𝑖+𝑚−1)𝑘𝑒^−𝛿𝜏𝑥_𝑖+1],

     𝜆₄^{𝑛−𝑖−1}=𝜆₄^𝑛−𝑖−ℎ[1+𝑝𝜆₂^𝑛−𝑖𝑦_𝑖+1+𝜆₄^𝑛−𝑖(𝑏−𝑐𝑦_𝑖+1)],

     𝑅₁^𝑖+1=(𝑘/𝐴₁)(𝜆₂^{𝑛−𝑖−1}𝑒^−𝛿𝜏𝑣_{𝑖−𝑚+1}𝑥_{𝑖−𝑚+1}−𝜆₁^{𝑛−𝑖−1}𝑣_𝑖+1𝑥_𝑖+1),

     𝑅₂^𝑖+1=(1/𝐴₂)𝜆₃^{𝑛−𝑖−1}𝑁𝛿𝑦_𝑖+1,

     𝑢₁^𝑖+1=min(1,max(𝑅₁^𝑖+1,0)),

     𝑢₂^𝑖+1=min(1,max(𝑅₂^𝑖+1,0)).

   end for

Step 3: for 𝑖=1,,𝑛,write

    𝑥^∗(𝑡_𝑖)=𝑥_𝑖, 𝑦^∗(𝑡_𝑖)=𝑦^∗_𝑖, 𝑣^∗(𝑡_𝑖)=𝑣_𝑖, 𝑧^∗(𝑡_𝑖)=𝑧_𝑖, 𝑢^∗₁(𝑡_𝑖)=𝑢^𝑖₁, 𝑢^∗₂(𝑡_𝑖)=𝑢^𝑖₂.

   end for