Wednesday, December 15, 2010

The SIR Model of the Antivirus-controlled Propagation

Note. The text below is part 8 of the paper titled "Epidemiological Modeling of Computer Virus Propagation." 
When the antivirus exists, the number of infected I computers (aside from being augmented by those coming in from susceptible S population) is being reduced by disinfecting action of the antivirus. As previously noted, the SIR model can be used to for this scenario.

With infected I computers now being decremented as these become removed R computers, new equations for s(t) and i(t) must be derived. This is in addition to the new variable r(t) – the number of computers being disinfected. The equation for r(t) is derived first.

           In the derivation of the r(t) equation, it is first assumed that the removal rate at time t is proportional to the number of infected computers at that time. Thus, the equation

                   dr/dt= α • i(t)

where α, like λ, is an appropriate nonnegative constant determined by the physical parameters of the situation.
If n represents the number of computers at time t = 0, the total S population should consists of n + 1 computers. One is added because at t = 0, one computer must be infected to initiate the epidemic.
Since the total population always consists of n +1 computers, the assumptions involving ds/dt and dr/dt also determines di/dt. Thus, the following system of first-order nonlinear differential equations:

                   ds/dt= -λ • s(t)• i(t),               λ>0     Equation 7
                   di/dt= λ • s(t)• i(t)  – α •i(t),   α>0     Equation 8
                   dr/dt= α • i(t),                                  Equation

These equations can be solved exactly for s, i, and r as functions of time. In pursuing these equations, the constant ρ is introduced. It represents the relative rate of removal of infected computer, or for short, the relative removal rate. This ρ should not be confused with the constant α which represents the removal rate of infected computers from the I population. This ρ is the factor derived by dividing α with α. To wit,

                           ρ= α/λ          Equation 10
Again, α is removal (or disinfection) rate I and λ as infection rate. By dividing the two rates, the constant ρ is actually an indicator the speed of disinfection versus infection. With this notation, the above equation for di/dt can be written as
                                   di/dt= λ • i(t)  [s(t)-ρ]

From the above equation, an epidemic develops when di/dt is positive for some values of t.

Expressing Equation 10 in terms of λ, and substituting this to Equation 9 and then re-arranging the resulting equation such λ•i(t) is on the left side, the equation below is obtained.
                               λ•i(t)= α/ρ  dr/dt
Substituting the above to Equation 7, the equation below is obtained.
                                    ds/dt=-α/ρ s(t)dr/dt
                                    1/s  ds/dt=-α/ρ  dr/dt

The above equation can be solved by a straightforward integration. Taking into account the initial condition r(0) = 0, an equation relating the functions s and r is obtained. To wit,
Since s(t) + i(t) + r(t) = n +1 for all values of t, the equation below is obtained.
Substituting the equation for i(t) into Equation 8, the differential equation involving only the function r is obtained.

           dr/dt= α [n+1-r(t)-s(0)e^(-r(t)/ρ)]        Equation 11
Deriving an r equation18 in terms of time is made complex by the occurrence of the exponential term e^(-r/ρ). One method of coping with this difficulty is to expand the exponential term in power series of r. Expanding the exponential, the equation below is obtained.

             e^(-r/ρ)=1- r/ρ+1/2 (r/ρ)^2- 1/3! (r/ρ)^3+⋯
Substituting this expression for e^(-r/ρ) in Equation 11, the equation below is obtained.

           dr/dt= α [n+1-s(0)+(s(0)/ρ-1)r- s(0)/2 (r/ρ)^2+ …]

          Note that by considering only fewer values of e^(-r/ρ), the equation for the r will just be an approximation of the real r(t). Thus we represent this variable as ᵲ. For this study, we consider only up to the second power of e^(-r/ρ). Now the above equation can be re-written as

  dᵲ/dt= α [n+1-s(0)+(s(0)/ρ-1)ᵲ- s(0)/(2ρ^2 ) ᵲ^2]  Equation 12

Although the algebra is somewhat involved, the above Equation 12 presents no unusual problems. It can be solved more easily by assigning the letters a, b, and c to the following group of terms.
                         a = α[n+1-s(0)]
                                 b= α (s(0)/ρ-1)
                                 c= α s(0)/(2ρ^2)                    

Substituting these letters, the Equation 12 becomes

                             dᵲ/dt= a+bᵲ-cᵲ^2
                            dᵲ/(a+bᵲ-cᵲ^2 )=  dt
The left side is in quadratic equation form which, when integrated, yields the inverse hyperbolic tangent equation. Thus, 

                     -2/q  tanh^(-1) [(-2cȓ+b)/q]= t+ c1

Here q is used for the quantity (b^2+ 4ac)^(1/2), and the constant c1  can be derived by setting r(0) = 0.
Using the definition of the inverse hyperbolic tangent, the equation for ȓ(t) even if it is only an approximation of the exact r(t) is shown below.

                     ȓ(t)= 1/2c  [b-q tanh⁡((-q/q2 x t
)+ c2 )]

In terms of the exponential functions, the equation below will approximate exact r(t).

          ȓ(t)= 1/(2c) [b-q (1-e^(-qt+ c_2))/(1+e^(-qt+ c3)))]  Equation 13

The constants c2 and c3 are to be chosen so that ȓ(0) = 0.
Differentiating Equation 13 gives dȓ/dt  which can be substituted to Equation 9 to derive the equation for i(t). Then, differentiating this equation for i(t) gives di/dt which can be substituted to Equation 8 to derive the equation for s(t). These mathematical manipulations are no longer presented here.
The equations for the s(t), i(t) and r(t) give an approximate information on the behavior of the functions s, i and r.  To see how these functions behave over time, the analytic behavior of these functions needs to be plotted. 

Unfortunately, the graph cannot be imported into the blog at the moment.  Anyway, the such graph would show that the curve of the infected I computers reaches a certain peak after which it starts to decrease. After some time, all susceptible S members eventually become removed R members. At this time, the virus is already contained indicating that the epidemic has ended. 

18Daniel P. Maki, Mathematical Models and Applications (New Jersey: Prentice Hall, 1973), 367.

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