In the SIS epidemic models the variable X = ( S , I ) T where S = S ( t ) is the susceptible population size and I = I ( t ) is the infected population size. In this model an individual starts off susceptible, at some stage catches the disease and after a short infectious period become susceptible again. Such a model is appropriate for a bacterial disease such as pneumococcus.

The SIS model without demography is considerably simpler than the one considered here (see for example [17] and its references). The SIS model with demography was proposed in [20] or [21], in particular we consider only death of infected individuals with rate β . The changes and their probabilities to the first order in Δ t are given in Table 1 with x = ( S , I ) T .

Following ( [11], p. 148) we obtain the following deterministic model:

{ d S d t   =   μ 1 ( S , I )   = − α S I S + I + γ I , d I d t   =   μ 2 ( S , I )   =   α S I S + I − ( γ + β ) I , (4)

This model has a fixed point at I = 0 because μ 1 ( S , 0 ) = μ 2 ( S , 0 ) = 0 and the solution approaches the disease-free equilibrium lim t → ∞ I ( t ) = 0 when a

new basic reproduction number satisfies R = α / ( γ + β ) ≤ 1 (see [17] or [22] ).

The covariance matrix

D ( S , I ) =   ( d ( S , I ) − d ( S , I ) − d ( S , I ) d ( S , I )   +   β   I ) , (5)

with d ( S , I ) = α S I S + I + γ I .

Then if S ( 0 ) = x , I ( 0 ) = y its backward Kolmogoronv equation is

μ 1 ( x , y ) ∂ τ ∂ x + μ 2 ( x , y ) ∂ τ ∂ y + 1 2 d ( x , y ) ( ∂ 2 τ ∂ x 2 − 2 ∂ 2 τ ∂ x ∂ y + ∂ 2 τ ∂ y 2 ) + β   y 2 ∂ 2 τ ∂ y 2 = − 1 , (6)

with a singularity at the boundary y = 0 because for this value the coefficient of the second derivative is cancelled, which is why boundary payer will arise. The boundary conditions are:

{ τ ( 0 , x ) = τ ( y , 0 ) = 0 , ∂ τ ∂ x ( M x , y ) = 0 , ∂ τ ∂ y ( x , M y ) = 0 , (7)

provided that the number of x and y cannot exceed some values M x and M y respectively.

In the SIR epidemic models the variable X = ( S , I , R ) T where S = S ( t ) is the susceptible population size, I = I ( t ) is the infected population size and R ( t ) the recovered individuals because are assumed that individuals are immune after recovering from the disease. For example, [7] [8] [9] [10] proposes a model that in view of the results of [13] does not have any practical value, in my opinion.

Now, the changes and their probabilities to the first order in Δ t are given in Table 2 with x = ( S , I , R ) T .

Quite similar the model in [23] plus a new parameter β . The deterministic model is

{ d S d t   =   − α S I S + I , d I d t   =   α S I S + I   −   ( γ + β ) I , d R d t   =   γ I , (8)

In this differential system, the surface I = 0 is fix and its jacobian has three eigenvalues: λ = 0 doble and λ = α − ( γ + β ) and then when

γ   +   β   −   1   < α <   1   +   γ   +   β ,

and the matrix covariance

D ( S , I , R ) =   ( α S I S + I − α S I S + I 0 − α S I S + I α S I S + I +   ( γ + β ) I − γ I 0 − γ I γ I ) . (9)

Now assuming S ( 0 ) = x , I ( 0 ) = y , R ( 0 ) = z the backward Kolmogoronv equation is

− d ( x , y ) ∂ τ ∂ x + ( d ( x , y ) − ( γ + β ) y )   ∂ τ ∂ y + γ y ∂ τ ∂ z + 1 2   d ( x , y ) ( ∂ 2 τ ∂ x 2 − 2 ∂ 2 τ ∂ x ∂ y + ∂ 2 τ ∂ y 2 ) + 1 2   ( γ + β ) y ∂ 2 τ ∂ y 2 + 1 2   γ y ∂ 2 τ ∂ z 2 − γ y ∂ 2 τ ∂ y ∂ z = − 1 (10)

where d ( x , y ) = α x y x + y , and the boundary conditions

{ τ ( 0 , y , z ) = τ ( x , 0 , z ) = τ ( x , y , 0 ) = 0 ,     on     Γ 1 , ∂ τ ∂ x ( M x , y , z ) = 0 ,     on     Γ 2 , ∂ τ ∂ y ( x , M y , z ) = 0 ,     on     Γ 3 , ∂ τ ∂ z ( x , y , M z ) = 0 ,     on     Γ 4 , (11)

provided that the number of x, y and z cannot exceed some values M x , M y and M z respectively.

In the deterministic models, the value of the basic reproductive number ℜ 0 = α β + γ determines persistence or extinction of the disease. If ℜ 0 < 1 , the disease is eliminated, whereas if ℜ 0 > 1 , the disease persists in the population (see for example [22] when the total population size remains constant). However, this parameter does not inform about the speed towards one situation or another.

Because these kinds of models are appropriate for a bacterial disease such as pneumococcus consider in [21], they study the pneumococcus in the population of Scottish children under two years, first we test this realistic example (Case 1) and later two example with similar value for ℜ 0 .

Let’s consider the following example.

Case 1: In [17] we studied the pneumococcus in the population of Scottish children with α = 0.0214 , γ = 0.0211 , β = 0.00137 ⇒ ℜ 0 = 0.9986 < 1 .

Case 2: For α = 0.1997 , γ = 0.1 , β = 0.1 ⇒ ℜ 0 = 0.9986 < 1 .

Case 3: For α = 0.3994 , γ = 0.3 , β = 0.1 ⇒ ℜ 0 = 0.9986 < 1 .

We solved the two deterministic models (4) and (8) using Matlab. In Figure 1 we have plotted two paths for 0 ≤ t ≤ 25000 in blue is the SIS Model with S ( 0 ) = 1000 , I ( 0 ) = 10 and in red the SIR Model with S ( 0 ) = 1000 , I ( 0 ) = 10 , R ( 0 ) = 1 , in both path I ( t ) → 0 but in very different directions, a result that we would expect. Moreover, it would be more natural to ask Matlab to return the time t * such that 0 < I ( t * ) = I * ≪ 1 , which can be done using the event location facility. The highlighted numerical results are summarized in Table 3. In this table we have noted the time t 0 where I ( t * ) ≈ 0.99 in the first line, and I ( t * ) ≈ 10 − 3 in the second.

These three examples with the same reproduction number already have slight differences in their extinction, the third case is by far the fastest as well as the second model SIR.

These authors have resolved the boundary problems (6) - (7) using the Finite Element Method (FEM) and the software FreeFem++ which is freely available [19], this is a standard technique hence we will skip the details [17]. In Figure 2 we can see the numerical solution of (6) - (7) with M x = 1000 , M y = 100 . Note that the three graphs are very similar but the scale is quite different, the value of its peaks can be read in Table 4.

On the other hand, the boundary problem (10) - (11) has three independent variables which further complicates its numerical study, although the technique is similar. Let us multiply multiple (10) by a regular function ϕ ( x , y , z ) satisfying the homogeneous Dirichlet boundary conditions on Γ 1 . Integrating over the domain Ω = [ 0 , M x ] × [ 0 , M y ] × [ 0 , M z ] , the following terms with will appear:

∫ Ω     d ( x , y ) ∂ 2 τ ∂ x 2 ϕ = − ∫ Ω     d ∂ τ ∂ x ∂ ϕ ∂ x   − ∫ Ω ∂ d ∂ x ∂ τ ∂ x ϕ   + ∫ ∂ Ω     d ( x , y ) ϕ ( ∂ τ ∂ x n x ) ,

∫ Ω     c ( x , y ) ∂ 2 τ ∂ y 2 ϕ = − ∫ Ω     c ∂ τ ∂ y ∂ ϕ ∂ y   − ∫ Ω ∂ c ∂ y ∂ τ ∂ y ϕ + ∫ ∂ Ω     c ( x , y ) ϕ ( ∂ τ ∂ y n y ) ,

∫ Ω     γ y ∂ 2 τ ∂ z 2 ϕ = − ∫ Ω     γ y ∂ τ ∂ z ∂ ϕ ∂ z + ∫ ∂ Ω     γ y ϕ ( ∂ τ ∂ z n z ) ,

− ∫ Ω     d ( x , y ) ∂ 2 τ ∂ x ∂ y ϕ = ∫ Ω     d ∂ τ ∂ y ∂ ϕ ∂ x   + ∫ Ω ∂ d ∂ x ∂ τ ∂ y ϕ   − ∫ ∂ Ω     d ϕ ( ∂ τ ∂ y n x ) ,

− ∫ Ω     γ y ∂ 2 τ ∂ y ∂ z ϕ = ∫ Ω     γ y ∂ τ ∂ z ∂ ϕ ∂ y   + ∫ Ω     γ ∂ τ ∂ z ϕ   − ∫ ∂ Ω     γ y ϕ ( ∂ τ ∂ z n y ) ,

and c ( x , y ) = d ( x , y ) + ( γ + β ) y and where the boundary terms are of the following form:

∫ ∂ Ω     d ϕ ( ∂ τ ∂ x n x ) = ∫ ∂ Ω     c ϕ ( ∂ τ ∂ y n y )   =   ∫ ∂ Ω     γ y ϕ ( ∂ τ ∂ z n z ) = 0 , ∫ ∂ Ω     d ϕ ( ∂ τ ∂ y n x ) = ∫ Γ 2     d ϕ ∂ τ ∂ y , ∫ ∂ Ω     γ y ϕ ( ∂ τ ∂ z n y ) = ∫ Γ 3     γ y ϕ ∂ τ ∂ z . .

In Figure 3 we can see the numerical solution (10) - (11) in S ( 0 ) = x = 1000 and M y = 100 , M z = 100 , again the three graphs are very similar but the scale is quite different and the value of its peaks can be read in Table 4. Note that the value of its peaks are close to half of the SIS model, it now appears that the distance between the two models is greater than the deterministic case.

In view of the results of the two previous subsections: the great difference in extinction time between deterministic and stochastic models, we thought it reasonable to use some numerical simulation of stochastic differential equations in order to have more information. Our numerical simulations for the two stochastic

models have been done using the classic Euler-Maruyama numerical method, although it has strong order 1/2 and weak order 1 (we refer to [24] or [17] for a review on the numerical solutions of SDEs). This method applied to stochastic SIS model is straightforward and may be described as follows:

Algorithm 3.1. Given α , β and γ , let Δ t be the time-step, nrun, the number of simulations, and Tol the tolerances. Then, we have:

This Algorithm obtains the mean and standard deviation of the stopping time T * = i n f { I n   or   S n   < T o l } . The numerical results are in Table 5 with S ( 0 ) = 1000 , I ( 0 ) = 10 , Δ t = 10 − 3 , n r u m = 100 . Note first that deviations from mean are very high, to obtain conclusions it would be necessary to decrease the step and increase the number of trials, which would cost a lot of computing time with doubtful results due to possible on numerical stability problems. In other words, we needed more precise and reliable numerical methods. However, note that the times are less than the estimates in the previous subsection and much further away from the deterministic problem. For the SIR model the results are very similar.