Natalia Ovsyannikova

Northern (Arctic) Federal University named after M.V. Lomonosov

A stochastic model of epidemic

Determined model, describing uncontrolled process of the spread of epidemic, is described by a system of differential equations:

                                                                          (1)

                      ,  , ,   ,                                (2)  

where  the rate of change in the number of people exposed to the disease,

       the rate of change in the number of infected people,

      function characterizing the number of meetings of people exposed to the disease and infected ones per unit of time.

       - the number of people who regained their health per unit of time without the influence of external means: quarantine, vaccination and others (-average time of natural healing),

      - the growth coefficient, which characterizes the frequency of meetings of healthy people with infected people (in general case it can be considered as a function ),

- the coefficient of natural mortality of people,

- the coefficient of mortality from this infection,

- average birthrate (reproduction).

The considered mathematical model is determined and allows to calculate in advance the change of a condition of the studied system, on an interesting time segment by solving the Cauchy problem (1)-(2).  We can assume that the values ​​of some of the coefficients of the system in the moment (of time)  are not uniquely defined, for example, because of their dependence on many unpredictable factors, and they can be regarded as random processes, the mathematical expectations of which are known.

Assume that the coefficient of growth has a random component, i.e. it can be represented as:

                                        ,                                                      (3)

where  - mathematical expectation (mean) of the coefficient , set it permanent, i.e.  ;  - random process;  - constant characterizing the degree of influence of the random perturbation on the value of the coefficient.

In this case, the mathematical model (1)-(2) takes the following form:

                                                                        (4)

                                                                              (5)

In this case, the state of the system is no longer a deterministic vector-function but is a vector random (stochastic) process, .

In general (in a general view), the system (4)-(5) can be written:

                                                                (6)

                      ,                                                                                   (7)

where ; ;  - scalar Wiener process; , , , .

The obtained stochastic differential equation will be solved numerically, for this we use a stochastic analogue of Taylor's formula. Apply the unified stochastic Taylor-Ito expansion in iterated stochastic integrals and also the approximation of iterated stochastic integrals by means of the polynomial system of functions. [1]

Formulate a theorem on Ito process expansion (decomposition) , where R:, in the unified Taylor-Ito series in iterated stochastic integrals

Theorem 1. Let the process  be Ito continuously differentiable  times in the mean-square sense on  along trajectories of the equation (6). Then for all,  it decomposes into a unified Taylor-Ito series of the following type:

                                                                            

                                         (8)

and there exists such a constant  that

,  ,

where ,                                  (9)

,        (10)

,                                                     (11)

,                                                    (12)

                                                        (13)

,                                                              (14)

,                                                                                      (15)

  ,                                                            (16)

,                                              (17)

                                                                          (18)

,                                        (19)

,                                                                       (20)

,                                           (21)

equality (8) is just(is fair, is true, takes place, holds) with probability 1, right parts of (8)-(10) exist in the mean-square sense.

We construct a unified Taylor-Ito expansion for the components of the solution   of the system (4)-(5)  for infinitesimal of order of , i. e. we will construct expansion of Ito process :

 

     (22)

 

       (23)

        

Relations (22) - (23) on a uniform discrete grid   constructed for the segment  , such that ,  are selected as a numerical method for modeling the system (6)-(7). Denote , , and then by putting , ,  in expansions (22)-(23)  and using the expansions of the iterated stochastic integrals   in terms of a polynomial basis, the  following expressions for the numerical method are obtained:

                (24)

 

                      (25)

 

where                  

  

                                             (26)

 

 

 - a system of independent Gaussian random variables with zero mean (expectation) and variance of one, which is generated on a step of integration with a number of k and is independent with the analogous systems of random variables that are generated on all the preceding steps of integration towards (with respect to) the step of integration with the number of k;  - step of integration of the numerical method; the number  is chosen from the condition ([1], p.199):

                               (27)

where constant  must be given (set). We choose it for the sake of simplicity to be unity (to be equal to one). The value of  increases with a decrease in the value of the step of integration. Consider the results of the choice of number  with the help of the relation (13). These results are placed in the following table:

0,004

0,001

0,0005

Q

1

2

4

I. e., it is enough to let q be equal to 1 that  would be 0,004, then the expansions for  and  take the form:

Make the numerical modeling of the solution of the system (6)-(7) by means of relations (24)-(25) on the time interval Ò=10 with the step  with the following initial data:   . The result of the numerical modeling is presented on fig. 1. Now introduce the stochastic perturbation . The evolution of processes, which characterize the process of evolution of  epidemic of the system (6)-(7) for values  is presented on figs. 1-4 respectively. The values of the maximal trajectories deviations of the perturbed system from the trajectories of the determined system are listed in table 1, from which direct dependence of maximal deviations of the solution of a perturbed system on the value of the perturbed parameter  is well seen.

 

   Fig.1 Determined and stochastic models                              Fig.2 Determined and stochastic models

            of epidemic ()                                               of epidemic ()

 

  

  Fig.3 Determined and stochastic models                         Fig.4 Determined and stochastic models

      of epidemic ()                                              of epidemic ()

 

Fig.5 Determined model and mean of the solution of                  Fig.6 Determined model and mean of the solution of

system (6, 7) found in 5 realizations ()                     system (6, 7) found in 10 realizations ()

 

 

Fig.7 Determined model and mean of the solution of                 Fig.8 Determined model and mean of the solution of system (6, 7) found in 15 realizations ()                         system (6, 7) found in 50 realizations ()

 

Table 1

Value

of

Maximum deviation of the trajectories of the perturbed system from the trajectories of the determined system depending on the value of the s.

Deviation X,   %

Deviation Y,  %

Practically none

2,5

Practically none

3,5

0,1

4

0,12

4,5

0,15

5

      For σ≤stochastic model is almost equal to the determined one, so it is necessary to take the determined model for description of the system; for σ> the stochastic model significantly (by more than 5%) is different from the determined one; therefore the perturbed coefficient needs to be taken ranging from to .

       For different realizations of the system of independent Gaussian values  we obtain different realizations of the solution of the system of stochastic differential equations (6). These trajectories for small perturbations lie inside of a tube constructed in a small neighborhood of the solution of determined system (1). Find mean of the solution of system (6) in 5, 10 realizations for , in 5, 10, 15 and 50 realizations for . On figs. 5-8 it is shown a comparison of means with the solution of the determined system of differential equations (1). One can conclude that , which is confirmed by numerical experiments, where - solution of determined system (1), (2),  - solution of stochastic system (4), (5).

 

References:

1.                 Kuznetsov D. F. Numerical modeling of stochastic differential equations and stochastic integrals St. Petersburg: Science, 1999, 459s.

2.                 Dmitrieva O.N. A stochastic model of the dynamics of the forests.- Collection of proceedings - Tver, 2006, 187 p.