-Modeling various phenomena occurring in nature [9]-[11], whether the deployment of service stations in some allows us to predict future effects in reality. One of such area due to existing roads and their traffic [12]. example is modeling the growth of infection in a given Another application is artificial neural networks, where sphoopwulathtieonusdeepoefnddiisncgreotne vMaarirokuosv pcahraaimnsetienrso.rIdnerthtios amrtoidcleel, twhee there is a specific variation thereof with a stochastic note. epidemic with distinction into four states in which individuals In [13], [14], the authors model such networks and analyze in the population may be. More accurately - healthy, infected, the flow of information. Hybrids are also created, such as the sick and recovered. The article presents a mathematical model connection of heuristics with neural networks [15]. Not only describing the phenomenon together with a calculation example neural networks, but other artificial intelligence algorithms are adnisdcusssiemdu.lations. The obtained results were described and used in practical terms, an example of which is optimization green computing awareness [16]. Another interesting idea
Stochastic processes are a family of random variables defined in a certain probabilistic space. It is the field of the probability theory, which in today’s science has become one of the most dynamically developed fields due to the numerous applications in optimization techniques or artificial intelligence. Such processes enable defining various phenomena based on a certain probability of its occurrence, as well as its components.
The field of stochastics found its place in the theory of
optimization, where Lévy and Poisson processes were used
in modeling various types of phenomena occurring in nature.
Optimization theory has gained the most, where techniques
inspired by nature have been created to this day. An example
is the cuckoo algorithm intended originally to search for
extremes of function. The movement of these birds was modeled
using the Lévy flights. In [1]–[3], the author presented the
idea of searching key-points on images which can be used
for feature extractions. Proposed idea is important in security
because using stochastic algorithms almost guarantees finding
other features due to the random placement of individuals
in the population at the beginning of the algorithm. With
similar motives in [4], [5], the author showed the use of
these algorithms in two-dimensional games, where such an
algorithm can be used to model the opponent’s movement.
Additionally, in [6]–[8], heuristic approach was used in
different games. Similar algorithms are genetic ones based on
chromosomes. Their use is visible in route planning programs
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Commons License Attribution 4.0 International (CC BY 4.0)
(
The process fXn; n 0g of state space S is called the discrete Markov chain, if for each n 2 f0; 1; : : : g, the following equation occurs
P fXn+1 = in+1jXn = in; : : : ; X0 = i0g
= P fXn+1 = in+1jXn = ing for all possible states i0; : : : ; in+1 2 S.
It is a mathematical model of a random phenomenon evolving over time in such a way that the past affects the future only through the present. This model has state space S, where we can give the following properties 1) S it is a finite or at most a countable set of states, 2) S = f0; 1; 2; 3; : : :g, 3) Xn = i, what means, that at time n, the process is in the state of i.
Let us define the initial distribution of states, i.e. the distribution of the variable X0. A probability vector as = [p0; p1; : : : ]. For each state i 2 S, we have
pi = P fX0 = ig:
A. Numerical example
Let us assume that at the beginning of our experiment, the
population is made up of a thousand people, where 495 people
are healthy, 3 are infected and 2 are sick. Hence, the initial
vector is
(
P0 = 495 3 2 0 : The stochastic matrix is defined as follows 0 0:8 0:1 0:1 0
0 0:35 0:4 P = @BB 0 0 0:85 0 0:05 0:02 00::2155 CCA : After the second iteration, there is 396 51 52
1 = 317 57 104 21 : of going from one state i to another j in one step. Generalizing this, we have where probabilities are independent of n, pij (k) is an element at position (i; j) in Pk.
III. MARKOV CHAINS FOR EPIDEMIC SIMULATIONS The spread of infection can be represented using a mathematical model – a discrete Markov chain. In this model, we divide the population into four groups marked as states S=f0; 1; 2; 3g, i.e.
0 – susceptible sus, 1 – infected inf , 2 – sick sick, 3 – recovered rec.
Unfortunately, there is no way to create a perfect model, that is why we create several assumptions that simplify this model a healthy person can become infected with an infection, an individual in the infected group may go only to the disease state, a sick subject can leave a group of patients only through complete recovery, recovery guarantees immunity, immunity is not inherited age, sex and social status do not affect the probability of infection, climate or demographic changes do not affect the epidemic.
For such assumptions, the stochastic matrix can be defined as
0 psus;sick psus;rec 1
ppisnicf;;ssiicckk ppsiincfk;;rreecc CCA (
prec;sus prec;inf where p is the probability of changing states at t to t + 1. An initial vector can be represented as
P0 =
nsus
ninf
nsick
nrec
;
(
Both matrices are similar but differ in the selected values. In the second matrix, the probability of transition from the infected to recovered and sick to the recovered state drastically changed, which means minimal chance of recovery. In all simulations, we assumed that the population is composed of 500 individuals. The effect of the difference in people needed for the spread of the disease on the entire population was examined. Tests were performed for the population composed of 1000000 individuals in population and 1 sick for two different step (time) values – f25; 150g. The results are shown in the form of diagrams in the Fig. 1–4. It is easy to see that the charts are identical regardless of the initial vector parameters (for the same probability matrix). Hence, the simple conclusion that Markov models allow simulation of the epidemic phenomenon, but the initial vector has little effect. Mainly stochastic matrices play a role, thus estimating the probability of transition between selected states.
Note that in the probability of transition from infected and sick to healed state are 25% and 15%. In the population the average number of healthy is around 80% people and it stays independent of the step. Large differences in jumps between the population can be seen in the first 20 steps, where the population is infected and gets ill. The further step, the more stable the chart is, which may be due to the lack of changes in the stochastic matrix. From a practical point of view, the infection can evolve, and then these matrices can change. Unfortunately, the classic approach to Markov’s chains does not modify the matrix during operation, although there are models that can do it. Impact on these matrices will result in a much better realignment of the model.
For the experiments we have carried out, we have performed statistical tests. On the significance level = 0:1, we also verify the hypothesis about the compatibility of the data distribution with the Gamma distribution with the shape parameter equal to 1 and the scale parameter equal to 2, which results are presented in Tab. I–IV. According to the tests carried out, in the case of each table presenting the process of healthy, infected, sick, recovered, at the significance level = 0:1, we can reject the hypothesis about the compatibility of the distribution of data with the Gamma distribution with the parameter equal to 1 and scale parameter 2.
V. CONCLUSION
Discrete Markov chains allow us to model phenomena occurring in nature. Each step depends on the previous one, although there is no change in probabilities during operation. This is a quite serious shortcoming in predicting the future effects of the model. In the analyzed case of epidemic spread, such action resulted in the absence of a possible mutation of the disease. However, it is worth noting that if the disease started to be fatal after a certain amount of time, in the case of the second matrix, it could kill almost the entire population, as opposed to the first one.
This type of modeling of phenomena may allow us to improve the predictions of some phenomena that depend only on selected states and the table of probabilities. The number of states can be huge, but then the problem arises to create such a matrix and find its coefficients. The simulations showed that there is a large impact of even a small change in the main matrix.