Cellular Network with Complex Connections for the Modeling of Epidemic Spreading

ROBERT A. KOSIŃSKI a,b

a Central Institute for Labour Protection – National Research Institute,

Czerniakowska 16, 00-701 Warszawa, Poland

bFaculty of Physics, Warsaw University of Technology,

Koszykowa 75, 00-662 Warszawa, Poland

Abstract.The model of an epidemic spreading in a two dimensional cellular network is described. Interpersonal contacts of short range type, between the nearest neighbors, as well as random long range, typical for small world complex networks, are taken into account as a ways for the spreading process. A number of parameters introduced in the model enable to describe different types of epidemics and different features of individuals. The range of epidemics as a function of time and epidemic curves are found numerically. The influence of the preventional vaccinations in the population, the density of long range connections and the localization of the initial source of epidemic for the epidemic spreading are discussed.

Key-Words: epidemic spreading, complex networks, small world, cellular networks

1 Introduction

In the recent years a large number of papers devoted to the epidemics spreading phenomena has appeared in the literature because of the hazards connected with such an epidemics like HIV, anthrax or SARS. In the significant part of these works complex networks were used for modeling human populations [1-9]. It was found lately, that in the social contacts of each person, having a character of a complex network, a “small world” type of connections are present [10,11], i.e. each person, besides a short range contacts within its nearest social neighborhood, has a number of a long range contacts with distant persons [10-12].

In the number of papers investigations of the spreading of epidemics were modeled using cellular networks, with the standard short range connections to the nearest neighbors and the random, long range connections (short cuts) [2-4,6,8,9]. The spreading process can be described using numerically modeled stochastic process, where the transmission of the infecting factor from the infected cell to the neighboring cells (and changing of the states of each individual in time) occurs with certain probabilities, which is typical for the spreading of viral or bacterial infection over the human population [13]. Such an approach is used in the present paper for the case of 2-dimensional regular lattice with the additional small world long range connections.

2 The model of population

In our paper human population is represented by the regular, square lattice of N individuals. Each individual can be in one of 5 possible states: 1- health and suscepted, 2- infected, 3 – ill, 4 – health and unsusceptible, 5- died. It is worth while to notice, that the state 4 means that the individuals became unsusceptible due to the passed infection, due to the inborn resistivity, or due to the preventive vaccinations. The sequence of the states, which occurs in the time evolution of each individuals are: 1-2-3-4 or 1-2-3-5, where the transition between the states 1-2 occur with the probability F (see below), between the states 3-5 occur with the probability M , between 3-4 with the probability 1-M, while the transition between other states occur with the probability equals 1. Thus, our assumption is a modification of the SIR – model (Susceptible - Ill – Recovered) widely used in theoretical epidemiology [13,14]. It is assumed that the epidemics in the network is transmitted from the infected (in state 2) or ill (in the state 3) individual to the adjacent susceptible individuals (of the first, second or higher order) with the probability F (in percents), depending on the intensity of infectious factor. The value F = 100 means that the appearance of infected or ill individuals in the network results in an infection of all neighboringindividuals. Besides this infectious process in the nearest neighborhood, a number n of short cuts, which connect randomly chosen pairs of individuals, cause that the infection can be transmitted to the long range distance, with the probability 1. Such a structure of interpersonal contacts were found in the human society [15,16].

The connections of the individuals located in the position i,j with other individuals, is

(1)

where the first term denotes the set of the short range connections of the i,j- individual with the individuals in the neighborhood of the range k (where k=1 means 8 nearest neighbors), denotes the long range connections of this individual with other, randomly chosen individuals, - is the random variable connected with the number of the short cuts, which appear with the probability  = n/m, where n and m are the number of short cuts and all short range connections in the network, respectively.

Unsusceptibility for the infection in the population, inborn or coming from the preventive vaccinations, is described by the parameter S, which defines the number of the unsusceptible individuals (in the state 4) in the population in t = 0. Spatial distribution of these individuals may be random or localized in a certain area, which corresponds to the preventive vaccinations applied randomly or to a certain group of populations e.g. customs-house and frontier officers.

Additional parameter W describe the threshold of the susceptibility of an individual for infectious factor. For W =1 connection of an individual in the state 1 with only one other infected/ill individual is sufficient for infection of this individual. When W =2 contact with two infected/ill individuals is necessary for infection a new individual, etc.

It is worth while to notice that in the present model it is possible to individualize properties of all individuals as well as the contacts of each individuals by variations of the probabilities M, F or k and W, respectively. In such a case, the properties of the model of the population would be more similar to the properties of a real human population, where each person is in other social surrounding and has its own threshold of susceptibility for the infections. However there would be difficult to propose a reasonable distribution of individual properties and local social contacts for a members of population because of a lack of proper medical and social data [13,14]. Moreover simulations of such diversified population would be much more time consuming.

In our calculations populations with N= 2500 and N=10000 individuals were examined. The parameters k, W, characterizing each individuals, their contacts and the infection F, M, were assumed uniform in the whole population. The case with ( =0) was examined in p. 3 and 4; the effect of short cuts ( >0) is investigated in p. 5.

3 Spreading of infections with a different factors of intensity

The value of parameter F, defining the effectiveness of transmission of infectious factor between the infected individuals and health individuals, has a very distinct influence on the spreading of infections. We start from the one infected individual located in the middle of the population. Large values of F (e.g. F = 40, for N=10000) cause that after some time t = tsthe whole population is prevailed by infection, only some single individuals stay in the initial state 1. On the other hand for F< 34 only a part of network isprevailed by the infection. For such values of F, after t > ts i.e. the dying out of the infection, randomly distributed “islands” individuals free of infections are observed in the population.

The number of individuals L prevailed by the infection as a function of time, for different values of F and the population N=10000 is shown in Fig.1. The critical value of F = 34 parameter, for which the infection began to comprise the whole population, corresponds to the critical probability pcin 2-dimensional lattice which is the threshold value for an appearance of a giant cluster in the site-percolating system [17].

In epidemiology an important relation is the epidemic curve. It shows the number of newly infected individuals in the unit time as a function of time D(t) for different values of F (Fig.2). As we see when the intensity of the infectious factor is greater the curve reaches the maximum faster and it has a higher value than for the case of the smaller values of F (c.f. curves a with b and c in Fig.2). Dying out of the infection, represented by the decreasing part of the maximum, is more sudden for the curve a and results from the fact that for high values of F infection quickly spreads over the whole population. For the times times t > tmaxthe epidemic curve decreases because almost all individuals had passed the infection and there is no newly infected individuals.

4 Epidemic prevention

One of main procedures used in the prevention of epidemics is an application of the vaccinations in the population. This process may be modeled by an introduction in t = 0, in the population the S individuals being in the state 4 (unsusceptible). In our calculations random distribution of unsusceptible individuals in the network was assumed, however arbitrary space distribution of unsusceptible individuals may be assumed. Increasing value of S limits the spreading process and when a critical value S=Sc is reached the spreading process is stopped. In Fig. 3. the number of infected individuals L as a function of time is shown for increasing values of S and population N = 10000. As we see (curve a in Fig.3), for a small value of S = 1000 almost all population became infected after t = 72, thecurve saturates, which mean that the infection died out. For the greater values of S only a part of population is prevailed by the epidemic and the time in which the epidemic died out increases also (see curves b,c,d in Fig.3).

For the number of unsusceptible individuals large enough, i.e. S Sc , the spreading process is not initialized (see curve e in Fig.3). In our case, in which unsusceptible individuals are distributed randomly and the probability of infections of neighbors equals 1 (i.e. F=100), for the network with N = 10000 individuals, it was found that Sc = 5555, which gives Sc/N = 0.555. (Note that Fig.3 is calculated for other value of F equals 40). The value Sc may be referred to the percolation threshold pc for the 2-dimensional lattice which equals 0.407 for the 8 neighbors of the second order [17].

5 Influence of the short cuts for spreading process

It was assumed in our computations that in the population there is n pairs of individuals connected by the short cuts, defined by the values of the random variable (see eq.1). where i -individual has to be already reached by the epidemic while the j-individual has to be located in the part of the network free of epidemic. Thus, only short cuts effective in epidemic spreading are taken into account. Epidemic curves for the case N= 2500 and F=26 are shown in Fig.4, for the case of  = 0.025 (circles),  = 0.00625 (triangles) and  = 0.0025 (crosses), respectively. It can be seen that the increasing density of short cuts in the network cause the faster reaching of global maximum of D(t) curves. The local maxima of the curves are connected with the reaching successive short cuts by the front of epidemic and creation of a secondary sources of infection combined with the superimposing of infected areas coming from different sources. This means that the reduction of the traveling, or new individuals coming to the population from outside, slow down the spreading process of epidemic.

6 Influence of the localization of the initial source

The position of the initial source of infection has significant influence on the spreading process, especially in the case of absence of short cuts in the population. The fastest spreading of infection occurs when the initial source is localized in the middle of the lattice (population). For the localization of the initial source of infection in the corner or in the edge of the lattice the spreading is limited to two or three directions, respectively. On the other hand when the infection front reaches the borders of the lattice the spreading process is slowed down. It is possible to calculate analytically the influence of these factors on the epidemic spreading [18]. In Fig.5 the influence of the borders of the lattice on the spreading process is shown. It can be seen that the spreading is slowed down after the times t1, t2 , t3 and t4 when the infection front reaches the succesive edges of the lattice.

7 Conclusions

The model presented here enable to control the spreading of epidemics of a different types in the populations of different sizes. It should be noted however the approximation of a human society with a two dimensional lattice is better for the case of a large N. In particular in our model it is possible to observe the influence of short cuts for the spreading process. It cause the creation of the secondary sources of epidemic coming from the individuals traveling in the poulation or appearing from outside. It is also possible to examine the influence of preventional vaccinations in the population for the spreading process as well as the influence of the localization of the initial. source of epidemic on the spreading process.

The model presented here may be treated as another example of an application of complex cellular networks (see e.g.[19]).

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Fig.1. Number of infected individuals L in the population with N = 10000 as a function of time t for different values of F. F=35 - curve a, F=30 - curve b, F=26 - curve c and F=24 - curve d.

Fig.2. Epidemic curves D(t) for N = 2500, k=1 and F = 40 – curve a, F = 26 – curve b, F = 24 – curve c.

Fig.3. Number of infected individuals L in the population with N = 10000 as a function of time t for different values of preventionaly vaccinated individuals S. S= 1000 - curve a, S=3000 - curve b, S=3200 - curve c, S= 3300 - curve d and S= 3500 - curve e.

Fig.4. Epidemic curves D(t) for N = 2500, k=1, F = 26 and different values of density of short cuts:  =0.025 (circles),  = 0.00625 (triangles) and = 0.0025 (crosses).

Fig.5. Epidemic curve showing the times in which the epidemic front reached the borders of the lattice (N = 10000; initial source localized in x=17, y=32)