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In this paper, we study the epidemic model: Here Sk, Ek, Ik denote the k-th total population may be infected population, the incidence of infection but not the number of incidence. All parameters in this model are non- negative. representative of the practical significance of the various parameters is as follows: βkj: Sk and transmission coefficients between Ij, dks, dkE, dkl: populations in the k S, E, I mortality, Λk: total population of the k populations growth, εk: k-th population is infected, after a period of incubation after the onset of disease probability. γk: k-th population incidence rate of population recovery. Specifically, βkj ≥ 0, βkj = 0 if and Only when the disease can not be transmitted between Sk and Ij when [1] For the propagation model matrix B corresponding network diagram, this paper considers the following five network diagrams, they are: five populations regular network, The latter three view, they have a greater randomness, which has practical significance is greater. (1) by computer simulation, we got these five total fluctuation of the network system changes with time relationship which we , the system fluctuations are small, and the equilibrium of the time required is a few in the equilibrium state, the network diagram of the system of these three values ??are stable at a total fluctuation of 1, that is, both initial and random matrix (βkj) how to select, these final equilibrium system is a fixed point, ie, the initial value and the random matrix (βkj) the selection of the final equilibrium of the system is not affected while for the 10 species in probability distribution of the random network, 5 populations in probability distribution of the random network, the equilibrium state, the system's overall fluctuations are relatively intense, and the time required for equilibrium are also relatively long, in equilibrium, the fluctuation of the total system is not stable at 1, that is, the two systems is not a final equilibrium point, the initial value and the random matrix (βkj) the selection of the final equilibrium of the system point is influential, but in equilibrium, the total system is stable at a point a few fluctuations, indicating that the selection of random matrix initial value and the final equilibrium point of the system impact is not great. Investigate its reason may population is 10 conventional network systems and five populations of conventional disease transmission network system diagram are strongly associated, 10 species average degree of 4 random network [2] System 100 simulations, diseases spread diagram is strongly associated probability of a large case, and for 10 populations in probability distribution of random network systems and five populations in probability distribution of the random network system in the 100 simulations, their disease transmission diagram is a strong probability that is not associated great, but when the disease spread graph is strongly associated, the whole system becomes more complex, the results are more difficult to predict which explains what to ask both the system's total volatile situation is more complex, and the final equilibrium point Indefinite problem for these systems, the overall fluctuation of the transition period is different, as illustrated by several different systems, so we can determine the volatility of a different system to distinguish between indicators (2) As for the previous transmission disease model results are built on the basis of the conventional network. therefore their results with great restrictive, such as: when B is the conventional network, the infectious disease status of the system ultimately only two populations have tended to be an all contagious equilibrium point; All populations have tended to balance the disease disappear, but when B is a random network, the infectious status of the system eventually becomes more complicated. populations tend to occur some contagious equilibrium point Some species tend to disappear disease state of equilibrium by 200 simulations, we found that the spread of disease population map 10 average degree of 4 random network [2] of the system, the final equilibrium state is infectious and disease disappear equilibrium probability of simultaneous 21/200 for the spread of disease by the population figure is 10 probability distribution of the random network system, the final status of infectious diseases disappear equilibrium and equilibrium probability of simultaneous 63 / 200 is a diagram for the transmission of disease from 5 populations in probability distribution of the random network system, the final state of equilibrium and infectious diseases exist equilibrium disappearance probability was 107/200 (3) Because the system reaches equilibrium state , the S value of each population is a constant, all the populations of the correlation coefficients are 1, that is, all species can be seen as a group, but before the system reaches equilibrium, the correlation coefficient of each of the two populations (considering each correlation coefficient between the two S) go through a process of change, we found that, no matter what network diagram, the system changes the number of groups are from more to less, finally tends to a group, but for normal network, since the system reaches equilibrium shorter time, so the system has reached a desired set time is shorter. (4) When we change the parameters εk when either the system the time required to reach equilibrium , or the system to reach equilibrium before the distribution of the number of groups, and system final state distribution (balance tends contagious, have tended to disappear balance disorders, both tend to become infectious diseases have disappeared equilibrium point equilibrium point) basically not changed, mainly because εk value changes, the value of the size R0 no effect (due to the formula R0, the item is a εk εk / (εk dkI)). When we change the parameters γk , we found that γk change the whole system to reach equilibrium, the average time required for a greater impact. Overall, with γk increases, the system reaches equilibrium, the average time required to render first increases and then decreases, finally tends to a constant trend from a practical sense, this phenomenon is very obvious, when γk is very small, which shows a more difficult disease to cure, so the system will soon become contagious population equilibrium point, With γk becomes large, required to cure a disease, fewer number of years, then tends to the equilibrium point contagious population Ik value γk is small than this value is smaller, because the disease is no longer as γk hour so difficult to cure. the Ik becomes smaller number of disease when γk continues to increase, the disease becomes better cure, so the system has reached equilibrium the average time required to decrease when γk continues to increase, the disease become very good cure. therefore all populations will tend to quickly disappear balance disorders. general, when γk from small to large, difficult to cure the disease will experience from → generally → good cure, that is, the system reaches equilibrium, the average time required for a small → large → small for conventional networks, the system reaches equilibrium with the average time required is that different random networks, with their average time required for changing the basic rendering γk linear first increases and then decreases the final constant relations, and for random network, the average time over γk change would render the curve style first and then decreases after the unaltered state, this is mainly because, for the random network, inside contains a lot of uncertain factors, it is difficult relationship between two variables showed a linear relationship when γk is changed, the five systems over periods of volatility and stabilization period the median volatility has a significant change in the median When γk changes, each system fluctuations transitional period of stability in the median and the median volatility changes essentially the same (each color dotted and solid lines overlap), but for two conventional networks, Whether or equilibrium phase transition period, the median vibrations are very close to 1, while for the other three random network, whether or equilibrium phase transition, volatility decreases and then the median is unchanged, which is mainly because the latter three network, the matrix (βkj) random selection of great when γk is very small, the randomness dominates, so volatile. but γk is large, the randomness of as γk great impact, so in this case very small fluctuations, remaining at around 1 Parameter βkj becomes large, the average degree population 10 random network of 4 [2] the system reaches equilibrium the average time becomes gradually smaller while the other four systems, the average time increases with βkj first increases and then decreases. This is mainly because when βkj increases, infectious diseases spread in various populations began to get a little messy, so the system reaches equilibrium the average time is longer, but βkj large, the disease spread among the various populations to become quickly, so the system reaches equilibrium the average time is shorter when βkj changes, 59 a system over a median period of volatility and stability during fluctuations have a significant change in the median when βkj small becomes large, each system fluctuations transitional period of stability in the median and the median volatility basically the same as the number of changes (each color dotted and solid lines overlap), but for two regular network, whether or equilibrium phase transition period, the median volatility is very close to 1, while for the other three random network , whether or equilibrium phase transition, the underlying trend median fluctuations are increased, that is, βkj increase, make the whole system becomes complicated in general, (1) both for the 5 populations or populations of 10, when the final state of the system when the disease disappears, the required transition time is very short, it is basically the same; (2) γk is the meaning of: γk small description of the disease is difficult to cure, so the system reaches equilibrium required a longer time, γk very large description cure the disease, the system reaches equilibrium time is short, when γk is large, all of the population will tend to balance the disease disappear, the system is very easy to reach equilibrium . (3) βkj meaning: βkj small description slow the spread of disease, the system is relatively easy to achieve equilibrium, but this time the number of incidence will not be great. With βkj becomes large, the system would become more complex, it is difficult to achieve balance, when βkj is large, βkj accelerate the speed of the system reaches equilibrium state βkj greater extent than the extent of the system becomes confused, the system reaches equilibrium state at this time is the average time becomes smaller.
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