| In the thesis, we consider the following epidemic model: Here Sk,Ek,andIk denote the population in the k-th group that are susceptible to the disease, infected but non-infectious, and infectious, respectively. The parameters in the model are non-negtive constants and summarized in the following list:βkj :transmission coefficient between compartments SkandIj, dsk, dkE, dkI:natural death rates of S,E,I compartments in the k-th group, respectively,Λk:influx of individuals into the k-th group,εk:rate of becoming infectious after latent period in the k-th group,γk:recovery rate of infectious individuals in the k-th group, In particular,βkj≥0, andβkj=0, if there is no transmission of the disease between compart-ments Sk and Ij[1].For the matrix B in this model, we mainly consider five types of network, which are regular network with 5 points; regular network with 10 points[2], random network with 10 points and 20 edges[2],5 points random network based on probability,10 points random net-work based on probability.For the last three networks, they have so much randomness that they have more appli-cability.(1)By simulations on computer, we get the relationship between total amplitude of the five networks and time. We find that for regular network with 5 points, regular network with 10 points[2], random network with 10 points and 20 edges [2],they have similar variation of total amplitude. Before they reach the equilibrium, their total amplitudes fluctuate only a little and their transit time is very short. After they reach the equilibriums, the total ampli-tudes of these three network stays in 1, which means that their equilibriums are fixed points whatever the matrix (βkj) is. That is to say the choice of matrix (βkj) has no effect on the equilibrium of the epidemic system. However, for 5 points random network based on prob-ability and 10 points random network based on probability, before the systems reach their equilibriums, their total amplitudes fluctuate intensively and their transit time is very long. After they reach the equilibriums, their total amplitudes fluctuate around 1, which means that their equilibriums are not fixed points and the choice of matrix (βkj) as well as initial value has effect on the equilibriums. Their total amplitudes stay in the value which is only a little bigger than one, which means that the choice of matrix (βkj) as well as initial value has only a little effect on the equilibriums. The reason for the above phenomenon might be: for regular network with 5 points and regular network with 10 points[2], the graphs of dis-ease transition are strongly connected. For random network with 10 points and 20 edges[2], the probability that the graph of disease transition is strongly connected is very large. But for 5 points random network based on probability and 10 points random network based on probability, the probability that the graphs of disease transition are strongly connected is not very large. When the graph of disease transition is not strongly connected, the system of epidemic model becomes complicated and the final state of the model is not easy to predict, which is the reason why the amplitudes of the last two systems fluctuate intensively and why their equilibriums are not fixed points. For these five systems, differences in their fluctuation of amplitude in transit stage proves differences among the five systems, so we can choose fluctuation of amplitude as a norm for us to distinct different systems.(2)Most conclusions for epidemic model are based on regular networks, so they have limited applicability. For example, when B is a regular network, the final state of epidemic model has only two options-all groups tend to their endemic equilibrium or all groups tend to their disease free equilibrium. However, when B is a random network, the final state of epidemic model becomes very complicated, such as some groups tending to their epidemic equilibrium while other groups tending to their disease free equilibrium. By 200 simulations on computer, we find that the probability of some groups tending to their epidemic equi- librium while other groups tending to their disease free equilibrium, is 21/200 when B is a random network with 10 points and 20 edges[2], is 63/200 when B is a 10 points random network based on probability, is 107/200 when B is 5 points random network based on prob-ability.(3) After the system of epidemic model reaches its asymptotic point, the value of S does not change for all groups, so the coefficient of any two groups is 1, which means that all groups belongs to one cluster. However, before the system reaches its asymptotic point, the coefficient of any two groups varies. We notice that the number of cluster becomes smaller and smaller when time becomes longer and longer. Finally all groups belong to one cluster. For regular network, the transit time for epidemic model is very short, so it takes short time for all groups tending to one cluster.(4) when we change the value ofεk, it has little effect on the length of transit time, the variation in the number of cluster and the final state of epidemic model (all groups tending to endemic equilibrium or disease free equilibrium, some groups tending to endemic equi-librium while other tending to disease free equilibrium). The reason for the above result is that the change ofεk has little effect on the value of R0·When we change the value ofγk, it has much effect on the length of transit time. It turns out that whenγk becomes bigger and bigger, the length of transit time first increases and the n decreases, finally it almost stays in one fixed value. It is easy to explain the result practically, whenγk is small, which means that the disease is hard to cure, so it takes not much time for all groups tending to endemic equilibrium. Whenγk becomes bigger, it takes fewer time for the disease to cure and the value of Ik is smaller than the value of Ik whenγkis very small just because the disease is not so hard to cure as whenγk is very small. When the value ofγk continues to increase, it is easy to cure the disease so it takes only a little time for all groups tending to disease free equilibrium. Overall whenγk becomes bigger, the disease goes from hard to cure to normal situation to easy to cure, which means that the transit time goes from small to big to small again. For regular network, the transit time of epidemic model has a linear relationship with time, while for random network, the transit time of epidemic model has a nonlinear relationship with time. The reason for the above result is that for systems with random network, they contain so much randomness that it is hard for two variable to have a linear relationship.The change ofγk has much effect on the median total amplitude during transit and asymptotic phase. For every network or system, the median total amplitude during transit phase is similar to that during asymptotic phase (solid lines and dashed lines coincide). We notice that for regular network, median total amplitude stays around 1 both in transit and asymptotic phase, but for random network, median total amplitude first decreases and then stays on one fixed value both in transit and asymptotic phase. The reason for the above result is that for the last three networks, there is so much randomness in the choosing of B that it plays a major role whenγk is very small. But whenγk becomes bigger, it plays a bigger role than the randomness so that the median total amplitude stays around 1.For random network with 10 points and 20 edges, mean transit time decreases whenβkj increases. For the other four network, mean transit time first increases and the decreases whenβkj increases. The reason for this is that whenβkj is big, the transition of disease be-comes chaotic so it takes a long time for system to reach asymptotic point, but whenβkj becomes bigger, the transition of disease becomes so fast that it takes only a short time for system to reach asymptotic time. The change ofβkj has much effect on the median total amplitude during transit and asymptotic phase. For every network or system, the median total amplitude during transit phase is similar to that during asymptotic phase (solid lines and dashed lines coincide) whenβkj increases. For regular network, median total amplitudes during transit and asymptotic phase are around 1. While for random network, median total amplitudes during transit and asymptotic phase increase whenβkj increases, which means that the increase ofβkj complicates the system.Overall, (1) when the final state of a group is its disease free equilibrium, its transit time will be very short whatever the network is(2) The meaning ofγk:whenγk is small, the disease is so hard to cure that the transit time is very long. Whenγk is big, the disease is easy to cure so the transit time is short. Whenγk is very large, all of the groups will go to their disease free equilibrium so the transit time is very short.(3) the meaning ofβkj: whenβkj is small, the transition of disease is so slow that it is easy for the system to reach its asymptotic point, what's more, the number of infectious people will be small. Whenβkj is large, the system becomes complicated so it is hard for the system to reach its asymptotic point. Whenβkj is very large, it plays a bigger role in speeding the system to reach its asymptotic point than in complicating the system, so the transit time is small.
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