Summary and Conclusion - EPIDEMICS OVER COMPLEX NETWORKS: ANALYSIS OF EXACT AND APPROXIMATE MOD

EPIDEMICS OVER COMPLEX NETWORKS: ANALYSIS OF EXACT AND APPROXIMATE MODELS

2.7 Summary and Conclusion

We studied the networked SIS, SIRS, SEIRS, SIV, and SEIV epidemics and their variants, using their exact Markov chain models, and their well-known linear and nonlinear mean-field approximations. Below a threshold, the disease-free fixed point is globally stable for the nonlinear model, and also the mixing time of the exact Markov chain is𝑂(log𝑛), which means the epidemic dies out fast. Furthermore,

10⁰ 10¹ 10² 10³ 10⁰

10¹ 10² 10³

Time Step

Number of Infected Nodes

βkAk δ = 1.2>1

βkAk

δ = 0.99<1

10⁰ 10¹ 10² 10³

Time Step

Number of Infected Nodes

γ γ+θ

βkAk δ = 1.2>1

γ γ+θ

βkAk

δ = 0.99<1

10⁰ 10¹ 10² 10³

Time Step

Number of Infected Nodes

(1−θ)_γ+θ^γ ^βkAk_δ = 1.2>1 (1−θ)_γ+θ^γ ^βk_δ^A^k= 0.99<1

Figure 2.5: The evolution of (a) SIS/SIRS/SEIRS, (b) SIV/SEIV (infection-dominant), (c) SIV/SIEV (vaccination-dominant) epidemics over an Erdős-Rényi graph with 𝑛 = 2000 nodes. The blue curves show fast extinction of the epidemic. The red curves show epidemic spread around the nontrivial fixed point.

above a threshold, the disease-free fixed point is not stable for the linear and nonlinear models, and there exists a second unique fixed point, which corresponds to the endemic state. This nontrivial fixed point is also stable in most cases. Figure 2.3 summarizes all the results.

Typical examples of the spread of the epidemic for all different propagation models studied throughout the chapter have been demonstrated in Figure 2.5. For the SIRS and SEIRS models, the threshold condition is ^𝛽k_𝛿^𝐴k <1, which is the same as that of the SIS one, and it means having an additional recovered state does not necessarily make the system more stable. For the infection-dominant SIV and SEIV models, we observe the same exponential decay of the infection when _𝛾^𝛾₊_𝜃^𝛽^k_𝛿^𝐴^k <1 (e.g., when k𝐴k = 16.232 and 𝛽 = 0.11, 𝛾 =0.5 and𝜃 =0.5), which means the vaccination indeed makes the system more stable. Furthermore, for the vaccination-dominant models, under(1−𝜃)_𝛾+𝜃^𝛾 ^𝛽k𝐴k

𝛿 < 1 (e.g., 𝛽=0.22), we observe the fast convergence again, which confirms that the system is even more stable when vaccination is dominant. As plots show, for above-the-threshold cases (e.g., 𝛽 = 0.07 for SIRS, 0.13 for SIV-infection-dominant, and 0.29 for SIV-vaccination-dominant), we do not observe epidemic extinction in any reasonable time, and effectively, the epidemic remains endemic.

Finally, we should remark that characterizing the exact epidemic threshold of the Markov chain model is still an open problem. Extensive numerical simulations suggest the existence of such a threshold and a phase transition behavior. Even though in certain networks, such as the Erdős-Rényi random graphs, the epidemic threshold seems to coincide with the condition for local stability of the nonlinear mean-field model (global stability of the linear model), it is different from that condition in general. For this reason, pairwise (and even higher-order) approximations may be sought, which provide tighter bounds on the epidemic threshold.

2.A Additional Models

We review additional epidemic models, namely the immune-admitting variant of the SIS model, the SIERS model, the vaccination-dominant variant of the SIV model, and the SEIV model.

2.A.1 Immune-Admitting SIS 2.A.1.1 Exact Markov Chain Model

A variant of the SIS model is the “immune-admitting” SIS, which is similar to the previous model except that a node does not get infected from its neighbors if it has just recovered from the disease (see Fig. 2.1). In other words, the probability of recovering from the disease is𝛿. That is

P(𝜉_𝑖(𝑡+1) =𝑌_𝑖|𝜉(𝑡) = 𝑋)













(1− 𝛽)^𝑚^𝑖 if(𝑋_𝑖, 𝑌_𝑖)= (0,0), |𝑁_𝑖∩S(𝑋) | =𝑚_𝑖, 1− (1− 𝛽)^𝑚^𝑖 if(𝑋_𝑖, 𝑌_𝑖)= (0,1), |𝑁_𝑖∩S(𝑋) | =𝑚_𝑖,

𝛿 if(𝑋_𝑖, 𝑌_𝑖)= (1,0), 1−𝛿 if(𝑋_𝑖, 𝑌_𝑖)= (1,1).

(2.34)

and, as before, the elements of the transition matrix are defined as P(𝜉(𝑡+1) =𝑌|𝜉(𝑡) = 𝑋) =

𝑛

𝑖=1

P(𝜉_𝑖(𝑡+1) =𝑌_𝑖|𝜉(𝑡) = 𝑋). (2.35) In this model, the probability that a node becomes healthy from infected (𝛿) is larger than that of the “immune-free” model (𝛿(1−𝛽)^𝑚^𝑖). Therefore, roughly speaking, the immune-admitting model is more likely than the immune-free model to hit the absorbing state.

2.A.1.2 Nonlinear Model

A mean-field approximation for the immune-admitting model can be studied as well, which is defined as

𝑃_𝑖(𝑡+1) = (1−𝛿)𝑃_𝑖(𝑡) + (1−𝑃_𝑖(𝑡)) 1− Ö

𝑗∈𝑁_𝑖

(1− 𝛽 𝑃_𝑗(𝑡))

. (2.36)

2.A.1.3 Linear Model

The nonlinear model has the same Jacobian matrix as that of the previous section, which is

𝑃(𝑡+1) =( (1−𝛿)𝐼_𝑛+𝛽 𝐴)𝑃˜(𝑡). (2.37)

2.A.2 Susceptible-Exposed-Infected-Recovered-Susceptible (SEIRS) 2.A.2.1 Exact Markov Chain Model

This model has an extra "exposed" state. The state of the nodes can take one of the following values: 0 forSusceptible, 1 forExposed, 2 forInfected(or Infectious), and 3 forRecovered(see Fig. 2.1). The 4^𝑛×4^𝑛state transition matrix𝑆of the Markov chain has elements of the form

𝑆_{𝑋 ,𝑌} =P(𝜉(𝑡+1) =𝑌 |𝜉(𝑡) =𝑋) =

𝑛

𝑖=1

P(𝜉_𝑖(𝑡+1) =𝑌_𝑖 | 𝜉(𝑡) = 𝑋), (2.38) as before. Here we have

P(𝜉_𝑖(𝑡+1) =𝑌_𝑖 | 𝜉(𝑡) = 𝑋) =













(1−𝛽)^𝑚^𝑖, if (𝑋_𝑖, 𝑌_𝑖) =(0,0) 1− (1−𝛽)^𝑚^𝑖, if (𝑋_𝑖, 𝑌_𝑖) =(0,1) 1−𝜖 , if (𝑋_𝑖, 𝑌_𝑖) =(1,1) 𝜖 , if (𝑋_𝑖, 𝑌_𝑖) =(1,2) 1−𝛿, if (𝑋_𝑖, 𝑌_𝑖) =(2,2) 𝛿, if (𝑋_𝑖, 𝑌_𝑖) =(2,3) 𝛾 , if (𝑋_𝑖, 𝑌_𝑖) =(3,0) 1−𝛾 , if (𝑋_𝑖, 𝑌_𝑖) =(3,3)

0, otherwise

, (2.39)

where𝑚_𝑖 =|𝑁_𝑖∩𝐼(𝑡) |. 2.A.2.2 Nonlinear Model

The nonlinear mean-field approximation is 𝑃_{𝐸 ,𝑖}(𝑡+1) =(1−𝜖)𝑃_{𝐸 ,𝑖}(𝑡)+

1− Ö

𝑗∈𝑁𝑖

(1−𝛽 𝑃_{𝐼 , 𝑗}(𝑡))

(1−𝑃_{𝐸 ,𝑖}(𝑡) −𝑃_{𝐼 ,𝑖}(𝑡) −𝑃_𝑅,𝑖(𝑡)) 𝑃_{𝐼 ,𝑖}(𝑡+1) =𝜖 𝑃_{𝐸 ,𝑖}(𝑡) + (1−𝛿)𝑃_{𝐼 ,𝑖}(𝑡)

𝑃_𝑅,𝑖(𝑡+1) =(1−𝛾)𝑃_𝑅,𝑖(𝑡) +𝛿 𝑃_{𝐼 ,𝑖}(𝑡).

2.A.2.3 Linear Model

The linearization of the above equations around the origin is







𝑃_𝐸(𝑡 +1) 𝑃˜_𝐼(𝑡+1)

𝑃_𝑅(𝑡+1)







=𝑀







˜ 𝑃_𝐸(𝑡)

𝑃˜_𝐼(𝑡)

˜ 𝑃_𝑅(𝑡)





 , where

𝑀 =







(1−𝜖)𝐼_𝑛 𝛽 𝐴 0𝑛×𝑛

𝜖 𝐼_𝑛 (1−𝛿)𝐼_𝑛 0𝑛×𝑛

−𝜃 𝐼_𝑛 𝛿 𝐼_𝑛 (1−𝛾)𝐼_𝑛





 .

2.A.3 Vaccination-Dominant SIV 2.A.3.1 Exact Markov Chain Model

In the vaccination-dominant variant of the model, the assumption is that if a susceptible node receives both infection and vaccine at the same time, it becomes vaccinated. Although in the context of contagious diseases, this variation might make less sense, in other applications, there are scenarios for which this model is more relevant. The transition probabilities of the Markov chain are again

𝑆_{𝑋 ,𝑌} =P(𝜉(𝑡+1) =𝑌 |𝜉(𝑡) =𝑋) =

𝑛

𝑖=1

P(𝜉_𝑖(𝑡+1) =𝑌_𝑖 | 𝜉(𝑡) = 𝑋), (2.40) with the change that

P(𝜉_𝑖(𝑡+1) =𝑌_𝑖 |𝜉(𝑡) =𝑋) =













(1−𝛽)^𝑚^𝑖(1−𝜃), if (𝑋_𝑖, 𝑌_𝑖) =(0,0) (1− (1−𝛽)^𝑚^𝑖) (1−𝜃), if (𝑋_𝑖, 𝑌_𝑖) =(0,1) 𝜃 , if (𝑋_𝑖, 𝑌_𝑖) =(0,2) 0, if (𝑋_𝑖, 𝑌_𝑖) =(1,0) 1−𝛿, if (𝑋_𝑖, 𝑌_𝑖) =(1,1) 𝛿, if (𝑋_𝑖, 𝑌_𝑖) =(1,2) 𝛾 , if (𝑋_𝑖, 𝑌_𝑖) =(2,0) 0, if (𝑋_𝑖, 𝑌_𝑖) =(2,1) 1−𝛾 , if (𝑋_𝑖, 𝑌_𝑖) =(2,2) ,

(2.41) where𝑚_𝑖 =

𝑗 ∈ 𝑁_𝑖 | 𝑋_𝑗 =1

=|𝑁_𝑖∩𝐼(𝑡) |, as before.

2.A.3.2 Nonlinear Model

The nonlinear map, or the mean-field approximation, can be stated as:

𝑃_𝑅,𝑖(𝑡+1) =(1−𝛾)𝑃_𝑅,𝑖(𝑡) +𝛿 𝑃_{𝐼 ,𝑖}(𝑡)

+𝜃(1−𝑃_𝑅,𝑖(𝑡) −𝑃_{𝐼 ,𝑖}(𝑡)), (2.42) 𝑃_{𝐼 ,𝑖}(𝑡+1) =(1−𝛿)𝑃_{𝐼 ,𝑖}(𝑡) + (1−𝜃)

1− Ö

𝑗∈𝑁𝑖

(1− 𝛽 𝑃_{𝐼 , 𝑗}(𝑡))

(1−𝑃_𝑅,𝑖(𝑡) −𝑃_{𝐼 ,𝑖}(𝑡)). (2.43)

2.A.3.3 Linear Model

As a result, the first order (linear) model is:

𝑃_𝑅(𝑡+1)

𝑃_𝐼(𝑡+1)

𝑃^∗

𝑅1𝑛

0𝑛

# +𝑀

𝑃_𝑅(𝑡) −𝑃^∗

𝑅1𝑛

𝑃_𝐼(𝑡) −0𝑛

# , where

𝑀 =

(1−𝛾−𝜃)𝐼_𝑛 (𝛿−𝜃)𝐼_𝑛−𝜃 𝑃^∗

𝑆𝛽 𝐴 0𝑛×𝑛 (1−𝛿)𝐼_𝑛+ (1−𝜃)𝑃^∗

𝑆𝛽 𝐴

# .

We should note that for the vaccination-dominant model, the steady state of the Markov chain and the main fixed point of the mapping are exactly the same as those of the infection-dominant model. However, as one may expect, it turns out that the vaccination-dominant model is more stable.

2.A.4 SEIV (Infection-Dominant)

The Markov chain model in this case has the following transition probabilities (see Fig. 2.1).

2.A.4.1 Exact Markov Chain Model

P(𝜉_𝑖(𝑡+1) =𝑌_𝑖 |𝜉(𝑡)= 𝑋) =













(1−𝛽)^𝑚^𝑖(1−𝜃), if(𝑋_𝑖, 𝑌_𝑖) =(0,0) 1− (1−𝛽)^𝑚^𝑖, if(𝑋_𝑖, 𝑌_𝑖) =(0,1) (1−𝛽)^𝑚^𝑖𝜃 , if(𝑋_𝑖, 𝑌_𝑖)= (0,3) 1−𝜖 , if(𝑋_𝑖, 𝑌_𝑖)= (1,1) 𝜖 , if(𝑋_𝑖, 𝑌_𝑖)= (1,2) 1−𝛿, if(𝑋_𝑖, 𝑌_𝑖)= (2,2) 𝛿, if(𝑋_𝑖, 𝑌_𝑖)= (2,3) 𝛾 , if(𝑋_𝑖, 𝑌_𝑖)= (3,0) 1−𝛾 , if(𝑋_𝑖, 𝑌_𝑖)= (3,3)

0, otherwise

, (2.44)

where𝑚_𝑖 =|𝑁_𝑖∩𝐼(𝑡) |. 2.A.4.2 Nonlinear Model

The nonlinear approximation in this case is 𝑃_{𝐸 ,𝑖}(𝑡+1) =(1−𝜖)𝑃_{𝐸 ,𝑖}(𝑡)+

1− Ö

𝑗∈𝑁𝑖

(1− 𝛽 𝑃_{𝐼 , 𝑗}(𝑡))

(1−𝑃_{𝐸 ,𝑖}(𝑡) −𝑃_{𝐼 ,𝑖}(𝑡) −𝑃_𝑅,𝑖(𝑡)) 𝑃_{𝐼 ,𝑖}(𝑡+1) =𝜖 𝑃_{𝐸 ,𝑖}(𝑡) + (1−𝛿)𝑃_{𝐼 ,𝑖}(𝑡)

𝑃_𝑅,𝑖(𝑡+1) =(1−𝛾)𝑃_𝑅,𝑖(𝑡) +𝛿 𝑃_{𝐼 ,𝑖}(𝑡)

+ Ö

𝑗∈𝑁_𝑖

(1−𝛽 𝑃_{𝐼 , 𝑗}(𝑡))

𝜃(1−𝑃_{𝐸 ,𝑖}(𝑡) −𝑃_{𝐼 ,𝑖}(𝑡) −𝑃_𝑅,𝑖(𝑡)).

2.A.4.3 Linear Model

The linearization around the main fixed point is as follows







𝑃_𝐸(𝑡+1)

𝑃_𝐼(𝑡+1) 𝑃˜_𝑅(𝑡+1)











 0𝑛

0𝑛

𝑃^∗

𝑅1𝑛





 +𝑀







˜ 𝑃_𝐸(𝑡)

˜ 𝑃_𝐼(𝑡) 𝑃˜_𝑅(𝑡) −𝑃^∗

𝑅1𝑛





 , where

𝑀 =







(1−𝜖)𝐼_𝑛 𝑃^∗

𝑆

𝛽 𝐴 0𝑛×𝑛

𝜖 𝐼_𝑛 (1−𝛿)𝐼_𝑛 0𝑛×𝑛

−𝜃 𝐼_𝑛 (𝛿−𝜃)𝐼_𝑛−𝜃 𝑃^∗

𝑆

𝛽 𝐴 (1−𝛾−𝜃)𝐼_𝑛





 .

2.A.5 SEIV (Vaccination-Dominant) 2.A.5.1 Exact Markov Chain Model

The vaccination-dominant variant of the model has the following transition probabilities.

P(𝜉_𝑖(𝑡+1) =𝑌_𝑖 |𝜉(𝑡) =𝑋) =













(1−𝛽)^𝑚^𝑖(1−𝜃), if (𝑋_𝑖, 𝑌_𝑖) =(0,0) (1− (1−𝛽)^𝑚^𝑖) (1−𝜃), if (𝑋_𝑖, 𝑌_𝑖) =(0,1) 𝜃 , if (𝑋_𝑖, 𝑌_𝑖) =(0,3) 1−𝜖 , if (𝑋_𝑖, 𝑌_𝑖) =(1,1) 𝜖 , if (𝑋_𝑖, 𝑌_𝑖) =(1,2) 1−𝛿, if (𝑋_𝑖, 𝑌_𝑖) =(2,2) 𝛿, if (𝑋_𝑖, 𝑌_𝑖) =(2,3) 𝛾 , if (𝑋_𝑖, 𝑌_𝑖) =(3,0) 1−𝛾 , if (𝑋_𝑖, 𝑌_𝑖) =(3,3)

0, otherwise

(2.45) where𝑚_𝑖 =|𝑁_𝑖∩𝐼(𝑡) |.

2.A.5.2 Nonlinear Model

The nonlinear mean-field approximation can be expressed as 𝑃_{𝐸 ,𝑖}(𝑡+1) = (1−𝜖)𝑃_{𝐸 ,𝑖}(𝑡) + (1−𝜃)×

1− Ö

𝑗∈𝑁_𝑖

(1−𝛽 𝑃_{𝐼 , 𝑗}(𝑡))

(1−𝑃_{𝐸 ,𝑖}(𝑡) −𝑃_{𝐼 ,𝑖}(𝑡) −𝑃_𝑅,𝑖(𝑡)) 𝑃_{𝐼 ,𝑖}(𝑡+1) =𝜖 𝑃_{𝐸 ,𝑖}(𝑡) + (1−𝛿)𝑃_{𝐼 ,𝑖}(𝑡)

𝑃_𝑅,𝑖(𝑡+1) =(1−𝛾)𝑃_𝑅,𝑖(𝑡) +𝛿 𝑃_{𝐼 ,𝑖}(𝑡) +𝜃(1−𝑃_{𝐸 ,𝑖}(𝑡) −𝑃_{𝐼 ,𝑖}(𝑡) −𝑃_𝑅,𝑖(𝑡)).

2.A.5.3 Linear Model The linearized model is







𝑃_𝐸(𝑡+1)

𝑃_𝐼(𝑡+1)

𝑃_𝑅(𝑡+1)











 0𝑛

0𝑛

𝑃^∗

𝑅1𝑛





 +𝑀







˜ 𝑃_𝐸(𝑡)

˜ 𝑃_𝐼(𝑡)

𝑃_𝑅(𝑡) −𝑃^∗

𝑅1𝑛





 , where

𝑀 =







(1−𝜖)𝐼_𝑛 (1−𝜃)𝑃^∗

𝑆

𝛽 𝐴 0𝑛×𝑛

𝜖 𝐼_𝑛 (1−𝛿)𝐼_𝑛 0𝑛×𝑛

−𝜃 𝐼_𝑛 (𝛿−𝜃)𝐼_𝑛 (1−𝛾−𝜃)𝐼_𝑛





 .

C h a p t e r 3

IMPROVED BOUNDS ON THE EPIDEMIC THRESHOLD OF THE

Dalam dokumen From Network Dynamics to Optimization Algorithms (Halaman 50-60)