Results on the Nonlinear MFA Model

EPIDEMICS OVER COMPLEX NETWORKS: ANALYSIS OF EXACT AND APPROXIMATE MODELS

2.3 Results on the Nonlinear MFA Model

The nonlinear mean-field approximation has been extensively studied in the literature for different propagation models. We review the most important results here, starting from the SIS epidemics.

2.3.1 SIS

It is straightforward to see that the linear model upper bounds the nonlinear one, as follows.

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

𝑗∈𝑁𝑖

(1−𝛽 𝑃_𝑗(𝑡))

!

≤(1−𝛿)𝑃_𝑖(𝑡) + 1− Ö

𝑗∈𝑁𝑖

(1− 𝛽 𝑃_𝑗(𝑡))

!

≤(1−𝛿)𝑃_𝑖(𝑡) +𝛽 Õ

𝑗∈𝑁𝑖

𝑃_𝑗(𝑡)

!

For two real-valued column vectors𝑢, 𝑣 ∈R^𝑛, we use the notation𝑢 𝑣to indicate 𝑢_𝑖 ≤ 𝑣_𝑖 for all 𝑖 ∈ {1, . . . , 𝑛}, and 𝑢 ≺ 𝑣, if the inequalities are strict. Defining 𝑃(𝑡) = [𝑃₁(𝑡), . . . , 𝑃_𝑛(𝑡)]^𝑇, we have

𝑃(𝑡+1) ( (1−𝛿)𝐼_𝑛+ 𝛽 𝐴)𝑃(𝑡), (2.26) which leads to the following well-known result.

Proposition 1. If ^𝛽𝜆max𝛿^(𝐴) < 1, the origin is a globally asymptotically stable fixed point for both the linear SIS model(2.8)and the nonlinear SIS model(2.5).

The origin, the trivial fixed point of the nonlinear model, is unstable when𝜆_{𝑚 𝑎𝑥}( (1− 𝛿)𝐼_𝑛+𝛽 𝐴) > 1. Moreover, if so, it is not clear in general whether there exists any

Figure 2.3: Summary of known results for different models. The results have been illustrated as a function of ^{𝛽𝜆𝑚𝑎 𝑥}_𝛿 ^(𝐴). MC stands for the Markov chain model. MFA stands for the mean-field approximation (the nonlinear model).

other fixed point, or how many fixed points there are. It has been shown in the literature (e.g., in [5]) that there exists a unique nontrivial fixed point, and it is stable.

Theorem 2. If ^𝛽𝜆max𝛿^(𝐴) > 1, the nonlinear SIS model(2.5)has a second unique fixed point. Furthermore, the fixed point is globally asymptotically stable from all initial points (except the origin).

2.3.2 SIRS/SEIRS/Immune-Admitting-SIS

Similar to the previous (immune-free) SIS model, for the immune-admitting-SIS, SIRS, and SEIRS epidemics, the linear model is an upper-bound on the nonlinear one, and therefore the origin is stable for the nonlinear model when the linear model is stable.

Proposition 3. If ^𝛽𝜆max𝛿^(𝐴) < 1, the origin is a globally asymptotically stable fixed point for both the linear model and the nonlinear model for immune-admitting-SIS, SIRS, and SEIRS epidemics.

In this case, when the origin is not stable, even though there still exists a unique nontrivial fixed point, it is not stable in general [5, 16].

Theorem 4. If ^𝛽𝜆max𝛿^(𝐴)

> 1, the nonlinear model for immune-admitting-SIS, SIRS, and SEIRS epidemics has a second unique fixed point.

The following is an example of an unstable nontrivial fixed point for the immune- admitting-SIS model [4, p. 64].

A=©

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0 1 1 1 0 0 1 0 0

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¬

𝛽=0.9 𝛿 =0.9 (2.27)

The nontrivial fixed point of the system above is𝑃^∗ =(0.286,0.222,0.222)^𝑇. The linearized model around𝑃^∗ is

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−0.260 0.514 0.514 0.700 −0.157 0 0.700 0 −0.157

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¬ ,

which has an eigenvalue of−1.059, which means that𝑃^∗ is not locally stable. It can be shown that for any initial condition other the origin and𝑃^∗,𝑃(𝑡)converges to a cycle.

Even though the nontrivial fixed point is not stable in general, it is known to be stable with high probability for a general family of random graphs [4, p. 66].

Theorem 5 (Ahn and Hassibi [5]). Suppose that 𝐺^(𝑛) is a random graph with 𝑛 vertices, and let𝑑⁽

𝑛)

min and𝑑⁽

𝑛)

maxdenote the minimum and maximum degree of𝐺^(𝑛). If, for any fixed𝑎 >0,P( (𝑑⁽

𝑛)

min)² > 𝑎·𝑑⁽

𝑛)

max)goes to1as𝑛goes to infinity, then, with high probability, the origin is unstable, and the second fixed point is locally stable, for any fixed 𝛽and𝛿.

One can think of several random graph models that satisfy the condition of Theorem 5. For example, if the random graph has a uniform degree that grows with𝑛, then the minimum degree and maximum degree are identical and the ratio

𝑑²

min

𝑑_max

=𝑑will grow with any𝑛and exceed𝑎with high probability. Similarly, for random graphs where the degree distribution of the nodes are identical and concentrate, so that we can expect the maximum degree and the minimum degree to be proportional to the expected degree,

𝑑²

min

𝑑_max grows if the expected degree increases unbounded with𝑛. In particular, the Erdös-Rényi random graph𝐺^(𝑛) =𝐺(𝑛, 𝑝(𝑛))has identical degree distribution, and we have the following result [4, p. 69].

Corollary 6(Ahn and Hassibi [5]). Consider an Erdös-Rényi random graph𝐺^(𝑛) = 𝐺(𝑛, 𝑝(𝑛))with 𝑝(𝑛)=𝑐

log𝑛

𝑛 where𝑐 > 1is a constant. The nonlinear model is

locally unstable at the origin and has a locally stable nontrivial fixed point with high probability for any fixed𝛽and𝛿.

Since 𝑝 = 𝑐log𝑛 𝑛

for 𝑐 = 1 is also the threshold for connectivity, we can say that connected Erdös-Rényi graphs have a nontrivial stable fixed point with high probability.

The random geometric graph𝐺^(𝑛) =𝐺(𝑛, 𝑟(𝑛))also has identical degree distribution if each node is distributed uniformly. Such random graphs have maximum and minimum degrees which are proportional to the expected degree with high probability if𝑟(𝑛)is smaller than the threshold of connectivity [167], and, similar to Erdös-Rényi graphs, it has high probability of having a nontrivial stable fixed point if the degree grows with𝑛.

2.3.3 SIV/SEIV (Infection-Dominant)

Since the linear model is always the Jacobian of the nonlinear one, its stability determines the local stability of the nonlinear model. However, global stability is harder to show. The following result summarizes the stability of the nonlinear model.

Proposition 7. The disease-free fixed point of the nonlinear model for the infection- dominant SIV and the infection-dominant SEIV epidemics is

a) locally stable, if _𝛾+𝜃^{𝛾 𝛽}𝛿

𝜆_{𝑚 𝑎𝑥}(𝐴) <1, and b) globally stable, if ^𝛽𝛿𝜆_{𝑚 𝑎𝑥}(𝐴) < 1.

When _𝛾^𝛾_+𝜃^𝛽𝜆max_𝛿^(𝐴) >1, the main fixed point of the nonlinear map is not stable, and again, there exists a unique non-trivial fixed point. This result has been proven in [16] for the SIV case, and it extends to the SEIV model in a similar fashion.

Theorem 8. If 𝛾^𝛾+𝜃

𝛽𝜆_max(𝐴)

𝛿 > 1, the nonlinear model for the infection-dominant SIV and the infection-dominant SEIV epidemics has a second unique fixed point.

The middle panel in Figure 2.3 shows a summary of the above results.

2.3.4 SIV/SEIV (Vaccination-Dominant)

It is natural to expect the vaccination-dominant models to be more stable than the infection-dominant ones. It turns out that these models are indeed more stable by a factor of 1/(1−𝜃).

Proposition 9. The disease-free fixed point of the nonlinear model for the vaccination- dominant SIV and the vaccination-dominant SEIV epidemics is

a) locally stable, if(1−𝜃)_𝛾+𝜃^{𝛾 𝛽}

𝛿𝜆_{𝑚 𝑎𝑥}(𝐴) < 1, and b) globally stable, if(1−𝜃)^𝛽

𝛿𝜆_{𝑚 𝑎𝑥}(𝐴) < 1. Theorem 10. If (1−𝜃) ^𝛾

𝛾+𝜃 𝛽

𝛿𝜆_{𝑚 𝑎𝑥}(𝐴) > 1, the nonlinear model for the vaccination- dominant SIV and the vaccination-dominant SEIV epidemics has a second unique fixed point.

Both of the above results have been proven in [16] for the SIV case, and the proof extends to the SEIV case with some modification. The lower panel in Figure 2.3 shows a summary of these results.