Network model formalized by J.C. Miller, A.C. Slim & E.M. Volz(2011). Result agreed with a recent paper by RomanescuEtAL(2023) and Novozhilov(2008)
Quick Summary: approaches.md
Consider a random network model that model a population
For random distribution given a degree distribution with PDF: \(\mathbb{P}(K=d)=p_d\). The probability generating function(PGF) of degree distribution is denoted by:\[G_p(x)=\sum_{d=0}^{\infty}p_d x^{d}\] - A useful expression would be the mean degree \(\delta\) is given by:\[\delta=\sum_{d=0}^{\infty}p_d d=\frac{d}{dx}G_p(x)|_{x=1}=G_p'(1)\]
Also for network model, Excess degree could also be important, as during the outbreak, any newly infected vertex with degree \(k\) could only infect at most \(k-1\) of its susceptible neighbors, as its infection must come from one neighbor that already being infected. Based on given degree distribution with PDF \(p_d\), we could define the distribution of excess degree with PDF denoted by \(q\), such that:\[\mathbb{P}(\text{excess degree}=d-1)=q_{d-1}=\frac{p_d d}{\sum_{k=0}^{\infty}p_k k}=\frac{p_d d}{\delta}\] The corresponding PGF for excess degree is \[G_q(x)=\frac{G_p'(x)}{\delta}\]
Following J.C. Miller, A.C. Slim & E.M. Volz(2011)(MSV), at any moment \(t\) we define \(\theta(t)\): - (MSV definition) the probability that randomly chosen neighbor vertex \(b\) of a randomly chosen vertex \(a\) has not yet transmit the infectious to \(a\). - (JD definition) the probability of a randomly chosen edge in the network has not yet transmit infection. Let \(\psi=1-\theta\) be the probability that the infection has transmitted.
==MSV claim for a random network with large enough size, neighbors of
a randomly chosen vertex \(a\) are
independent, which could be a key assumption for the framework to work
in large size network.== Given its degree \(d\), vertex \(a\) is susceptible at time \(t\) with probability \(s(d; \theta(t)) = \theta(t)^d\). Therefore,
the proportion of susceptible vertex \(S(t)\) at time \(t\) is given by:\[S(t)=G_p(\theta(t))=\sum_{d}p_d
\theta(t)^d\] Now for the vertices compartment, one can write a
system such that: \[
\begin{cases}
S(t)=G_p(\theta(t))
\\
I(t)=1-S(t)-R(t)
\\
\dot{R}(t)=\gamma I(t)
\end{cases}
\]
Now to find the ODE that governs \(\theta(t)\), we consider the 4 compartment of all edges/neighbor of a randomly chosen vertex \(a\): - \(\psi(t)=1-\theta(t)\): the proportion/probability that a neighbor \(b\) is infected and the infection has transmitted to \(a\). - \(\phi_S\): the proportion/probability that a neighbor \(b\) is susceptible. Given degree \(d\), \(b\) is susceptible with probability \(\theta(t)^{d-1}\) (not consider transmission from \(a\) so \(d-1\) nodes can infect \(b\)). A weighted average based on excess degree gives\[\phi_S=\sum_dq_{d-1} \theta(t)^{d-1}{}=\frac{G'(\theta)}{G'(1)}=\frac{G'(\theta)}{\delta}\] - \(\phi_I\): the proportion/probability that a neighbor \(b\) is infected but the infection has not yet transmitted to \(a\). - \(\phi_R\): the proportion/probability that a neighbor \(b\) is recovered but the infection has not transmitted to \(a\). - As we have a closed system, we have \[1=(1-\theta)+\phi_S+\phi_I+\phi_R \Leftrightarrow\theta=\phi_S+\phi_I+\phi_R\] - Consider the infection rate \(\beta\), the flux from \(\phi_I\) to \(1-\theta\) will be \(\beta \phi_I\), this gives: \[\frac{d\theta(t)}{dt}=\dot{\theta}=-\beta\phi_I=-\beta(\theta-\phi_S-\phi_R)\] - Consider the recovery rate \(\gamma\), the flux from \(\phi_I\) to \(\phi_R\) will be \(\gamma \phi_I\), this gives:\[\dot{\phi}_R=\gamma\phi_I\] - As \(\phi_R(0)\) and \(1-\theta(0)\) both close to \(0\), with the large network size, we have the integration:\[\phi_R=\frac{\gamma}{\beta}(1-\theta)\] - Therefore we combine these results got to:\[\dot{\theta}=-\beta\phi_I=-\beta(\theta-\phi_S-\phi_R)=-\beta\theta+\beta\frac{G_p'(\theta)}{\delta}+\gamma(1-\theta)\]
MSV named \(\mathcal{R}_0\) as basic reproductive number and use the same notation as in homogeneous model. However, its idea and definition is differ from intuitive understanding of \(\mathcal{R}_0\) for homogeneous model.
They define of \(\mathcal{R}_0\) from network approach first: \(\mathcal{R}_0\) is the expected number of infections a ==newly== infected vertex causes. This gives the expression:
\[\begin{align} \mathcal{R}_0&=\sum_{d=1} q_{d-1} \times(d-1)\times \frac{\beta}{\beta+\gamma} \\ &= \frac{\beta}{\beta+\gamma} \sum_{d=1} \frac{p_d \times d}{\delta} \times(d-1) \\ &=\frac{\beta}{\beta+\gamma} \frac{\sum_{d=1} p_d \times d \times(d-1)}{\delta} \\ &=\frac{\beta}{\beta+\gamma} \times\frac{G''_p(1)}{\delta}=\frac{\beta}{\beta+\gamma} \times\frac{G''_p(1)}{G'_p(1)} \end{align}\]
They then provide an equivalent derivations based on the ODE of \(\theta\): Consider \(\theta(0)=1+\epsilon\) (\(\epsilon<0\))which is very close to \(1\):\[\dot{\theta}=\dot{\epsilon}=-\beta(1+\epsilon)+\beta\frac{G_p'(1+\epsilon)}{\delta}+\gamma (1-(1+\epsilon))\] Consider the first order:
\[\begin{align} G_p'(1+\epsilon)&=\sum_d p_d d(1+\epsilon)^{d-1} \\ &= \sum_d p_d d (1+(d-1)\epsilon+o(\epsilon)) \\ & \approx\sum_d p_d d+\sum_d p_d d(d-1)\epsilon \\ &= \delta+\epsilon\sum_d p_d d (d-1) \\ & = \delta+\epsilon G_p''(1) \end{align}\]
Then for the first order we have: \[\dot{\theta}=\dot{\epsilon}\approx-\beta(1+\epsilon)+\beta\frac{\delta+\epsilon G_p''(1)}{\delta}-\gamma\epsilon=[-(\beta+\gamma)+\beta\frac{G_p''(1)}{\delta}] \times \epsilon\] For the outbreak, \(\mathcal{R}_0= \frac{\beta}{\beta+\gamma} \frac{G''_p(1)}{\delta}>1 \Leftrightarrow \frac{\dot{\epsilon}}{\epsilon}=[-(\beta+\gamma)+\beta\frac{G_p''(1)}{\delta}]>0\)
More precisely, since \(\dot{\theta}=-\beta\phi_I\), we can further have: \[\dot{\phi_I}=[-(\beta+\gamma)+\beta\frac{G''_p(\theta)}{\delta}] \phi_I\]
This result is not really situated inside this framework, although it was inspired by it. If we imagine that both the number of susceptible and their mean susceptibility are shaped by a uniform probability of transmission on each edge, then we can solve for each of them in terms of this probability, and consider the relationship between number of susceptible and total susceptibility of the population as a possible functional form for phenomenological heterogeneity.
Consider the effective “incidence” term using \[\rho=\frac{\mathcal{R}_{\text{eff}}}{\mathcal{R}_0}\] Follow JD’s idea: \[\rho=\frac{\mathcal{R}_{\text{eff}}}{\mathcal{R}_0}=\frac{\sigma_{\phi}}{\sigma_0}\] - \(\sigma_{\phi}\): Expected number of edges of susceptible vertices\[\sigma_{\phi}=\sum_{d=0}^{\infty}p_d \times d \times(1-\phi)^{d}=\sum_{d=0}^{\infty}p_d \times d \times \theta^{d}=\theta\sum_{d=0}^{\infty}p_d \times d \times\theta^{d-1}=\theta G_p'(\theta)\] - At \(t=0\), we have \(\sigma_\phi(0)=\sigma_0\): \[\sigma_0=\lim_{t\rightarrow0}\sigma_{\phi}=\lim_{\theta\rightarrow1} \theta G_p'(\theta)=\delta\] - Therefore, \[\rho=\frac{\theta G_p'(\theta)}{\delta}\]
Consider the Negative binomial distribution as degree distribution, which could also be seen as a continuous mixture of Poisson distribution and Gamma distribution. For this family of distributions (Negative Binomial, Poisson, Geometric), the PGF \(G_p\) could be written easily in closed form and the distributions are connected such that other distributions could be deduced from the Negative Binomial distribution.
Typical PMF of negative binomial distribution is given by:\[p_d=\binom{d+r-1}{d}(1-p)^dp^r\] with mean \[\delta=\frac{r(1-p)}{p}\] and variance \[\text{Var}=\frac{r(1-p)}{p^2}\] For better understanding, we parameterize the distribution using mean \(\delta\) and Dushoff-style squared CV \(\kappa\), such that
\[\begin{align} r & =\frac{1}{\kappa} \\ p & =\frac{1}{1+\kappa\delta} \end{align}\]
This gives us the variance \(\text{Var}=\delta(1+\kappa\delta)>\delta\), as these family of distributions are always over-dispersed, and further never converge to homogeneous case.
Now consider the PGFs of this family: - Negative Binomial distribution:\[G_p(\theta)=(\frac{1}{1+\kappa\delta-\theta\times\kappa\delta})^{\frac{1}{\kappa}}\] - Poisson: As \(\kappa \rightarrow 0\), NegBinom converge to Poisson distribution with \[G_p(\theta)=e^{\delta(\theta-1)}\] - Verification: As \[\lim_{a\rightarrow \infty}(1+\frac{x}{a})^a=e^x =\lim_{b\rightarrow0}(1-x \times b)^{-\frac{1}{b}}\]we have\[\lim_{\kappa\rightarrow 0} (\frac{1}{1+\kappa\delta-\theta\times\kappa\delta})^{\frac{1}{\kappa}}=\lim_{\kappa\rightarrow 0} (1-\kappa \times\delta(\theta-1))^{-\frac{1}{\kappa}}=e^{\delta(\theta-1)}\] - Geometric: As \(\kappa=1\), NegBinom is Geometric distribution by definition, with PGF: \[G_p(\theta)=\frac{1}{1+\delta-\theta\times\delta}\]
Just consider the general Negative Binomial case, as from the MSV framework, we got \[S=G_p(\theta)=(\frac{1}{1+\kappa\delta-\theta\times\kappa\delta})^{\frac{1}{\kappa}}\Rightarrow\theta=G^{-1}_p(S)=1+\frac{1-S^{-\kappa}}{\kappa\delta}\] and \[G'_p(\theta)=(\frac{1}{1+\kappa\delta-\theta\times\kappa\delta})^{\frac{1}{\kappa}}\times\frac{\delta}{1+\kappa\delta-\theta\times\kappa\delta}=\frac{S\delta}{1+\kappa\delta-\theta\times\kappa\delta}\] Take this into \(\rho\) gives us for Negative Binomial:
\[\begin{align} \rho&=\frac{\mathcal{R}_{\text{eff}}}{\mathcal{R}_0}=\frac{\theta G'_p(\theta)}{\delta}=\frac{S\theta}{1+\kappa\delta-\theta\times\kappa\delta} \\ &=S\times \frac{1+\frac{1-S^{-\kappa}}{\kappa\delta}}{1+\kappa\delta-(1+\frac{1-S^{-\kappa}}{\kappa\delta})\kappa\delta} \\ &=S\times(S^{\kappa}+\frac{S^{\kappa}-1}{\kappa\delta}) \end{align}\]
For further illustration, we define \[\sigma(S)=\frac{\rho}{S}\]and present the \(\sigma(S)\) curve on \([0,1]\) for the distributions mentioned and different \(\delta\) and \(\kappa\) value. Note the horizontal line at \(\sigma\equiv1\) represent the homogeneous case.
****
However, JD and RZ doubt this results should only work in a directed network. ==TODO: what kind of network/assumptions do we need for this result to work, as it is related to call known results==
A conjecture now is this apply for a network where in-degree is negative-binomial distributed while the directed edges towards each nodes is uniformly randomly connected to all other nodes. But it might not be that interesting.
==Idea: Derive this from Novozhilov Framework, even if discrete distribution does not fit well with their assumption: parametric heterogeneity of susceptibility==
(??) How to connect \(\mathcal{R}_\text{eff}\) with incidence
term \(\frac{dS}{dt}\)? \[\mathcal{R}_\text{eff}=-\frac{\frac{dS(t)}{dt}}{I(t)}\times\frac{1}{\gamma}\]
- ==This does not work for Miller’s network model== - ==(TODO: We need
further understanding this and give explanation)== - Def corresponding to \(-\frac{\frac{dS(t)}{dt}}{I(t)}\times\frac{1}{\gamma}\)
- Cal corresponding to JR’s result for Poisson distribution.
==(TODO: How to estimate the Def/Black curve here?)== Observation: for Poisson degree distribution, the peak value seems to be \((\delta-1)\frac{\beta}{\gamma}\). See sdeas at the last part of this document.
Recent paper by G.A. Rampala(2023) discussed about that for Poisson network, the MSV dynamic is equivalent to dynamic from a homogeneous SIR-like ODE system , but with modification. I have verified his result.
\[\begin{align} \dot{S}& = -\beta\delta SX_D \\ \dot{X}_D & = -\dot{S}-(\beta+\gamma)X_D \\ \dot{I} & = -\dot{S}-\gamma I \\ \dot{R} & = \gamma I \end{align}\]
This indicate that incidence term \(\dot{S}\) in network model is not directly determined by \(I\). Note, \(X_d\) curve converge to \(I\) as \(\delta \rightarrow \infty\) with the same \(\mathcal{R}_0\).
For MSV network frame with configuration network, I don’t think the relationship of reproductive number/ratio and \(\dot{S}\) from homogeneous model, like: \[\mathcal{R}_\text{eff}=-\frac{\frac{dS(t)}{dt}}{I(t)}\times\frac{1}{\gamma}\] can be directly applied. \(\mathcal{R}_\text{eff}\) defined as (expected) number of infection a randomly chosen infected individual can cause, which affect the incidence of the system.
For homogeneous model, any individual in \(I(t)\) have same infectivity to infect susceptible nodes, because of the fully-mixed mass-action assumption for contact. In such case, every infected individual has the same infectivity like any individual in \(I(t)\), so the \(\mathcal{R}_\text{eff}\) is proportion to new incident \(\dot{S}(t)\) averaged on \(I(t)\), i.e. the incident is governed by \(\mathcal{R}_\text{eff} \times I(t)\)
(??) However, this assumption might be less justifiable on MSV network frame with heterogeneity in contact.
Consider transmission on configuration network, it is possible to have an vertex in the \(I(t)\) compartment while no longer being able to transmit infection to any of its neighbor at the moment \(t\), i.e. all of its neighbor are not susceptible at and after time \(t\). An obvious example would be infected vertex with degree one, whose only neighbor will be its infector and thus not being able to transmit the infection to any other vertices. Unlike homogeneous case, such vertices are counted in \(I(t)\) but contribute nothing to new infections, and also flow to \(R\) compartment with same rate as those infected vertices with transmission potentials. Similar arguments about contact heterogeneity applies to other vertices in \(I\), where their heterogeneity in degree affect the their infection potential, but cannot be represented by just the proportion \(I(t)\), as it is easily to see that degree distribution of infected nodes is not invariant with time.
Therefore, the incidence \[\dot{S}(t)=G_p'(\theta) \dot{\theta}=- \beta G_p'(\theta) \phi_I\]might not simply proportional to \(\mathcal{R}_\text{eff} \times I(t)\), but actually governed by \(\phi_I\) as \(\dot{\theta}=-\beta \phi_I\), where \(\phi_I\) is defined as the proportion that a neighbor \(b\) is infected but the infection has not yet transmitted to a randomly chosen vertex \(a\).
A more intuitive understanding for incidence of configuration network would be \[\dot{S}(t)=-\beta x_{SI}\], where \(x_{SI}\) is the proportion of \(S\)-\(I\) edge in the network.
Within MSV’s framework, we can write \[x_{SI}=G'_p(\theta) \phi_I=(\theta G'_p(\theta)) \times \frac{\phi_I}{\theta}\] The first term \(\theta G'_p(\theta)=\sigma_{\phi}=\sum_{d}p_d d \theta^{d}\) is expected number of edges of susceptible vertices. The second term \(\phi_I/\theta\) is the probability that an edge is connected to an \(I\) vertex given it has not yet transmitted the infection(connected to \(S\)).
This is the probability to forming an \(S\)-\(I\) pair as we consider the random network of being randomly forming edges(pairs) as infection transmitting.
An equivalent illustration would based on the \(\phi_S=\frac{G_p'(\theta)}{\delta}\), so \[\dot{S}=-\beta x_{SI}=-\beta\delta\times\frac{G_p'(\theta)}{\delta}\times\phi_I=-\beta\delta\phi_S\phi_I\] and \(\phi_I\) is governed by:\[\dot{\phi_I}=-\dot{\phi}_S-(\beta+\gamma)\phi_I=[-(\beta+\gamma)+\beta\frac{G''_p(\theta)}{\delta}] \phi_I\] As \(\phi_S\) is governed by:\[\dot{\phi}_S=-\beta\frac{G''_p(\theta)}{\delta} \phi_I\]
A problem for Jonathan-Richard formula is the derivation of \(\rho\) might not fit with the MSV network framework properly as it lacks of locality.
RomanescuEtAL(2023) and RomanescueDeardon(2017) provides an alternative derivation, which derive the same \(\rho\) results based on MSV network for the same family of degree distribution with consideration of network locality. (These papers are not really well-written, especially the 2017 one which contains more derivation.)
However, equivalent result of the two approaches only happens on the negative-binomial family, as \(\frac{G_p(x)}{G'_p(x)}\) is linear on \(x\), which is the property of this distribution family.
After reading their derivation, RZ think they might made some mistake but the idea is still valuable. #### Follow RomanescuEtAL(2023): Define \(\mathcal{R}^*_{\text{eff}}(t)\) to be the expected number of secondary infections for a ==newly?== infected individual \(X_t\) at time \(t\), s.t. \(\mathcal{R}^*_{\text{eff}}(t)=\mathbb{E}[X_t]\).
To derive \(X_t\) - \(K_S(t)\) is the random variable of degree of a random susceptible vertex at time \(t\) - Proportion of susceptible vertices with degree \(k\) at time \(t\) in the entire population \(p^S_k(t)=p_k \theta^k\) - Total proportion of susceptible nodes at time \(t\) is \(S(t)=G_p(\theta(t))\) - Corresponding PGF of \(K_S(t)\) is \(G_{S}(x)=\frac{G_p(x \theta)}{G_p(\theta)}=\frac{G_p(x \theta)}{S}\) - Observe the process by which a susceptible individual becomes infected. - Consider a random edge that has the potential to transmit infection at time \(t\), which is an \(S-I\) pair: The uninfected individual at the end of this edge is chosen from the susceptible set, but not at random: an individual’s chance of being selected is proportional to their degree, in the absence of higher-order features. - Thus, the relative frequency of an individual of degree \(k\) becoming infected at the next time step is proportional to \(k p^S_k(t)=k p_k \theta^k\). - \(K_I(t)\) is the random variable of degree of a random infective vertex at time \(t\) - Corresponding PGF of \(K_I(t)\) is \[\frac{\sum_k k p_k\theta^kx^k}{\sum_k k p_k\theta^k}=\frac{xG'_S(x)}{G '_S(1)}=G_I(x)\] - Expectation of \(K_I(t)\) is:\[\mathbb{E}(K_I(t))=G_I'(1)=\frac{\theta G_p''(\theta)}{G'_p(\theta)}\] - Since neighbors are assumed to be independent, given the a ==newly infected== vertex have degree \(K_I(t)\), the number \(X_t\) of new infective vertices infected by this vertex is distributed as a Binomial(\(n=K_I(t)-1,\mu\)). - \(n=K_I(t)-1\) is because we know for any infective vertex, it can no longer infect its infector. - RomanescuEtAL(2023) claim that \(\mu= \frac{\beta}{\beta+\gamma} S(t)\) where \(\frac{\beta}{\beta+\gamma}\) is the per-edge infection probability/transmissibility and on average, only a fraction \(S(t)\) of its contacts will still be susceptible. - ==RZ believe here they might make a mistake here, since this \(\mu\) probability should correspond to edge-forming process, where the probability should related to the proportion of edges connected to \(S\) vertices, not the proportion of \(S\) vertices.== - ==They did not explicitly state that this derivation also assume that every edge connect to this \(I\) vertex are still being able to transmit the infection other than the known one connect to its infector. This is the same to assume the vertex is newly infected and not yet infect any others. - An correction would be \(\mu=\frac{\beta}{\beta+\gamma}\times \frac{\phi_S}{\theta}\) if we considering that the vertex is newly infected, where \(\frac{\phi_S}{\theta}\) is the probability that an edge is connected to a susceptible node given that it has not transmitted infection. - Therefore, with law of total expectation, for a randomly newly infected node we know the expectation number \(X_t\) of infected vertices it can generate would be:
\[\begin{align} \mathcal{R}^*_{\text{eff}}=\mathbb{E}[X_t]&=\mathbb{E}_{K_I}[\mathbb{E}[X_t|K_I]] \\ & =\mathbb{E}_{K_I}[\mu(K_I-1)] \\ & =\mu \mathbb{E}[K_I-1] \\ & = \mu (\mathbb{E}[K_I]-1) \\ & = \mu \theta\frac{G''_p(\theta)}{G'_p(\theta)} \end{align}\]
Take into the idea that \[\mu=\frac{\beta}{\beta+\gamma}\times \frac{\phi_S}{\theta}=\frac{\beta}{\beta+\gamma}\times\frac{G'_p(\theta)}{\theta G'_p(1)}\], we have:\[\mathcal{R}^*_{\text{eff}}=\frac{\beta}{\beta+\gamma}\frac{G''_p(\theta)}{\delta}\] As \(t\rightarrow 0 \Leftrightarrow \theta \rightarrow 1\), \(\mathcal{R}^*_{\text{eff}}\) converge to \[\mathcal{R}^*_0=\frac{\beta}{\beta+\gamma}\frac{G''_p(1)}{\delta}\] Similar as how MSV verifying their \(\mathcal{R}^*_0\) with dynamic, \(\mathcal{R}^*_{\text{eff}}=1\) is where \(\dot{\phi}_I=0\). which verify the definition.
Consider rate of change for the incidence term
\[\begin{align} \frac{d}{dt}(-\dot{S}(t))& =-\ddot{S}(t) \\ & =\beta\delta[\phi_S\dot{\phi}_I+\dot{\phi}_S\phi_I] \\ & =\beta\delta[\phi_S\phi_I(\beta+\gamma)(\mathcal{R}^*_\text{eff}-1)-\beta\frac{G''_p(\theta)}{\delta}\phi_I^2] \\ & =\beta\delta\phi_I[\phi_S(\beta+\gamma)(\mathcal{R}^*_\text{eff}-1)-\beta\frac{G''_p(\theta)}{\delta}\phi_I] \end{align}\]
If and only if \((\mathcal{R}^*_\text{eff}-1)\) >0, the rate of change \(-\ddot{S}(t)\) have positive term (increasing force).
(TO DO: Applying this for all NegBinom distribution family)
Similarly with previous Jonathan-Richard result, for Negative binomial degree distribution with PGF:\[G_p(\theta)=(\frac{1}{1+\kappa\delta-\theta\times\kappa\delta})^{\frac{1}{\kappa}}\]we have: \[\sigma=\frac{\mathcal{R}^*_\text{eff}}{\mathcal{R}^*_0S}=\frac{\frac{\beta}{\beta+\gamma}\frac{G''_p(\theta)}{\delta}}{\frac{\beta}{\beta+\gamma}\frac{G''_p(1)}{\delta}\times G_p(\theta)}=\frac{G''_p(\theta)}{G''_p(1) G_p(\theta)}=\frac{\delta^2(\kappa+1)G_p(\theta)^{2\kappa+1}}{\delta^2(\kappa+1)G_p(\theta)}=S^{2\kappa}\] (TODO) more detailed derivation?
==(TODO) verify the gamma case==
Also, consider the Bayesian Formula and a randomly chose edge/stub \(u\):
\[\begin{align} \mathbb{P}(u\in\theta \Leftrightarrow u \in\phi_I|u \text{ connect to a vertex }\in I) & = \frac{\mathbb{P}(u\in\phi_I|u\in\theta)\mathbb{P}(u\in\theta)}{\mathbb{P}(u \text{ connect to a vertex }\in I)} \\ & =\frac{\frac{\phi_I}{\theta}\times \theta}{\frac{NI\times \mathbb{E}(K_I)}{N\delta}} \\ & =\frac{\phi_I \delta}{I(t) \mathbb{E}[K_I]} \end{align}\]
Following previous idea for newly infected vertex, we could slightly modify this probability argument by replacing \(K_I\) with \(K_I-1\) as we are sure for each infected (other than the initial patient-zero) vertex, there is one and only one edge comes from its infector, thus can no longer transmit the infection.
We name this new probability \(\eta=\frac{\phi_I \delta}{I(t) \mathbb{E}[K_I-1]}\) which is the probability that a randomly chosen edge \(u\) is not yet transmit the infection, given it is connected to an infected focal vertex and not the edge infected the focal node. Following the previous binomial distribution process, we might be able to construct the expected number of new infection caused by a RANDOM infected vertex instead of a NEWLY infected vertex.
For a random edge \(u\) connected infected focal vertex at time \(t\) while not connected to its infector, the probability \(u\) can transmit the infection is \[\eta \times \phi_S \times \alpha\] - \(\eta\) as defined earlier is probability \(u\) still not yet transmit disease. - \(\eta\times\phi_S\) is the probability that such \(u\) is connected to a susceptible node. Note we no longer need to conditional on \(\theta\) like in \(\mu\) as it is already considered in \(\eta\). - \(\alpha\) is the probability such edge could transmit the infection before the infected focal vertex recovering.
Then following the binomial distribution, the expected number of new cases that a randomly infected vertex will causes at time \(t\) is given by: \[\mathcal{R}_\text{random}=\alpha \phi_S\eta (\mathbb{E}[K_I]-1)=\alpha \phi_S \frac{\phi_I \delta}{I(t) \mathbb{E}[K_I-1]} (\mathbb{E}[K_I]-1)=\alpha \times \frac{\delta \phi_S \phi_I}{I(t)}\] Now we observe the traditional expression for \(\mathcal{R}_\text{eff}\): \[\mathcal{R}_\text{eff}=-\frac{\frac{dS(t)}{dt}}{I(t)}\times\frac{1}{\gamma}=-\frac{-\beta \delta\phi_S\phi_I}{I(t)}\times\frac{1}{\gamma}=\frac{\beta}{\gamma}\times \frac{\delta \phi_S \phi_I}{I(t)}\]
So they merge if \(\alpha=\frac{\beta}{\gamma}\)!
But my problem is for a newly chosen \(S-I\) edge in the whole network, the probability it transmit the infection before \(I\) recovered or \(S\) is infected by others is \(\frac{\beta}{\beta+\gamma}\). There is some inconsistency here and we need to better understand \(\alpha\).
By def, \(\alpha\) should be a probability while \(\frac{\beta}{\gamma}\) could be larger than 1 while still feasible.
Also, for \(\mathcal{R}_\text{eff}=-\frac{\frac{dS(t)}{dt}}{I(t)}\times\frac{1}{\gamma}\), we observe it is peaked at/near \((\delta-1)\frac{\beta}{\gamma}\) for Poisson degree distributed network at some time near but not equal to \(t=0\). I come up with an estimation to this peak value with some problem:
At the peak point, we must have \(\dot{\mathcal{R}}_\text{eff}=0\), which leads to \[0=\dot{\mathcal{R}}_\text{eff}=\frac{1}{\gamma}\times\frac{\ddot{S}I-\dot{S}\dot{I}}{I^2}\] For non-zero \(I(t)\), this just requires the numerator:\[0=\ddot{S}I-\dot{S}\dot{I}=\ddot{S}I-\dot{S}(-\dot{S}-\gamma I) \Leftrightarrow I_\text{max}=-\frac{\dot{S}^2}{\ddot{S}+\gamma\dot{S}}\] As we could represent \(S\) and its derivatives with \(\theta\) and PGFs but have no explicit expression for \(I\), we could take this relationship at peak back into \(\mathcal{R}_\text{eff}\): \[\max(\mathcal{R}_\text{eff})=-\frac{\dot{S}}{I_\text{max}}\times\frac{1}{\gamma}=-\frac{\dot{S}}{-\frac{\dot{S}^2}{\ddot{S}+\gamma\dot{S}}}\times\frac{1}{\gamma}=\frac{\ddot{S}+\gamma\dot{S}}{\gamma\dot{S}}\] Take into the previous relationships for \(\dot{S}\) and \(\ddot{S}\) we have:
\[\begin{align} \max(\mathcal{R}_\text{eff}) & =\frac{\ddot{S}+\gamma\dot{S}}{\gamma\dot{S}} \\ &=\frac{\beta}{\gamma}(\frac{G''_p(\theta)}{\delta}\times\frac{\phi_S-\phi_I}{\phi_S}-1) \\ &=\frac{\beta}{\gamma}[\frac{G''_p(\theta)}{\delta}(1-\frac{\phi_I}{\phi_S})-1] \\ &=\frac{\beta}{\gamma}[\frac{G''_p(\theta)}{\delta}(1-\frac{\theta-\frac{\gamma}{\beta}(1-\theta)-\frac{G'_p(\theta)}{\delta}}{\frac{G'_p(\theta)}{\delta}})-1] \\ &=\frac{\beta}{\gamma}[\frac{G''_p(\theta)}{\delta}(2-\delta\times\frac{\beta\theta-\gamma(1-\theta)}{\beta G'_p(\theta)})-1] \end{align}\]
If \(\theta=1\), this function equals to: \[\frac{\ddot{S}+\gamma\dot{S}}{\gamma\dot{S}}|_{\theta=1}=\frac{\beta}{\gamma}[\frac{G''_p(1)}{\delta}(2-\delta\times\frac{\beta-\gamma(1-1)}{\beta G'_p(1)})-1]=\frac{\beta}{\gamma}[\frac{G''_p(1)}{\delta}-1]\] This amount equals to 1 iff \(\mathcal{R}^*_0=\frac{\beta}{\beta+\gamma}\frac{G''_p(1)}{\delta}=1\).
Unlike the \(\mathcal{R}^*_0\) or \(\mathcal{R}^*_\text{eff}\) , \(\max(\mathcal{R}_\text{eff})|_{\theta=1}\) this amount could easily be larger than the maximum degree as there is no bond for the ratio \(\frac{\beta}{\gamma}\). - E.g. consider a network with every nodes has degree \(k=5\), then \(\delta=5\) and \(G''_p(1)=k^2-k=20\), assume \(\beta=0.2\) and \(\gamma=0.1\), then \[\max(\mathcal{R}_\text{eff})|_{\theta=1}=\frac{\beta}{\gamma}[\frac{G''_p(1)}{\delta}-1]==\frac{0.2}{0.1}[\frac{20}{5}-1]=6>k=5\] - We need a better definition for this! - Volz 2008 paper: The number of new infections in a small time interval is proportional to \(\phi_I\) . This is in contrast to compartment models in which the number of new infections is proportional the current number of infectious. ##### Poisson For Poisson distribution with: \[G_p(\theta)=e^{-\delta(1-\theta)}\] \[G'_p(\theta)=\delta e^{-\delta(1-\theta)}\] \[G''_p(\theta)=\delta^2 e^{-\delta(1-\theta)}\]we have
\[\begin{align} \max(\mathcal{R}_\text{eff}) & =\frac{\beta}{\gamma}[\frac{G''_p(\theta)}{\delta}(2-\delta\times\frac{\beta\theta-\gamma(1-\theta)}{\beta G'_p(\theta)})-1] \\ & =\frac{\beta}{\gamma}[\delta e^{-\delta(1-\theta)}(2-\frac{\beta\theta-\gamma(1-\theta)}{\beta e^{-\delta(1-\theta)}})-1] \\ & = \frac{\beta}{\gamma}[2\delta e^{-\delta(1-\theta)}-\delta\theta+\frac{\gamma}{\beta}(1-\theta)\delta-1] \end{align}\]
If we consider \(\theta \rightarrow 1\) we have \(max(\mathcal{R}_\text{eff})\) converge to our observation: \[\lim_{\theta\rightarrow1}{\max(\mathcal{R}_\text{eff})}=\frac{\beta}{\gamma}(2\delta -\delta-1)=\frac{\beta}{\gamma}(\delta-1)\] But I have not figure out why it converge to our observation at some \(t>0+\epsilon\). A guess would be the initial condition need some time to reach eigenvector direction?
For general NB distribution with: \[S=G_p(\theta)=(\frac{1}{1+\kappa\delta-\theta\times\kappa\delta})^{\frac{1}{\kappa}}\] \[G'_p(\theta)=(\frac{1}{1+\kappa\delta-\theta\times\kappa\delta})^{\frac{1}{\kappa}}\times\frac{\delta}{1+\kappa\delta-\theta\times\kappa\delta}=\frac{S\delta}{1+\kappa\delta-\theta\times\kappa\delta}=\delta S^{\kappa+1}\] \[G''_p(\theta)=\delta^2(\kappa+1)S^{2\kappa+1}\]we have
\[\begin{align} \max(\mathcal{R}_\text{eff}) & =\frac{\beta}{\gamma}[\frac{G''_p(\theta)}{\delta}(2-\delta\times\frac{\beta\theta-\gamma(1-\theta)}{\beta G'_p(\theta)})-1] \\ & =\frac{\beta}{\gamma}[\delta (\kappa+1)S^{2\kappa+1}(2-\frac{\beta\theta-\gamma(1-\theta)}{\beta S^{\kappa+1}})-1] \\ & =\frac{\beta}{\gamma}[\delta (\kappa+1)(2S^{2\kappa+1}-S^{\kappa}(\theta-\frac{\gamma}{\beta}(1-\theta)))-1] \end{align}\]
If we consider \(\theta \rightarrow 1 \Leftrightarrow S \rightarrow 1\) we have \(max(\mathcal{R}_\text{eff})\) converge to: \[\lim_{\theta\rightarrow1}{\max(\mathcal{R}_\text{eff})}=\frac{\beta}{\gamma}(\delta(\kappa+1)-1)\] ??? Does not work well for Gamma distribution…