The authors investigate the spatial dynamics of airborne disease transmission, which requires fine aerosol droplets small enough to remain suspended in air, yet large enough to contain non-neglibible pathogen load.
In an unventilated environment, diffusion depends on movement of people (a droplet will only diffuse 5.7mm)
Ventilation allows more rapid droplet diffusion, which can lead to secondary outbreaks. Above a critical level of ventilation, however, disease transmission is impared as droplets are transported out of the domain before they can cause infection.
Disease outbreaks, such as those of Severe Acute Respiratory Syndrome in 2003 and the 2009 pandemic A(H1N1) influenza, have highlighted the potential for airborne transmission in indoor environments. Respirable pathogen-carrying droplets provide a vector for the spatial spread of infection with droplet transport determined by diffusive and convective processes. An epidemiological model describing the spatial dynamics of disease transmission is presented. The effects of an ambient airflow, as an infection control, are incorporated leading to a delay equation, with droplet density dependent on the infectious density at a previous time. It is found that small droplets ($\sim 0.4\ \mu$m) generate a negligible infectious force due to the small viral load and the associated duration they require to transmit infection. In contrast, larger droplets ($\sim 4\ \mu$m) can lead to an infectious wave propagating through a fully susceptible population or a secondary infection outbreak for a localised susceptible population. Droplet diffusion is found to be an inefficient mode of droplet transport leading to minimal spatial spread of infection. A threshold air velocity is derived, above which disease transmission is impaired even when the basic reproduction number exceeds unity.
Allesina and Tang do a more detailed analysis of May’s seminal work on stability matrices.
May’s approach defines a community matrix M of size SxS, where S is the number of species, and Mi,j is the effect of species j on species i. Entries Mi,j are with prob C, zero otherwise. May showed that when the complexity the probability of stability is near null. Thus rich (high S) or highly connected (high C) communities should be rare.
The current authors allow the community matrix to have structure. Preditor-prey networks are structured so that Mi,j and Mj,i have opposite signs. A mixture of competition and mutulalism arises when Mi,j and Mj,i are constrained to have the same sign.
Stability arises when the eigen values have negative real parts. Random matrices constrain their eigenvalues to a circle, pred-prey to a vertical ellipse, and comp/mut to a horizontal ellipse. More formally, the difference in stability is driven exclusively by the arrangement of the coefficients into pairs with random, opposite, and same signs. Intermediate cases can be formed by linear combinations.
Imposing realistic food webs decreases stability.
What if Mi,j are not normally distributed? Imagine many weak interactions: preditor-prey networks become less stable, competition/mutulalism networks become more stable, and random are unchanged.
Very interesting work.
“Stability criteria for complex ecosystems“, Stefano Allesina,Si Tang. Nature (2012)
The authors focus on scale-free networks,
with reference to Moore et al’s model which includes annihilation of nodes. They analyze a Japanese business relationship network. Companies are born, grow, die, and merge. Firms have (observed and measurable) preferential attachment to firms with higher degree.
The cumulative distribution of company lifetimes is exponential where years. (we see similar patterns in speciation). A semi-log plot shows degree grows exponentially with lifetime, where expected degree is and $A=0.017$.
This suggests the degree distribution should be power-law with exponent , or 3.1. The observed degree distribution has exponent 1.3. Conclusion: random annihilation and growth by preferential attachment do not explain the real world.
To explain this, the authors turn to coagulation models of aerosols and colloids; the mass of these clouds follows a power law. They propose a model where network nodes can merge. More specifically, starting with N nodes, evolve the network by stochastically choosing annihilation, creation, or coagulation at each time step, with probabilities a, b, c (). With 10^5 nodes, the model converges after 10^7 iterations, suggesting one year in the real world == 10600 iterations.
They show, both analytically and by simulation that the coagulation probability controls the final shape of the degree distribution. Small gives an exponential distribution, large gives power law.
Phys. Rev. Lett. 108, 168701 (2012) Effect of Coagulation of Nodes in an Evolving Complex Network. Wataru Miura, Hideki Takayasu, and Misako Takayasu
The “news and views” write up summaries the paper by comparing to earlier results on controlling node dynamics.
“Controlling edge dynamics in complex networks” Nepusz and Vicsek, Nature Physics (2012)
Under node dynamics, a time-evolving state variable is associated with each node, with the evolution of the state is dependent on the state of the neighbours. Think metabolite concentrations, gene expression levels, or formation of opinions.
Under edge dynamics, a state variable is associated with each edge. Nodes act as switchboards, reading the state of upstream neighbours, processing it, and passing it to downstream neighbours. Think load-balancing routers (node=router, edge=route, state=load).
Under nodal dynamics, the minimum set of driver nodes can be identified by the minimum-inputs theorem of Liu, Slotine, and Barabasi (2011, Nature). Under edge dynamics, one still wants to find the minimum set of driver nodes (since nodes set the state of their edges). Nepusz and Vicsek’s algorithm solves this.
The two representations are duals in many ways. Most obviously, edge dynamics on a network can be represented by node dynamics on the network’s line graph. More interestingly,
Role of hubs:
Nodal. A hub relays a common signal to a large portion of the network, creating symmetries which restrict the state space. Thus not in minimal driver set.
Edge. A hub can set each associated edge state separately. The large number of edges thus controlled means they fit in the minimal driver set.
Sparse vs dense networks:
Nodal. Sparse, inhomogeneous networks are the hardest to control.
Edge. Sparse, inhomogeneous networks are the easiest to control.
Nodal. Enhances controllability.
Edge. No effect.
Correlated in and out degree:
Nodal. No effect.
Edge. enhances controllability.
The work may be relevant to evolution, under the assumption that evolution is based on ancient, optimized components whose connections are the main target of natural selection.