## Motivation:

To understand the potential effectiveness of a peer-to-peer not-a-tracing system against an unknown pathogen, in this case zombies, we may consider its effect on scientists residing at a technology park. I believe this is a potentially useful thought experiment for sub-populations that can choose who they interact with. This would include people that can work from home.

We shall assume that there is a one-day incubation period and that:

1. There is a population of N=500 people.
2. 1% of the population is infected i.e. 5 scientists.
3. The reproduction number, $R_0$, is greater than one.
4. The incubation period is somewhere between one to two days.
5. The social graph is assumed to be a small-world network so given two distinct vertices $v_i$ and $v_j$:

$$\mathbb{E}[d(v_i,v_j)] \leq \log N = \log 500 \approx 3$$

Furthermore, we shall assume that each scientist has an ‘R U a zombie’ app on their smartphone that is powered by Google AI and collects data on symptoms such as anosmia on a daily basis.

## The social network graph approximates the transmission network:

When ‘R U a zombie’ with its advanced machine learning software identifies that a particular scientist is at risk of being infected, they and their friends are encouraged to self-isolate. The system operates entirely through daily self-reporting and information is transferred in a peer-to-peer manner without a centralised databased. This should work in principle because the social network approximates the transmission network.

It’s worth adding that during a zombie pandemic individuals naturally minimise their risk surface area so they mainly socialise with friends, if at all.

## Invertible surgical operations on small-world networks:

Given that $R_0 > 1$ and the average path length between nodes is less than 3.0, the health consequences of this outbreak would quickly explode if not for preventive measures that anticipate future zombie infections.

If $v_i$ is infected, a simple and effective mechanism for reducing the spread of infection would be to modify the network $G=(V,E)$ by removing vertices from the vertex set $V$ as follows:

$$S \subset v_i \cup N(v_i) \cup \bigcup_{v \in N(v_i)} N(v)$$

$$V \mapsto V \setminus S$$

$$V^* := V \setminus S$$

and we note that the residual graph $G^* =(V^*,E^*)$ is still a small-world network since the operations are local.

We also note that when members of $S$ recover from infection the operation is reversible so we have:

$$V^* \mapsto V^* \cup S$$

$$V := V^* \cup S$$

and these operations should be sufficient to halt outbreaks provided that the false negative rate of ‘R U a zombie’ is close to zero and $S$ is carefully chosen.

A simple and robust algorithm for choosing $S$ is as follows:

1. If ‘R U a zombie’ determines that $v_i$ is at risk of being zombie-positive given their symptoms, a message is broadcast to $N(v_i)$ to self-isolate for a week.

2. $v_i \cup N(v_i)$ are then tested for infection on every day of self-isolation.

3. If any of $v_i \cup N(v_i)$ are zombie-positive then they are to be tested every two days until they are zombie-negative. Now, $\bigcup_{v \in N(v_i)} N(v)$ are also at risk of being zombie-positive so they may proceed with step (1).

## Asymptotic analysis of susceptible, infected and recovered populations:

The rationale for the previous section is quite simple. Let’s assume that at time $t$ we have:

$$S(t) = \text{number of susceptible scientists}$$

$$I(t) = \text{number of infected scientists}$$

$$R(t) = \text{number of recovered scientists}$$

Assuming that the zombie virus is non-lethal:

$$\forall t, N = S(t) + I(t) + R(t)$$

and the effect of ‘R U a zombie’ is to guarantee that $I(t)$ is a decreasing function of time so:

$$\frac{dI}{dt} \leq 0$$

and in principle $\lim_{t \rightarrow \infty} I(t) = 0$. So if the outbreak is identified early enough its spread may be halted relatively quickly at minimal cost.

## Discussion:

It is worth noting that in order to incentivise adoption of an app like ‘R U a zombie’ we need to assure the population that the system will work. In practice, this requires three things:

1. Effective daily monitoring of symptoms using a machine learning system such as the online survey proposed in [3].

2. A system for large-scale testing with results within 48 hours.

3. Paid sick leave during self-isolation.

It is worth noting that in most European countries we don’t even have two out of three. Small and medium sized businesses such as restaurants don’t have guaranteed incomes at all. As it is, occidental democracies aren’t even able to cater to the population that can afford to work from home which represents the most privileged sub-population.

Having said this, each of those three points I have noted corresponds to an engineering problem that may be solved so there is hope.

## References:

1. Alexander A. Alemi, Matthew Bierbaum, Christopher R. Myers, James P. Sethna. You Can Run, You Can Hide: The Epidemiology and Statistical Mechanics of Zombies. Arxiv. 2015.

2. Jussi Taipale, Paul Romer, Sten Linnarsson. Population-scale testing can suppress the spread of covid-19. medrxiv. 2020.

3. Hagai Rossman, Ayya Keshet, Smadar Shilo, Amir Gavrieli, Tal Bauman, Ori Cohen, Esti Shelly, Ran Balicer, Benjamin Geiger, Yuval Dor & Eran Segal. A framework for identifying regional outbreak and spread of COVID-19 from one-minute population-wide surveys. Nature. 2020.