Pandemic modeling with the game of life

John Horton Conway made many contributions during his lifetime but is best known for his mathematical game of life developed in the late 1960s. Conway's game of life^1,2 has become the most popular cellular automaton in history deserving of a dedicated Wikipedia page³ and mobile app.⁴ 

Conway's game of life is played on a two-dimensional (2D) grid with each cell either alive (1) or dead (0). A set of evolutionary rules define how each cell changes, or not, at each time step. The rules dictate for each cell whether a birth (⁠ $0 → 1$⁠), death (⁠ $1 → 0$⁠), or survival (⁠ $0 → 0 , 1 → 1$⁠) event takes place based on the number of live cells in the eight cells of its immediate neighborhood.

Conway's game of life and its variants have been the subject of study in statistics and statistical mechanics, nonlinear dynamics, critical behavior, chaos, and species ecology.^5–10 There have been numerous adaptations of Conway's original game^5,11–14 most notably quantization of the game by Flitney and co-workers,^15–17 who replaced dead and alive (0 and 1) by qubits that evolve by repeated applications of birth, death, and survival operators. The generalized semi-classical game of life (gSCGOL) adapted the quantized games to model the fortunes of species subjected to environmental stresses, predation, and disease.¹⁰ The gSCGOL is used here to model the response of the United Kingdom (UK) population to the COVID-19 pandemic.

In December 2019, a cluster of unexplained “viral pneumonia” cases were reported in Wuhan, China.¹⁸ Scientists determined that a new coronavirus was the cause, likely to have come from genetic recombination of bat viruses.^18,19 Less than three months later, in March 2020, the World Health Organization declared a pandemic. By late 2022, an excess of 630 × 10⁶ worldwide cases had claimed 6.6 × 10⁶ lives.²⁰ For the UK, the statistics stood at 24 × 10⁶ cases and nearly 197 000 deaths.²⁰ 

Despite these tragic losses, the timely intervention of epidemiologists was invaluable throughout the pandemic. The predictions of a team at Imperial College London, headed by Professor Ferguson, led to the first lockdown on March 23, 2020 in the UK.²¹ The Imperial Group used a susceptibility/infectious-removed/recovered (SIR) model,^22–24 a simple mathematical model requiring solutions to coupled differential equations. The SIR model moves a population through the stages Susceptible–Infectious–Recovered by solving the parametrized differential equations to reveal the population fraction at each stage of the infection. The SIR model can also be treated as a cellular automaton in which a cell represents a person who goes through each of the stages S-I-R while infecting susceptible neighboring cells during the infectious period. The simulation is useful in providing a visual representation of the S-I-R stages and the spread of the virus.

The SIR model does not predict deaths resulting from infection. One might argue that the peak of infections in March 2020, which can be modeled by SIR, leads to the observed maximum in deaths occurring later in April 2020. However, the infection numbers did not drop because humans recovered from the infection, as in the SIR model, rather the drop in infections stemmed from the change in behavior of humans principally through lockdown.

The SIR model does not model changes in human vulnerability to infection. By contrast, the gSCGOL models changes in human behavior through the vulnerability parameter V. The gSCGOL model adopted here is based on a modified SIR model in which a human infection is followed by an incubation period of three days before becoming infectious. The infection is removed after a further 13 days.²⁵ The post-infection time-of-death probability density distribution is also implemented allowing the number of daily deaths to be realistically modeled based on infections. Furthermore, previous work¹⁰ showed that the gSCGOL successfully models the slow recovery of species from severe decline and is, therefore, suitable for a pandemic simulation where the COVID-19 virus presents a similar dramatic drop in prevalence during summer 2020 before its autumn recovery.

Section II presents the gSCGOL model as a representation of the vulnerable human population. The rules of interaction between the vulnerable human system and a separate virus system are also described in Sec. II. Section III shows how changing the vulnerability parameter V leads to an excellent fit to the UK daily 7-day average COVID-19 deaths. The results are then interpreted in terms of the human response to the pandemic crisis during 2020.

The pandemic model comprises two separate systems: The vulnerable human population and a virus system that runs in parallel. Only the vulnerable human system evolves according to the rules of the gSCGOL and does so independently of the virus system. The virus system evolves on a cellular grid of the same size according to a distinct set of rules that depend on the cell liveness of the vulnerable human system.

Each cell evolves at each time step, or generation, following the application of a generation operator $G ̂$ according to the algorithm by Flitney and Abbott.¹⁵ Each generation represents a single day. The generation operator $G ̂$ associated with a specific cell is constructed from the birth, death, and survival operators, the Moore liveness A for that cell, the user-selected optimum Moore liveness A₀, and a user-selected population vulnerability parameter V, as shown in Table I. The choice of A₀ defines the steady-state vulnerable population density and is fixed at $A 0 = 3$ for all the simulations. The vulnerability parameter V defines when rule changes take place in terms of a change in A (see Table I) and must satisfy $A 0 − V ≥ 0$ and $A 0 + V ≤ 8$⁠. The value of V may change daily.

The vulnerable human gSCGOL simulation starts on day 1, assumed to be December 1, 2019. The response of the human population to the pandemic is characterized by the parameter V, representing the vulnerability of the population to infection (the virus-eye view of humans) on a particular day. The vulnerable population density is high if V is large, and the imposition of lockdown, for instance, is characterized by a drop in V.

The virus system is not modeled using gSCGOL. Instead, the evolution of the virus depends on the vulnerable human population according to the following rules. A virus is represented as a virus count (not a qubit) at each cell of a 2D grid, which is the same size as the 2D vulnerable human system.

1. A virus is added by setting $N i = 1.0$ on a random cell i of the virus grid. To achieve suitable statistical averaging in a single simulation, 100 viruses are added on day d_v at random positions on a virus grid large enough so that the probability of two viruses in close proximity is low. We chose a 400 × 400 grid supplying one virus per 1600 cells, equivalent to one virus per 666 000 people.

2. A number $N i = v P X i a i$ viruses (not necessarily an integer) are added to each cell i of the virus system on each subsequent day, where v_P is a virus potency scale factor and a_i is the vulnerable human liveness at the same cell location i of the human system. X_i is a measure of the exposure of the cell i to the virus defined by

   $X i = ∑ k = 1 8 v k , i ,$

   (5)

   where $v k , i$ is the virus count on cell k in the Moore neighborhood of cell i. This step models the spread of the virus to cells in the Moore neighborhood of an infected cell i. The number of viruses added to cell i is proportional to the number of viruses in its immediate neighborhood and to a_i, the human liveness at cell i.

3. A virus added on day n to cell i cannot infect neighboring cells until day $n + v e + 1$⁠, where v_e is the exposure time in days. The virus is then infectious until day $n + v e + v I$⁠, where v_I is the infectious duration. The N_i viruses added on day n to cell i are removed on day $n + v e + v I + 1$⁠.

4. If the vulnerable human liveness a_i = 0 on each of three consecutive days on a cell i, the virus count on cell i is set to zero. In other words, a virus cannot survive outside a body for more than 72 h.

5. Every new virus infection causes a fractional human death on each subsequent day determined by a probability density function. The probability of a death occurring is assumed to be described by a gamma distribution with a mean of 18.8 days and a standard deviation of 8.46 days.²⁶ The probability density x days after exposure to the virus is

   $P ( x ) = β α Γ ( α ) x α − 1 e − β x ,$

   (6)

   where $α = 4.938$ and $β = 0.2627$⁠.²⁶ The N_i new viruses at cell i lead to $N i P ( x )$ deaths on day x after exposure for 31 days (3 days before a positive test is possible plus 28 days for a COVID-19 recorded death). Note that no humans are removed from the vulnerable human system.

The first step is to parameterize both the human and virus systems. The vulnerable human system is set up on a 401 × 401 grid with $A 0 = 3.0$ and V = 1.0. The optimum Moore liveness A₀ is fixed throughout the simulation, but the vulnerability V may change daily.

The gSCGOL model is initially run for the human system for d_v days to establish equilibrium. The virus system is on a matching 401 × 401 grid, which is empty until day d_v when 100 viruses v_i = 1 are added at random positions on the virus grid. The model is executed for 336 days until October 31, 2020. The simulation outcome is presented in Fig. 1, and snapshots of the human and virus systems are shown in Fig. 2. Parameterization of the gSCGOL model is achieved by matching to the 7-day average COVID-19 deaths reported each day from March 2, 2020 in the UK. The position of the exponential rise in deaths during March 2020 is parameterized by d_v (the day the viruses are added to the virus system), the rate of rise yields the virus potency v_P, and the maximum number of viruses per cell v_m is chosen to provide the best fit to the April 2020 peak. The vulnerability V is displayed in Fig. 1, showing that V = 1.0 until March 16, 2020 when restrictions began. The simulation 7-day averages are scaled by a factor of s to best match the actual 7-day average COVID-19 deaths, and the vulnerability V is changed using a trial-and-error approach to achieve a good match to the reported 7-day average COVID-19 deaths. The quality of fit is assessed using a least-squares metric.

The final model parameters are listed in Table II, and the results are presented in Fig. 1. The match is excellent, in part because V is adjusted to produce a quality fit. The changes in V provide little insight into the evolution of the virus but do provide insight into the response of humans to the pandemic as the crisis evolved during 2020.

The key UK government actions²⁷ in response to the pandemic are labeled 1–10 in Fig. 1 and can be assessed in terms of the changing value of the vulnerability parameter V. The interventions are listed in Table III. Snapshots of the vulnerable human population and the virus concentrations are presented in Fig. 2.

Figure 2(a) shows a normally vulnerable human population and virus infections the day before the first restrictions. Figure 2(b) shows the decline in the vulnerable population two weeks later with the virus concentration now in check. Figure 1 shows that the population responds to restrictions immediately (label 1) but adheres stringently to the lockdown (label 2) for only four days before a steady-state vulnerability $V ≈ 0.86$ establishes. Approximately three weeks elapse from the first population response due to lockdown to the peak in the daily 7-day average count of deaths on April 8, 2020. The death rates fall significantly thereafter, despite the constancy of V from April 1, 2020 to June 14, 2020. There is no significant change in human behavior due to the easing of restrictions labeled 4.

The most significant factor affecting the spread of the virus post-lockdown, and hence vulnerability of the population, is schools. A phased reopening of schools in England took place from June 1, 2020 (labeled 5). The reopening process was slow due to the time required for schools to plan and manage the return, with a cautious phased increase in school population over the following weeks. The first significant period of increase in human vulnerability starts two weeks after the June 1, 2020 announcement and slows when the summer holidays started in England (labeled A). The second, dramatic, increase in vulnerability started when schools re-opened after the summer break during the week from September 1, 2020 (labeled B), followed by students returning to universities and colleges a few weeks later. Even in the absence of a pandemic, this is the time when viruses are shared. Here, the vulnerability in the model rises steeply.

The striking feature of the model is how effectively it tracks the low death counts over the summer period. The slow recovery of the virus from a population loss is reminiscent of the demise and slow recovery of species.¹⁰ There is no proportionality between vulnerability and death count. Figure 2(c) presents the vulnerable human and virus state on September 1, 2020, the first day of school for many in England. The virus is absent save for a few small pockets. The virus takes significant time to re-emerge as a second phase later in October 2020.

The first reported case of a COVID-19 infection in the UK was from an individual who traveled from Wuhan, China, to the UK, landing on January 23, 2020.^28–30 The virus must have been introduced into the UK before this date, and there is evidence that people were becoming ill with COVID-19 symptoms before this time.³¹ 

The model can be used to predict the day that the first virus arrived in the UK. To do so, we need a count of infections rather than deaths. Suppose that on day 52 (January 21, 2020), one cell of the virus system labeled cell k is set to 1 (see Table II). There are 401 × 401 cells representing the UK population of 67 × 10⁶ in 2020. Each cell of the system, therefore, represents 416 people. When the virus count v_k = 1 at cell k, 416 people are infected. Moreover, according to the model, each of these 416 infections causes a death.

To estimate the total number of virus infections, including those who do not lead to death, it is necessary to know the mortality rate.³² No data for January 2020 are available due to insufficient testing. The mortality rate data start from July 2020 and show 0.9–2.1 deaths from 100 000 people with 95% confidence. The mortality rate during the summer months is, however, considerably lower than during the winter months. The most representative value is 56.9 deaths per 100 000 individuals for January 2021. Notwithstanding that the delta variant was dominant at this time, we estimate 40–70 deaths per 100 000 for January 2020, which equates to 1429–2500 virus infections per recorded COVID-19 death.

The number of new infections per day is provided by the same simulation starting with one new infection added on infection day 1. The total number of infections on subsequent days for the entire system is found by summing the infection count over all cells. A virus lives for 16 days before succumbing to the human immune system, and the dead viruses are subtracted from the total number of infections. The result is that the simulation yields more than 1429 infections on day 30 and more than 2500 infections on day 32.

The result is that the model predicts that the first virus arrived in the UK on December 22, 2019 with an uncertainty of ±1 days. It is likely that a person on a flight from Wuhan to the UK on or about December 22, 2019 initiated the pandemic in the UK.

On March 23, 2020, the UK government implemented a national lockdown with measures legally enforced on March 26, 2020.³³ The government originally considered a herd immunity strategy, with the infection and subsequent immunity of healthy people preventing infection of more vulnerable people. However, epidemiologists' data discussed on March 13, 2020 by senior government advisors led to a scientific advisor stating “…that even on the best-case scenario with the official plan, you are going to completely smash through the capacity of the National Health Service.”³⁴ The government then changed its policy, and thousands of lives were saved.

Nonetheless, the UK government received sustained criticism for not implementing the lockdown a week earlier. As early as June 10, 2020, Professor Ferguson was quoted “Had we introduced lockdown measures a week earlier, we would have reduced the final death toll by at least a half.”³⁵ (Professor Ferguson led the Imperial Group, which was instrumental in advising the UK government to enact the first lockdown on March 23, 2020 based on its SIR model.^21,24) When Professor Ferguson made the statement, he did not know how the pandemic would evolve. The estimate of lives saved was based on modeling. With the actual data now available, the question can be revisited using the gSCGOL model.

If the start date of the lockdown of March 23, 2020 is changed to March 16, 2020, we find the result presented in Fig. 3. The first phase of the pandemic was deemed to have ended on September 3, 2020, and it is assumed that only the first phase of deaths would have been impacted by an earlier lockdown. Significantly, fewer deaths would have occurred during the alpha variant phase of the pandemic. The model predicts 40% fewer COVID-19 deaths, equivalent to saving nearly 17 000 lives. This conclusion is in good agreement with the prediction of 50% from the Imperial Group in June 2020.

Complex behavior emerges from simple rules enacted via a 2D cellular automaton. The gSCGOL is used to describe the human–virus interaction during the COVID-19 pandemic. The simulation describes the vulnerable human population with a separate non-gSCGOL virus model operating in parallel and surviving by infecting humans according to the set of rules presented in Sec. II B. The rules are parameterized by matching the simulations to the early stage of the UK daily 7-day average COVID-19 deaths.

The simulated UK daily 7-day average COVID-19 deaths are matched to the real data to October 31, 2020 by adjusting the vulnerability V daily. Changes in V are shown to reflect events such as lockdown and, especially, the opening and closing of schools. It is calculated that the first COVID-19 virus entered the UK on December 22, 2019 with an uncertainty of ±1 days. Finally, it is predicted that a lockdown one week earlier in the UK would have resulted in a 40% reduction in COVID-19-related deaths in the first phase, saving nearly 17 000 lives.

The matlab programs and datasets are presented in the supplementary material to facilitate problem-solving activities.³⁶ The most challenging aspect of the modeling is matching the simulation results to real data by changing the vulnerability V. The time lag makes the impact of a change difficult to predict, and so a trial-and-error approach was adopted.

Problem 1. The initial phase of the pandemic is easiest to match. For a country of your choice, obtain the official daily COVID-19 death count (the UK data are provided in Ref. 37). Guess a date for the introduction of the first virus (i.e., choose d_v) and run the simulation. Change d_v to match to the initial exponential rise in the death count. Trace back using the method described in Sec. III B to find the date of introduction of the virus.

Problem 2. Improve your results from problem 1 by matching to the initial peak in daily COVID-19 deaths. The rate of increase yields the virus potency parameter v_P, and the maximum number of viruses per cell v_m is chosen to provide the best fit to the initial peak. The simulated 7-day death counts are scaled by a factor of s determined by minimizing the squared difference between the simulated and real data. Compare with the UK parameters in Table II and discuss the differences.

Problem 3. The most straightforward extension for those interested in the management of the pandemic in the UK is to continue the present simulation beyond October 31, 2020. Early autumn 2020 saw the second phase of the alpha variant, but soon the delta variant appears with a higher virus potency (higher v_P) but possibly less deadly (lower s). The increasing impact of the vaccination program, which started just before Christmas 2020, may become apparent through 2021.

The authors have no conflicts of interest to disclose.