# Towards predicting COVID-19 infection waves: A random-walk Monte Carlo
simulation approach

D. P. Mahapatra<EMAIL_ADDRESS>S. Triambak<EMAIL_ADDRESS>Department of Physics, Utkal University, Vani Vihar, Bhubaneshwar 751004,
India  Department of Physics and Astronomy, University of the Western Cape,
P/B X17, Bellville 7535, South Africa

###### Abstract

Phenomenological and deterministic models are often used for the estimation of
transmission parameters in an epidemic and for the prediction of its growth
trajectory. Such analyses are usually based on single peak outbreak dynamics.
In light of the present COVID-19 pandemic, there is a pressing need to better
understand observed epidemic growth with multiple peak structures, preferably
using first-principles methods. Along the lines of our previous work [Physica
A 574, 126014 (2021)], here we apply 2D random-walk Monte Carlo calculations
to better understand COVID-19 spread through contact interactions. Lockdown
scenarios and all other control interventions are imposed through mobility
restrictions and a regulation of the infection rate within the stochastically
interacting population. The susceptible, infected and recovered populations
are tracked over time, with daily infection rates obtained without recourse to
the solution of differential equations.

The simulations were carried out for population densities corresponding to
four countries, India, Serbia, South Africa and USA. In all cases our results
capture the observed infection growth rates. More importantly, the simulation
model is shown to predict secondary and tertiary waves of infections with
reasonable accuracy. This predictive nature of multiple wave structures
provides a simple and effective tool that may be useful in planning mitigation
strategies during the present pandemic.

###### keywords:

COVID-19 , random walk , Monte Carlo simulations , epidemic waves

††journal: Chaos, Solitons & Fractals

##  1 Introduction

In the midst of the current COVID-19 pandemic, there is a continuing need to
accurately model region-specific infection and mortality data, so that
intervention methods and containment strategies can be planned accordingly.
The simplest picture of epidemic growth is at most times provided by
phenomenological models that are based on logistic growth [1, 2, 3, 4, 5, 6,
7]. However, real-time interventions that may affect the trajectory of the
growth curve are not incorporated in such models. More commonly,
epidemiological modeling uses compartmentalized populations based on the SIR
model [8] and its variants (see, for example [9, 10, 11]) that follow the time
evolution of susceptible $(S)$, infected $(I)$ and recovered $(R)$
populations, among others [12]. More often than not these deterministic models
do not involve a stochastic formulation, which takes into account the random
aspects of human mobility. Based on the early work by Bartlett [13], such
randomness can be incorporated through the equations [14, 15, 16]

$\displaystyle\frac{\partial S}{\partial t}=D_{S}\nabla^{2}S-\beta SI$ (1)
$\displaystyle\frac{\partial I}{\partial t}=D_{I}\nabla^{2}I+\beta SI-\mu I.$
(2)

Here, $\nabla^{2}=\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right)$, and $S(x,y,t)$ and
$I(x,y,t)$ are the spatial densities of the susceptible and infectious
components of the population. The $D_{S,I}\nabla^{2}(S,I)$ ‘diffusion’ terms
represent the spatial movement of both susceptibles as well as infectives. The
parameters $\beta$ and $\mu$ represent the infection and recovery rates
respectively. Such formulations have also been extended to biased [17] and
Lévy [18] random walks. More recently, stochastic models have been developed
using Markovian chains [19, 20], contact and community networks [21, 22],
Bayesian modeling [23] etc. The effects of mobility restrictions have also
been studied using Monte Carlo techniques [24, 25, 26] that inherently include
the diffusion terms mentioned above. In most studies, the analyses that are
solely based on single epidemic growth curves do not adequately explain the
multiple wave structures seen in global COVID-19 data. For example, it was
recently proposed that a superposition of epidemic waves [27] could be used to
describe COVID-19 growth curves. However this does not describe the observed
multiple waves of infection for individual countries, that are well separated
in time.

The present work uses an uncorrelated random walk approach to study COVID-19
infection spread via contact interactions. Our simulations show that the
number of successive waves of COVID-19 infection in specific countries depend
on their underlying effective population density, the intermixing rate, and
most importantly the timing and the duration of the control interventions
imposed on/by the population. The effects of these interventions are visible
as modulations in the infection rate. As test cases, we compare our simulation
results with reported data from India, Serbia, South Africa and USA. We show
that multiple peak structures of the pandemic waves are reasonably well
reproduced by the simulations.

##  2 The random walk Monte Carlo method

Our random walk simulation model is described in our previous paper [25] and
similar in approach to the work reported in Ref. [28]. Briefly, people
belonging to a region of population $N$, are described as points that execute
random walks on a 2-dimensional plane. The speed of infection growth depends
on the jump distance $l$ for each point, which we assume is a multiple of the
mean separation $\langle r\rangle$ between points. In Cartesian coordinates,
the jump components are simply $l\cos\theta$ and $l\sin\theta$, with $\theta$
generated randomly between $0$ and $2\pi$ from a uniform distribution. Unlike
other approaches [29, 30], we do not consider a lattice or apply periodic
boundary conditions. Instead, if a jump takes a point out of the area
considered, it is reflected back into the system in a random direction. In all
cases we consider particles in a 1 km2 unit area element. Therefore we can use
the words population and population density interchangeably. For $N$ random
walkers per unit area, $\langle r\rangle=\sqrt{1/N}$ in units of km. A
separation distance of $\leq 2$ m between individuals is taken as the
‘contact’ distance [25].

The simulation starts with the introduction of an infected individual in state
$I$. The disease then spreads through contact interactions, resulting in a
drop in the number of susceptible points, $S$, which at time zero equals
$N-1$. Each time step corresponds to one random jump executed by all the
points and we take that as one day. The number of infected points increase
with such steps, based on the number of contacts between susceptible and
infectious individuals. The populations in $S,I$ and $R$ states are
$N_{s},N_{i}$ and $N_{r}$ respectively, so that $N=N_{s}+N_{i}+N_{r}$ is
preserved at all times. This results in an SIR compartmentalization of the
population.

In the next step of the simulations, a recovery rate was incorporated in a
slightly different manner than used previously in Ref. [25]. We assume that
80% of the infected population (picked at random) ultimately recover, becoming
immune, and keep track of the number of time-steps (days) taken by an infected
point before it goes into a recovered state. The time period is taken to be 35
days, and determined from a comparison of our simulated results with data.
This choice of $\mu=1/35~{}{\rm day}^{-1}$ is not unfounded. The incubation
period of the coronavirus disease (after which symptoms develop) for infected
individuals is found to be in the range of 8.2–15.6 days at the $97.5^{\rm
th}$ percentile level [31]. Furthermore, recent studies have shown that
COVID-19 recovery times have an average value of about 25–28 days [32, 33].
Given that these recovery times are evaluated after symptoms develop, a total
of 35 days after infection is a reasonably accurate estimate.

From here on, one can follow two equivalent approaches to incorporate the
various intervention and mitigation strategies, usually employed after the
start of an epidemic such as COVID-19. A decisive criterion is the strict
imposition of a lockdown that places significant mobility constraints on a
majority of the population. This effectively moderates the growth in $N_{i}$,
so that it is compensated by the recovery rate $\mu$. Other interventions such
as vaccinations, mask mandates and behavioral changes by the population
(social distancing, bubbles, etc.) additionally contribute to stalling the
epidemic growth.

In our earlier work [25], we studied contact-interactions between random
walkers on a plane, with $\beta$ equal to unity. This approach can be modified
to incorporate the effects of all control interventions, exclusively via
mobility restrictions on the stochastic agents. Such restrictions would impede
the growth of the epidemic curve, during which time the number of recoveries
continue to rise. However this is never a permanent solution. In more
realistic scenarios the restrictions are relaxed from time to time (such as
with lockdowns). In such a situation the epidemic growth continues, usually
with a steep rise in the number of infections that is shifted in time.

An alternative approach would be to assume $\beta$ values that are $<1$ for
specific cases under consideration, during the times that the control
interventions and mitigation strategies are followed. For example,
vaccinations, mask mandates, etc. lead to a drop in the number of susceptible
individuals that can be infected. The effect of this drop can also be
incorporated within the parameter $\beta$. This fractional $\beta$ is
implemented by invoking a random number $r$ uniformly between 0 and 1, such
that a change in state from $S\to I$ only occurs when $r\leq\beta$. We show
below that both approaches yield similar results with regard to predicting
multiple infection waves. However we prefer the second approach, as the former
may mislead the reader in terms of the difference between actual lockdowns
imposed on a population and effective mobility restrictions on the random
walkers in the simulations.

The simulations were performed for comparison with reported data for India,
South Africa, Serbia and USA. The adjustable parameters were the population
density (two values were used for this work, $N=$ 5k and 10k), $\beta$, the
percentage of mobile walkers, the duration during which mobility was
restricted and the jump length $\l$. The recipe for the simulations was as
follows. We consider the start date of the growth curve to be $D_{0}$. The
intervention and mitigation strategies during this initial phase would
effectively produce a first wave peak in $N_{i}$ on date $D_{1}$. This is
obtained from reported data. The parameters in the simulation are then
adjusted, so that the Monte Carlo results reproduce the observed first wave
peak in the fractional daily infection rate.111We only consider events where
the disease trajectory extends beyond day 500. The simulated results are
obtained from $[N_{r}(t+\Delta t)-N_{r}(t)]/N$, for $\Delta t=1$ day. Here, it
should be noted that since $\partial N_{r}/\partial t\propto N_{i}$, the peaks
in $N_{i}$ roughly coincide with the daily infection rate peaks along the time
axis. These simulated results are compared to scaled-down data from the World
Health Organization (WHO) [34]. This comparison is then used to infer if
secondary or tertiary pandemic waves subsequently appear at later dates, due
to the control interventions imposed in previous time windows. This is based
on the premise that the control interventions and mitigation strategies would
reduce the slope of the growth curve in the time duration that they are
imposed (or followed). At the end of each such duration (when restrictions,
mandates etc. are relaxed) there would be growth again. This effectively
results in growth curves with multiple peaks in daily infection rates, that
are shifted in time and interpreted as epidemic waves. We discuss a few
country specific results below.

Figure 1: Simulated infected fractions for India, obtained for various
parameter sets. The curves for whom the $\beta$ values are not specifed in the
legend were generated assuming $\beta=1$. The others were generated with the
$l=1\langle r\rangle$ and $2\langle r\rangle$ combination, as described in the
text. All results are for $N=10$k and averaged from five independent
simulations.  Figure 2: Simulation results for India assuming two different
densities ($N=5$k and 10k), compared with normalized WHO data. The simulated
values were averaged over six sets. Figure 3: SIR fractions for the $N=10$k
results in Fig. 2.

##  3 Analysis

###  3.1 India

We first consider the case of India, since the reported data show two distinct
pandemic peaks. Furthermore, there has been a lot of speculation regarding the
appearance of a large third wave, following the devastating second wave in
2021. The data show a first wave peak around Sep 15, 2020 and a second wave
peak around May 06, 2021. These two structures contain in them all effects of
interventions and mitigation strategies, including imposed lockdowns. We take
$D_{0}$ to be April 22, 2020, with the first peak ($D_{1}$) around day 150
(September 15, 2020). Since this was the first country that we studied, we
performed several simulations to better understand the general shapes of the
curves, obtained for different parameter values. The first two sets of
simulations used jump steps of $l=1\langle r\rangle$ and $2\langle r\rangle$,
with $\beta=1$. In the third set, we used $l=1\langle r\rangle$ from $D_{0}$
to $D_{1}$ and $l=2\langle r\rangle$ beyond $D_{1}$. The fractional $N_{i}$
results for different parameter values and $N=10$k are shown in Fig. 1. As
evident in the figure, a larger amplitude infection peak is obtained when
$l=2\langle r\rangle$ step sizes are used, with a peak value of approximately
0.45 around day 140. For this case the number of infections drop to negligible
levels after about 200 days. In comparison, $l=1\langle r\rangle$ produces a
broader and smaller peak that is shifted to a later date (near day 230). On
the other hand, the $l=2\langle r\rangle$ and $l=1\langle r\rangle$
combination yields two distinct peak structures, as shown in the figure. In
the next set of simulations we used this combination, together with $\beta$
values $<1$ that account for intervention/mitigation strategies, which would
lead to a drop in the $N_{i}$ fraction and consequently the daily infection
rate. These results are also shown in Fig. 1, for different values of $\beta$
during the time period $D_{1}$ to $D_{2}$ (the falling part of the first
curve), with the later date representing the beginning of the second wave.
Beyond $D_{2}$ we assume $\beta=1$ for the rise of the second wave. Guided by
reported data we take $D_{2}$ to be day 300.

We observe that the shapes of the fractional $N_{i}$ curves, obtained with
$\beta=0.3$ from dates $D_{1}$ to $D_{2}$ (magenta curve in Fig. 1) are very
similar to scaled down daily infection rates obtained from the WHO, including
the relative amplitudes for the two peaks. This is further validated in Fig.
2, which shows a comparison between the reported daily rates and the simulated
results (assuming $\beta=0.3$). The excellent agreement between the
simulations and the reported data for the two waves is noteworthy. Similar
agreement is not obtained from other parameter value combinations (such as the
use of only $l=1\langle r\rangle$ or a population density $N=5$k for the
$1\langle r\rangle+2\langle r\rangle$ jump-step combination. The results from
the latter simulations are shown in Fig. 2 for comparison). The infection rate
$\beta=0.3$, used for time periods when effective mitigation/intervention
measures are assumed in the simulations, is consistent with those obtained
with a deterministic SIQRk model that also took into account the quarantined
and confirmed-recovered portions of the population (c.f. Table 2 in Ref.
[33].)

It is also important to point out here that in addition to the aforementioned
agreement with data, the second wave peak in Fig. 2 emerges naturally from our
simulations, as a result of interventions imposed previously, mainly through
the parameter $\beta$. Furthermore, the simulations also show no indication of
a third peak that may appear on applying additional interventions (through
$\beta$) in the second wave. This is supported by Fig. 3, which shows the time
development of the individual $S,I$ and $R$ components for these data. One can
see that after day 500 $N_{i}$ drops to nearly zero, while $N_{r}$ saturates
at $\sim 90\%$. It is apparent that at this point there are not enough
infectious agents remaining to drive the curve and infect the small remainder
of the susceptible population. This does not indicate a significant third
wave. However, it may be noted that we assume all recovered individuals to be
immune in our simulations. During the course of this work, several countries
reported a surge in COVID-19 infections due to the emergence of the SARS-CoV-2
Omicron variant [35]. This variant is characterized by an unusually large
number of mutations in the spike protein and an ability to escape vaccine
induced immunity [36]. In light of this recent development, we performed
additional simulations that allow reinfections. Our preliminary results show
that a small fraction of reinfections ($\lesssim 0.5\%$) would increase the
susceptible population significantly, enough to cause a third wave for
India.222For example, our preliminary simulations show that 0.2% of
reinfections would cause a third wave peak for India around day 600. Further
investigations that delve into this aspect and take into consideration the use
of a different infection probability $\beta$ (due to the waning of vaccine-
induced immunity) will be useful in this regard.

###  3.2 Serbia

Serbia is an interesting case study. The data show that the pandemic
trajectory comprises two small peaks followed by two large peaks [34]. The
country was grappling with another wave of infections during the time of this
work. It was therefore interesting to see whether the last wave is a
consequence of the interventions imposed earlier. Here we take $D_{0}$ as
March 04, 2020, following which there are four pandemic peaks in the daily
rates (excluding the latest wave). In our simulations these correspond to time
domains where $\beta$ values $<1$ are used. We identify these to be days
50–100, 130–180, 250–320 and 380–450. Since Serbia has a much lower population
density compared to India, we used $N=5$k/unit area and $l=2\langle r\rangle$.
Guided by the results for India, we again assumed $\beta=0.3$ for the periods
that correspond to systematic drops in the observed rates, following each
observed peak in the epidemic curve.

The simulation results are shown in Fig. 4, together with corresponding scaled
down data reported by the WHO. One can clearly see reasonable agreement
between the two, with the simulations adequately predicting the peak of the
fifth wave that was ongoing at the time of this work.

Figure 4: Simulated daily rates of Serbia, compared with scaled down WHO data.
The simulation results were averaged over 10 sets. Figure 5: Simulated results
for South Africa, compared with scaled down WHO data. The simulations assumed
$N=5$k and were averaged over 10 sets. The blue curve is generated assuming no
control interventions beyond day 500. The green curve assumes such
interventions ($\beta=0.3$) from day 500–600.

###  3.3 South Africa

Next we looked at the data for South Africa, whose growth trajectory is
interesting to follow, as the Omicron variant was first identified in the
region [35]. We took $D_{0}$ to be Mar 04, 2020. There are three pandemic
waves, with peaks around days 150, 300 and 500. To simulate the initial wave
we used333South Africa also has a lower population density, similar to Serbia.
$N=5$k and effective interventions ($\beta=0.3$) from day 150 to 250, with
$l=2\langle r\rangle$. On imposing a second period of lower infections from
day 320 to day 420, we obtain consistent results for the first two waves.
These are plotted in Fig. 5. In comparison, the peak around day 500 was found
to be too broad compared to the reported data from the WHO. We note that a
further intervention period from day 500 to day 600 (again with $\beta=0.3$)
yields daily rates for the third wave that are in reasonably good agreement
with WHO data. These results also show a clear fourth wave, whose position
depends on the imposed intervention in the previous wave. This Omicron driven
fourth COVID-19 wave has already been reported for South Africa [37]. It is
interesting to note that our predicted fourth wave is rather broad, similar to
the blue curve obtained on assuming $\beta=1$ beyond day 500. Although we have
not carried out simulations with imposed control interventions beyond day 700,
this similarity suggests the possibility of a fifth wave for South Africa.

Figure 6: Simulated rates for USA compared with scaled-down WHO data. These
simulations assumed $N=10{\rm k},l=0.7\langle r\rangle$ and were averaged over
5 runs.

###  3.4 USA

Our final set of simulations were for USA, which presents a different
challenge as each of its 50 states follow independent mitigation and
containment strategies. There are five observed waves of infections, with the
third peak being significantly larger than the others. A scaled down version
of the reported data is shown in Fig. 6. In our simulations we use $N=10$k and
take $D_{0}$ to be January 1, 2020. Based on the WHO data, we introduce three
intervention periods, between days 130–180, 230–280, and 400–550. For these
time periods we used $l=0.7\langle r\rangle$ and $\beta=0.3$. As Fig. 6 shows,
apart from the small peak around day 470, all other features in the data are
reasonably well reproduced by the simulations. The simulations also correctly
predict a wave after day 500, which was ongoing during the time of this work.
On the basis of our results one can also anticipate a fresh wave of infections
following the peak after day 600. Similar to the other cases studied, we
observe that control interventions following an increase in the daily rate of
infections invariably lead to a new epidemic wave later in time, provided
there are enough agents available to infect the remaining susceptible
fraction.

##  4 The equivalence between mobility restrictions on the random walkers and
imposition of a lower infection rate

One can get similar results as above by using the other approach that assumes
$\beta=1$ at all times. This is obtained by applying mobility restrictions at
different time ranges on most of the random walkers, so that the primary
infection peaks are reproduced. However, it is important to keep in mind that
these time ranges are not the same as periods of imposed lockdowns by
governing bodies. We discuss below the results obtained using this analogous
approach, for the case of India.

Here again two different sample populations with $N=5$k and 10k were used to
simulate the data for different jump lengths $l$. All calculations were
carried out for $\mu=1/35$ day-1 with a 20% mobile population during the time
period spanning from day 120 to day 300. The remaining 80% remain frozen at
their positions. Fig. 7 shows the fractional daily rates obtained for the two
populations for the intervention period mentioned above, together with a jump
distance of $l=2\langle r\rangle$. The 10k results show excellent agreement
with reported data for the second wave. Similarly consistent results are
obtained for the other countries studied in this work.

Figure 7: Simulated daily rates for India obtained using the mobility
restrictions as described in the text and $l=2\langle r\rangle$. The $N=10$k
results are averaged over 7 sets, while the $N=5k$ results are from a single
set, shown for comparison with normalized WHO data.

##  5 Discussion

The results presented in this work show that it is possible to predict the
trajectory of an epidemic using a random walk Monte Carlo approach that
includes time periods of control interventions. Two equivalent simulation
approaches (either using $\beta<1$ or keeping a percentage of the population
immobile when effective interventions were in place) can be followed, as they
yield similar results. We observe that for populous countries such as India
and USA reasonably good results are obtained on using a higher population
density (10k) in the simulations.444Although not shown explicitly, we find
that a higher population of $20$k per unit area leads to poor results. In
comparison, for countries such as Serbia and South Africa (whose population
densities are much smaller) a $N=5$k works best for the simulations.
Conservatively, both these numbers take into account higher floating densities
at localized places. In all cases, we show that the use of step-sizes
$\lesssim 2\langle r\rangle$ yield optimal results. The bound on this
parameter value can be justified as below.

In our earlier work [25] we showed that there are essentially two limits of
growth. If the jump step is too small then the growth is purely quadratic in
time [10] and if the jump step is too large the growth is exponential due to
homogeneous mixing [38]. The country-specific growth modes described here
occur between these two limits when there is sub-exponential behavior [9] due
to the containment strategies employed. The power-law nature of the growth is
lost for jump steps $\geq 3\langle r\rangle$. On the other hand when the
density becomes too high, since our simulations use jump lengths that are
proportional to the average inter-person separation, the epidemic growth
approaches the quadratic (lower) limit. In such a scenario an alternative
approach, such as in Ref. [24] may be better suited for studying the sub-
features in a single wave. This may offer an explanation as to why our
simulated results for large jump lengths or population densities fail to
reproduce the observed data.

It must also be added that in the present formalism the exact total number of
cases are not meaningful. Instead, the focus is on the rapid time-shifted
increase in the number of infections following intervention periods. These
correspond to new waves of infection. On this basis the simulations are used
to predict pandemic peaks at later dates, based on earlier data. If one
chooses to use a calibration factor based on previously observed peaks, it is
also possible to roughly estimate the total number of people infected.

##  6 Conclusions

To conclude, we use a random walk Monte Carlo simulation technique to better
understand contact-based epidemic spread. The model of independent random
walkers as infection carriers in two-dimensional space is intuitive and shows
promise in better understanding infectious disease outbreaks [25]. It is more
accessible than other models and has the ability to capture random
interactions that are missed through more conventional approaches. The
simulations are shown to predict reasonably accurate disease trajectories in
terms of secondary and tertiary waves, for COVID-19 data from four countries
with vastly different features.

##  7 Acknowledgements

We are grateful to Prof. Niranjan Barik for several helpful discussions and to
Dr. Ara Chutjian for useful remarks. ST acknowledges funding support from the
National Research Foundation (South Africa) under grant number 85100.

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