Philippa Kaur

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Data selection and handling: death data
For mortality due to COVID-19, we used time series provided by the New York Times12. We selected the New York Times dataset because it is rigorously curated. We analyzed separately only counties that had records of 100 or more deaths by 23 May, 2020. The threshold of 100 was a balance between including more counties and obtaining reliable estimates of r(t). Preliminary simulations showed that time series with low numbers of deaths would bias r(t) estimates (Supplementary Fig. 2). However, we did not want to use the maximum daily number of deaths as a selection criterion, because this could lead to selection of counties based on data from a single day. It would also involve some circularity, because the information obtained, r(t), would be directly related to the criterion used to select datasets. Therefore, we used the threshold of 100 cumulative deaths. The District of Columbia was treated as a county. Also, because the New York Times dataset aggregated the five boroughs of New York City, we treated them as a single county. For counties with fewer than 100 deaths, we aggregated mortality to the state level to create a single time series. For thirteen states (AK, DE, HI, ID, ME, MT, ND, NH, SD, UT, VM, WV, and WY), the aggregated time series did not contain 100 or more deaths and were therefore not analyzed.
Data selection and handling: explanatory county-level variables
County-level variables were collected from several public data sources36,37,38,39,40,41,42. We selected socio-economic variables a priori in part to represent a broad set of population characteristics.
Time series analysis: time series model
We used a time-varying autoregressive model15,56 designed explicitly to estimate the rate of increase of a variable using nonlinear, state-dependent error terms16. We assume in our analyses that the susceptible proportion of the population represented by a time series is close to one, and therefore there is no decrease in the infection rate caused by a pool of individuals who were infected, recovered, and were then immune to further infection.
The model is

$$xleft( t right) = rleft( {t-1} right) + xleft( {t-1} right)$$
(1a)

$$rleft( t right) = rleft( {t-1} right) + omega_rleft( t right)$$
(1b)

$$Dleft( t right) = {mathrm{exp}}(xleft( t right) + phi left( t right))$$
(1c)

Here, x(t) is the unobserved, log-transformed value of daily deaths at time t, and D(t) is the observed count that depends on the observation uncertainty described by the random variable ϕ(t). Because a few of the datasets that we analyzed had zeros, we replaced zeros with 0.5 before log-transformation. The model assumes that the death count increases exponentially at rate r(t), where the latent state variable r(t) changes through time as a random walk with ωr(t) ~ N(0, σ2r). We assume that the count data follow a quasi-Poisson distribution. Thus, the expectation of counts at time t is exp(x(t)), and the variance is proportional to this expectation.
We fit the model using the extended Kalman filter to compute the maximum likelihood57,58. In addition to the parameters σ2r and σ2ϕ, we estimated the initial value of r(t) at the start of the time series, r0, and the initial value of x(t), x0. The estimation also requires terms for the variances in x0 and r0, which we assumed were zero and σ2r, respectively. In the validation using simulated data (Supplementary Methods: Simulation model), we found that the estimation process tended to absorb σ2r to zero too often. To eliminate this absorption to zero, we imposed a minimum of 0.02 on σ2r.
Time series analysis: parametric bootstrapping
To generate approximate confidence intervals for the time-varying estimates of r(t) (Eq. 1b), we used a parametric bootstrap designed to simulate datasets with the same characteristics as the real data that are then refit using the autoregressive model. We used bootstrapping to obtain confidence intervals, because an initial simulation study showed that standard methods, such as obtaining the variance of r(t) from the Kalman filter, were too conservative (the confidence intervals too narrow) when the number of counts was small. Furthermore, parametric bootstrapping can reveal bias and other features of a model, such as the lags we found during model fitting (Supplementary Fig. 1a, b).
Changes in r(t) consist of unbiased day-to-day variation and the biased deviations that lead to longer-term changes in r(t). The bootstrap treats the day-to-day variation as a random variable while preserving the biased deviations that generate longer-term changes in r(t). Specifically, the bootstrap was performed by calculating the differences between successive estimates of r(t), Δr(t) = r(t) – r(t-1), and then standardizing to remove the bias, Δrs(t) = Δr(t) – E[Δr(t)], where E[] denotes the expected value. The sequence Δrs(t) was fit using an autoregressive time-series model with time lag 1, AR(1), to preserve any shorter-term autocorrelation in the data. For the bootstrap, a new time series was simulated from this AR(1) model, Δρ(t), and then standardized, Δρs(t) = Δρ(t) – E[Δρ(t)]. The simulated time series for the spread rate was constructed as ρ(t) = r(t) + Δρs(t)/21/2, where dividing by 21/2 accounts for the fact that Δρs(t) was calculated from the difference between successive values of r(t). A new time series of count data, ξ(t), was then generated using equation 1 with the parameters from fitting the data. Finally, the statistical model was fit to the reconstructed ξ(t). In this refitting, we fixed the variance in r(t), σ2r, to the same value as estimated from the data. Therefore, the bootstrap confidence intervals are conditional of the estimate of σ2r.
Time series analysis: calculating R0
We derived estimates of R(t) directly from r(t) using the Dublin-Lotka equation21 from demography. This equation is derived from a convolution of the distribution of births under the assumption of exponential population growth. In our case, the “birth” of COVID-19 is the secondary infection of susceptible hosts leading to death, and the assumption of exponential population growth is equivalent to assuming that the initial rate of spread of the disease is exponential, as is the case in equation 1. Thus,

$$Rleft( t right) = 1/mathop {sum}nolimits_{_tau} {{mathrm{e}}^{ – r(t)}} tau p(tau)$$
(2)

where p(τ) is the distribution of the proportion of secondary infections caused by a primary infection that occurred τ days previously. We used the distribution of p(τ) from Li et al.59 that had an average serial interval of T0 = 7.5 days; smaller or larger values of T0, and greater or lesser variance in p(τ), will decrease or increase R(t) but will not change the pattern in R(t) through time. Note that the uncertainty in the distribution of serial times for COVID-19 is a major reason why we focus on estimating r0, rather than R0: the estimates of r0 are not contingent on time distributions that are poorly known. Computing R(t) from r(t) also does not depend on the mean or variance in time between secondary infection and death. We report values of R(t) at dates that are offset by 18 days, the average length of time between initial infection and death given by Zhou et al.60.
Time series analysis: Initial date of the time series
Many time series consisted of initial periods containing zeros that were uninformative. As the initial date for the time series, we chose the day on which the estimated daily death count exceeded 1. To estimate the daily death count, we fit a Generalized Additive Mixed Model (GAMM) to the death data while accounting for autocorrelation and greater measurement error at low counts using the R package mgcv61. We used this procedure, rather than using a threshold of the raw death count, because the raw death count will include variability due to sampling small numbers of deaths. Applying the GAMM to “smooth” over the variation in count data gives a well-justified method for standardizing the initial dates for each time series.
Time series analysis: validation
We performed extensive simulations to validate the time-series analysis approach (Supplementary Methods: Simulation model).
Regression analysis for r 0
We applied a Generalized Least Squares (GLS) regression model to explain the variation in estimates of r0 from the 160 county and county-aggregate time series:

$$r_0 = b_0 + b_1start.date + b_2logleft( {pop.size} right) + b_3pop.den^{0.25} + varepsilon$$
(3)

where start.date is the Julian date of the start of the time series, log(pop.size) and pop.den0.25 are the log-transformed population size and 0.25 power-transformed population density of the county or county-aggregate, respectively, and ε is a multivariate Gaussian random variable with covariance matrix σ2Σ. We used the transforms log(pop.size) and pop.den0.25 to account for nonlinear relationships with r0; these transforms give the highest maximum likelihood of the overall regression. The covariance matrix contains a spatial correlation matrix of the form C = uI + (1–u)S(g) where u is the nugget and S(g) contains elements exp(−dij/g), where dij is the distance between spatial locations and g is the range62. To incorporate differences in the precision of the estimates of r0 among time series, we weighted by the vector of their standard errors, s, so that Σ = diag(s) * C * diag(s), where * denotes matrix multiplication. With this weighting, the overall scaling term for the variance, σ2, will equal 1 if the residual variance of the regression model matches the square of the standard errors of the estimates of r0 from the time series. We fit the regression model with the function gls() in the R package nlme63.
To make predictions for new values of r0, we used the relationship

$$hat e_{mathrm{i}} = bar e + {mathbf{v}}_{mathbf{i}} ast ,{mathbf{V}}^{ – 1}(epsilon_i – bar e)$$
(4)

where ει is the GLS residual for data i, (hat e)i is the predicted residual, (bar e) is the mean of the GLS residuals, V is the covariance matrix for data other than i, and vi is a row vector containing the covariances between data i and the other data in the dataset64. This equation was used for three purposes. First, we used it to compute R2pred for the regression model by removing each data point, recomputing (hat e)i, and using these values to compute the predicted residual variance23. Second, we used it to obtain predicted values of r0, and subsequently R0, for the 160 counties and county-aggregates for which r0 was also estimated from time series. Third, we used equation (4) to obtain predicted values of r0, and hence predicted R0, for all other counties. We also calculated the variance of the estimates from64

$$hat v_{mathrm{i}} = sigma^2-{mathbf{v}}_{mathbf{i}} ast ,{mathbf{V}}^{ – 1} ast v_i^{mathbf{t}}$$
(5)

Predicted values of R0 were mapped using the R package usmap65.
Regression analysis for SARS-CoV-2 effects on r0
The GISAID metadata27 for SARS-CoV-2 contains the clade and state-level location for strains in the USA; strains G, GH, and GR contain the G614 mutation. For each state, we limited the SARS-CoV-2 genomes to those collected no more than 30 days following the onset of outbreak that we used as the starting point for the time series from which we estimated r0; from these genomes (totaling 5290 from all states), we calculated the proportion that had the G614 mutation. We limited the analyses to the 28 states that had five or more genome samples. For each state, we selected the estimates of r0 from the county or county-aggregate representing the greatest number of deaths. We fit these estimates of r0 with the weighted Least Squares (LS) model as in equation (3) with additional variables for strain. Figure 3 was constructed using the R packages usmap65 and scatterpie66.
Statistics and reproducibility
The statistics for this study are summarized in the preceding sections of the “Methods”. No experiments were conducted, so experimental reproducibility is not an issue. Nonetheless, we repeated analyses using alternative datasets giving county-level characteristics, and also an alternative dataset on SARS-CoV-2 strains (Supplementary Methods: Analysis of Nextstrain metadata of SARS-CoV-2 strains), and all of the conclusions were the same.
Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article. More

As a wildlife ecologist, I study many species, but the platypus (Ornithorhynchus anatinus) is my speciality. I’ve worked with these duck-billed monotremes, which are indigenous to eastern Australia, for 13 years. So much remains unknown about them — as solitary animals, they are very hard to capture. The males have a venomous spur on each of their hind feet, so I always hold them by the tail. Luckily, I’ve never been spurred.
At the environmental consultancy firm Cesar Australia, we’ve focused on whether platypuses’ numbers are rising or falling. Their aquatic habitat has shrunk by 25% in the past 30 years — partly owing to reduced rainfall from climate change and partly to residential and industrial development, leaving less water, which is of poorer quality.
To monitor the population’s size and health, we do overnight trapping surveys — typically during the spring breeding season and in the autumn, when the babies emerge. For each survey, we set up nets along 5–10 kilometres of waterway and then drive round checking them through the night. It’s a long, sleepless process that usually yields just one or two catches. In the 15 minutes or so for which each animal is out of the water, we weigh and measure it. If it’s carrying an identification chip, we know we’ve caught it before. If not, we implant one in the animal. In this picture, I am releasing a platypus back into a stream outside Melbourne.
In the past five years, we’ve also been sampling environmental DNA for signatures of platypuses’ presence. It’s been a game-changer in terms of mapping their distribution. We can take a water sample, extract all the DNA and confidently identify whether platypus DNA is present. We’ve surveyed more than 2,000 sites spanning 500,000 square kilometres, enabling us to measure at scales never before possible.
We’ve also developed a citizen-science platform through which people can submit platypus sightings. This helps us still further. More

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Biological experiments
Cell culture
We obtained unlabelled parental (sensitive) and radiation-resistant populations from two prostate cell lines, PC3 and DU145, from the Liu laboratory (University of Toronto, Canada). Radioresistant cell populations comprised pooled cells from the parental line that survived a clinically-relevant course of radiotherapy23,24. To produce stable, fluorescent cell lines, we transduced cells using lentiviral particles containing the vectors, pCDH1-CMV-GFP-EF1-Hygro or pCHD1-CMV-DsRed-EF1-Hygro (Systems Biosciences), collected the top 30% of brightest cells by flow cytometry, and used hygromycin B (50 mg/mL, Gibco) for selection (200 µg/mL for PC3 and 250 µg/mL for DU145). Cell lines were cultured in DMEM medium (low glucose, pyruvate, GlutaMAX, Gibco) supplemented with 25 mM HEPES (Gibco), 10% foetal bovine serum (Sigma or Pan-Biotech), and 1% penicillin/streptomycin. Authentication was performed using STR profiling (Promega PowerPlex 21 PCR kit, Eurofins), and mycoplasma checks were performed routinely using MycoAlert Mycoplasma Detection Kit (Lonza). All cells were maintained in an incubator (37 °C, 5% CO2). Unlabelled cells were cultured for up to 10 passages (~6 weeks) for transduction; labelled cells were cultured for up to 10 passages (~6 weeks). No additional courses of radiation were used to maintain resistance. We measured clonogenic survival at 2 Gy (SF2) to verify that the labelled cells maintained the resistance phenotype for up to 12 passages (~8 weeks) in culture.
Monolayer growth experiments
Single cells (1 × 103 cells/well, 200 μL medium) were seeded in triplicate in flat-bottom 96-well plates, allowed to attach overnight, irradiated (0, 2, or 6 Gy), and imaged daily using brightfield (Incucyte Live Cell Imaging System, Sartorius). The medium was changed every 2 days. Cell confluence was determined using Incucyte Base Software (Sartorius).
Clonogenic assays
Survival of cell lines after radiation was measured using a clonogenic assay36. Briefly, cells were seeded in triplicate in six-well plates and irradiated using a Cs-137 (dose rate of 0.89 Gy/min) or an X-ray irradiator (195 kV, 10 mA). Surviving colonies were stained after 10 days with crystal violet and counted. The surviving fraction was calculated as (number of colonies/number of seeded cells) × plating efficiency.
Response to cisplatin
To check whether the RR cells had altered DNA damage response, we measured the cell viability of each population in response to a range of concentrations of cisplatin using a modified cytotoxicity assay37. Briefly, single cells (2 × 103 cells in 100 μL/well) were seeded as monolayers in triplicate in a 96-well plate and allowed to attach for 36 h before treatment. Increasing concentrations of cisplatin (made in 100 μL/well) were added to each well resulting in a final volume of 200 μL/well; cisplatin (Sigma) was prepared fresh for each treatment by dissolving the powder in 0.9% sterile-filtered saline to a stock concentration of 3.3 mM. Treated cells were then cultured for 72 h in cisplatin before they were fixed with 10% formalin. Cell confluence was determined using Incucyte Base Software (Sartorius) and reformatted to a concentration–response curve by normalising cell confluence values to the untreated well.
Spheroid generation and culture
Homogeneous and mixed spheroids were generated in 96-well, ultra-low attachment plates (7007, Corning) by seeding different ratios of parental and radioresistant cell populations (2 × 103 total cells/well) using Matrigel (5% v/v, Corning) to promote spheroid formation38. For all spheroid experiments, after a formation phase of 3 days, spheroids were fed every 2 days by replacing 50% of the medium in each well with fresh medium (200 μL total/well). Culture medium and incubation conditions were as described under ‘Cell culture’. Spheroid volumes were calculated using SpheroidSizer39.
Unirradiated spheroid growth experiments
Spheroids were generated as described in ‘Spheroid generation and culture’ and monitored for growth by brightfield imaging (Leica DM IRBE, Hamamatsu).
Flow cytometry
The proportions, survival, and cell cycle of each population from spheroids were measured by flow cytometry. Mixed unirradiated or irradiated spheroids (seeded 1:1 parental:RR, 6–8 pooled/group) were incubated with EdU (10 µM final concentration) 12 h prior to dissociation, dissociated (100 μL Accumax, Millipore) for 20 min at 37 °C, washed with phosphate-buffered saline (PBS), centrifuged (300 × g, 5 min), and incubated with efluor-780 (1 μL/mL PBS; ThermoFisher Scientific) for 30 min on ice in the dark to distinguish live/dead cells. After washing in PBS, samples were fixed for 10 min in IC Fixation Buffer (ThermoFisher Scientific), and permeabilized and stained with Click-iT Plus EdU Alexa Fluor 647 (ThermoFisher Scientific) according to manufacturer’s instructions. Following a wash in 1× saponin, cells were incubated 30 min with FxCycle Violet Stain (1:1000, 300 µL of 1× saponin; ThermoFisher Scientific) before being run on the BD LSR Fortessa X-20 Cytometer or the Attune NxT Flow Cytometer using the 405, 488, 561, and 633 lasers. Data were analysed using FlowJo (Treestar, Inc.) as described in Supplementary Figs. 2 and 4.
Spheroid growth experiments after radiation
To determine bulk radiation response of spheroids, PC3 cells were seeded as spheroids (n = 15 per dose per group) with 4 groups as described in ‘Spheroid generation and culture’: parental, 9:1 parental:RR, 1:1 parental:RR, and RR. After formation, spheroids were irradiated (0, 2.5, 5, 7.5, 10, 15, and 20 Gy) on day 4 and imaged for up to 48 days to monitor regrowth using brightfield (Celigo Imaging Cytometer, Nexelcom). After log-transforming the volume data, we calculated the radiation-induced growth delay (days) relative to untreated spheroids as the time for each irradiated spheroid to reach a volume endpoint (2.5 times the starting volume right before irradiation); we selected the lowest endpoint that was still within the exponential growth phase of all spheroids in the experiment. The average time for untreated spheroids to reach endpoint was estimated in R by local regression using the loess function with “direct” surface estimation to allow extrapolation for the parental spheroids (R project, v. 3.6.2).
Regrowth experiments were repeated by irradiating day 3 spheroids from three groups (parental, mixed and RR) of PC3 cells (6 Gy, n = 17–18/ group) and of DU145 cells (6 Gy, n = 12/group; 10 Gy, n = 20/group). Spheroids were imaged using brightfield (Leica DM IRBE, Hamamatsu) for up to 27 days (PC3) and up to 23 days (DU145). The radiation-induced growth delay (delays) was calculated as above, but with different endpoints (3.5 times starting volume for PC3 and 4 times starting volume for DU145) to ensure the endpoint was within the exponential growth phase. Data from Fig. 2 were used to estimate average time of untreated spheroids; the ‘span’ parameter of the loess function was reduced from the default of 0.75–0.5 for the unirradiated DU145 spheroids to better estimate the average time of reaching the endpoint.
To measure changes in the radiation response of PC3 cell populations within spheroids, untreated homogeneous and mixed spheroids were grown until day 5 or 11, dissociated using Accumax, seeded as single cells for clonogenic experiments, and allowed to attach for 6 h prior to radiation. Fluorescent colonies were counted using the Celigo Cytometer.
Immunofluorescence
For immunofluorescence and H&E experiments, spheroids were treated and fixed prior to staining40. To investigate the spatial distribution of fluorescent populations, sections were hydrated in PBS, stained for 10 min with Hoechst (1 μg/mL in PBS, Sigma) to visualise nuclei, and mounted using ProLong Diamond Antifade Mountant (ThermoFisher). For hypoxia, spheroids were pre-treated with 300 μM of the hypoxia drug EF5 (gift from Dr. Cameron Koch, University of Pennsylvania) prior to fixation. They were then permeabilized (PBS containing 0.3% Tween-20, 10 min), blocked (5% goat serum in PBS containing 0.1% Tween-20, 30 min), stained using anti-EF5 antibody (75 μg/mL; from Dr. Cameron Koch) overnight at 4 °C, washed (ice-cold PBS containing 0.3% Tween-20, 2 × 45 min)40, stained for nuclei as above, and mounted. For Ki67, spheroid sections were permeabilized (PBS containing 0.3% Tween-20, 10 min), blocked (5% goat serum in PBS containing 0.1% Tween-20, 30 min), and incubated overnight at 4 °C with primary antibody (clone SP6, 1:100, Vector Laboratories). After washing in PBS, sections were incubated for 1 h with goat anti-rabbit Alexa Fluor 647 (4 μg/mL, ThermoFisher), washed, and stained with Hoechst 33342 (5 μg/mL, Sigma) for 10 min. Slides were mounted and imaged using epifluorescence microscopy (20× objective; 0.30 NA; 0.64 μm resolution; excitation lasers: 395, 470, 555, and 640; Nikon Ti-E). Sections were stained using H&E and imaged using a Bright Field Slide Scanner (Aperio CS2, Leica) to visualise necrosis. To quantify the ratio of parental to radioresistant populations in spheroid cross-sections (n = 16 spheroids from 4 batches), we measured the number of pixels from each population (i.e., signal) by applying a threshold value 5 times higher than the median value of the background (i.e., noise) (Octave 4.4.1).
Oxygen consumption measurements
OCR was measured from each population using the Seahorse assay. Cells (1.2 × 104/well) were seeded in triplicate using the normal culture medium in a Seahorse XF 96-well microplate (Agilent) and allowed to attach overnight. Prior to the assay, cells were washed with and incubated in assay medium (DMEM basal medium containing 5 mM glucose, 4 mM glutamine, 5 mM pyruvate, pH 7.4; 200 μL/well) for 2 h at 37 °C without CO2 to degas the medium. Calibrant buffer (200 μL/well) was added to wells of the probe plate and also left at 37 °C without CO2 to degas. After OCR was measured on the Seahorse XF Analyser (Agilent Biosciences), cells were fixed using 4% paraformaldehyde, stained using Hoechst 33342, and counted (Celigo Cytometer, Nexelcom).
Transwell experiments
Co-culture experiments were performed to measure whether transferred factors between cell populations enhanced survival under hypoxia. Cells were seeded in triplicate (3.0 × 104/bottom well and 1.0 × 104/insert) in 12-well plates and in Transwell inserts, and allowed to attach overnight. Once the medium was changed, the plates were placed into normoxia or hypoxia (0.1% O2) for 24 and 120 h. Cells were fixed using 4% paraformaldehyde, stained with Hoechst (5 μg/mL), and counted (Celigo Cytometer, Nexelcom).
Statistics and reproducibility
Data were evaluated for equal variance using homoscedasticity plots (absolute value of residual vs predicted value) and for normality using Q–Q plots (Prism 8.0, GraphPad). Unless otherwise indicated, statistical significance was evaluated using one-way ANOVA, two-way ANOVA, or a mixed-effects model followed by multiple testing correction (α = 0.05). For clonogenic assays, the radiation protection factor was calculated as the area under the dose–response curve (AUC) for the RR cell populations divided by that of the parentals; AUC values were analysed for significance using a Student’s t test (unpaired, one-tailed, α = 0.05). For cisplatin cytotoxicity assays, IC50 values were calculated using a normalised response, variable slope, dose–response model (Prism, 8.0, GraphPad) and evaluated for statistical significance using extra sum-of-squares F-test. For post-radiation growth experiments, survival curves were analysed using the Mantel–Cox (log-rank) test and adjusted for multiple testing using Holm’s correction; spheroids that did not reach the endpoint during the timeframe of the experiment were marked as ‘censored’ on the final day of the experiment (please see Supplementary Methods section 3 for further details). Due to heteroscedasticity, cell counts from flow cytometry experiments involving cell cycle and death, and from co-culture Transwell assays were analysed using overdispersed Poisson or binomial regression models (please see Supplementary Methods section 3 for further details). For quantification of population proportions in microscopy images, pixel numbers were analysed using a two-tailed, Wilcoxon matched-pairs signed-rank test. Adjusted P values (Padj) are reported in the main text for experiments where multiple comparisons were performed.
Data points represent biological replicates; experiments were performed using at least two separate batches of cells. We note the following data exclusions: missing data from some time points due to technical failures in imaging (Fig. 1), one excluded mixed DU145 spheroid because its growth did not resemble that of the other 35 spheroids (Fig. 2a), and one excluded plate of PC3 spheroids from survival analysis (Fig. 4) because of irregular growth that did not match the other nine plates. Sample sizes were approximated using effect sizes from pilot studies to ensure power (approximate β = 0.8); randomisation and blinding were not possible.
Mathematical experiments
Non-spatial mathematical models
We used the logistic growth model to describe the growth of homogeneous tumour spheroids26. Thus, the rate of change of spheroid volume V at time t is given by

$$frac{{dV}}{{dt}} = rV left(1 – frac{V}{K}right),$$
(1)

where r  > 0 represents the growth rate, K  > 0 is the carrying capacity (the limiting volume of the spheroid) and V(t = 0) = V0 denotes the spheroid volume at t = 0. The analytical solution to the logistic model is given by

$$Vleft( t right) = frac{{V_0Ke^{rt}}}{{K + V_0(e^{rt} – 1)}}.$$
(2)

The Lotka–Volterra model was used to describe the growth of mixtures of parental and RR cell populations

$$left. {begin{array}{*{20}{c}} {frac{{dV_P}}{{dt}} = r_PK_Pleft( {1 – frac{{V_P}}{{K_P}} – lambda _{RR}frac{{V_{RR}}}{{K_P}}} right)} \ {frac{{dV_{RR}}}{{dt}} = r_{RR}K_{RR}left( {1 – frac{{V_{RR}}}{{K_{RR}}} – lambda _Pfrac{{V_P}}{{K_{RR}}}} right)} end{array}} right},$$
(3)

with VP(t = 0) = VP0 and VRR(t = 0) = VRR0. In these equations, VP and VRR represent respectively the volumes of parental and RR populations, rP and rRR their initial growth rates, KP and KRR their carrying capacities, and VP0 and VRR0 their initial volumes. The parameters λP and λRR describe the effect that parental cells have on RR cells, and vice versa. These type of interactions, found in ecology12, may be competitive (λP  > 0 and λRR  > 0), mutualistic (λP  More

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