Spatiotemporal Fidelity of a Metapopulational Model Evaluated on the COVID-19 Pandemic in Sweden

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Spatiotemporal Fidelity of a

Metapopulational Model Evaluated on the COVID-19 Pandemic in Sweden

Filip Skogh

Computer Science and Engineering, bachelor's level 2021

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

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Abstract | i

Abstract

A modified SEIR compartmental model is constructed for COVID-19 in a metapopulational setting with fine-scaled population data. The model is stochastically simulated with GLEAMviz that provides realistic short and long distance mobility based on real-world data. A retrospective evaluation in both the temporal and spatial dimensions is conducted with over one year of collected data from the COVID-19 pandemic in Sweden. We find that to reproduce the multimodal behavior a seasonal scaling factor of 0.4 is necessary, which is introduced to the model by scaling R0with corresponding sinusoidal. For the spatial dimension we divide Sweden into a southern, middle and northern region and the model is able to capture the dynamics in all regions. Additionally, we introduce compartmental models in a constructive manner and motivate metapopulational models, including how commuting is integrated by a force of infection. The next generation method for calculating the basic reproductive number for arbitrary compartmental models is also presented.

Keywords

Epidemiology, COVID-19, Sweden, Metapopulation, SEIR, Model

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ii | Abstract

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Acknowledgments | iii

Acknowledgments

I wish to express my gratitude to my supervisor András Bota for his ideas and infectious enthusiasm on epidemic modelling.

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iv | CONTENTS

Chapter 1 Introduction

Since coronavirus disease (COVID-19) emerged in December 2019 it has had immense economical implications and caused fatalities in the order of mil- lions [1]. The disease, caused by SARS-CoV-2 [2] can with access to global transportation spread fast as demonstrated when COVID-19 was declared a pandemic on 11 March 2020 by the World Health Organization [3].

To be prepared and quickly be able to make data driven decisions in future pandemics, models have to be adaptable to new diseases and reliable. In this thesis the latter is investigated by modelling the COVID-19 pandemic in Sweden and comparing it with recorded data from the COVID-19 pandemic in Sweden.

1.1 Background and Motivation

As we are substantially affected by diseases it is of great importance to be able to predict them. A model is a simple representation of reality and hence, cannot be expected to have perfect accuracy. Still their usefulness in predicting large-scale behavior in both temporal and spatial dimensions have made them popular [4].

With ever more computational capacity, in silico simulations are a useful addition to purely analytical models as real-world air travel, commuting and demographic data can be included. This allows the model to add real- world complexities and capture more intricacies. Common methods include metapopulational models where the population is divided into sub-populations

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2 | Introduction

with a homogeneous mixing assumption, agent based models where each individual is modelled individually [5] and combinations of them. These methods vary greatly in computational power and data requirements that one must consider if the simulation shall be accessible.

1.2 Problem

For epidemic models to be used in policy making and be part of predicting disease dynamics they must be complex enough to capture intricacies. But at the same time be parametrizable with limited disease data, as often is the case in real-world scenarios. In essence, can models using real mobility and population data produce realistic scenarios in both spatial and temporal dimensions?

1.3 Purpose and Goals

The purpose of this thesis is to show how a publicly accessible epidemic model using real-world population and mobility data can predict large-scale behavior of COVID-19 in both temporal and spatial dimensions. This is aligned with the United Nations’ third Sustainable Development Goal for good health and well-being. Moreover, the reliability of public models are especially useful in developing countries as they have less opportunity for other research and hence reduces inequalities among countries as goal 10 aspires to. Another aim is to acquaint the reader with some of the popular models used in epidemiology. To fulfill our purpose we set the following goals:

1. Construct and introduce a disease model for COVID-19.

2. Capture both the temporal and spatial dimensions of the COVID-19 pandemic in Sweden.

1.4 Methodology

We use the global epidemiology and mobility (GLEaM) model [6–8] that allows simulation in a metapopulation setting with real-world mobility and

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Introduction | 3

population data. The metapopulational approach introduces a geospatial dimension at the level of regions and enables a spatiotemporal comparison with real regions in Sweden. We choose to use a modified SEIR model as it is a simple and well-studied model. These characteristics are important at the onset of epidemics as when adding more compartments, more parameters are needed that often have great uncertainty, and thus provides no improved insights. As countermeasures in Sweden have been relatively mild, we can assume that the transmission rate is smooth and the only time dependent change is due to seasonal effects.

The metric used for evaluating the fidelity of the model is COVID-19 fatalities.

We choose this as testing has not been constant throughout the period and as a result the incidence data is not accurate. Another reasonable proxy is the number of hospitalizations.

1.5 Delimitations

Due to the complexity of reality we have to limit our scope. This is done through model choices such as disregarding age dependent variables and explicit modelling of super-spreaders. Fluctuations caused by differences in strains or vaccination is also not in our scope. We also choose to limit our evaluation to Sweden only.

1.6 Thesis Structure

Chapter 2 introduces disease models with temporal and spatial dependence that combined allows for a introduction to GLEaM. The chapter also includes related work on COVID-19 modelling. Chapter3illustrates how the disease model was constructed, parametrized and simulated. In chapter 4 the simulation results of Sweden is presented and finally, Chapter 5 concludes and reflects on the thesis.

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4 | Introduction

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Theory | 5

Chapter 2 Theory

Understanding the biological aspect of diseases has naturally led to modelling them. Some of the first known results in the field are due to Daniel Bernoulli, kinsman to the notable Bernoulli family. During the 18th century he argued for smallpox inoculation by developing a disease model for the disease [9].

More recently, focus have been on spatial heterogeneity by studying disease dynamics on complex networks.

2.1 Compartmental models

In 1927 Kermack and McKendrick introduced the SIR model [10]. It is insightful and extensible, allowing for modelling of a wide variety of diseases.

The model’s essential idea is to divide a population into three compartments:

susceptible (S) individuals that can get infected by infectious individuals, infectious (I) individuals that can spread the disease and removed (R) individuals who either are immune or dead. The transition dynamics that move individuals between these compartment is what determines its usefulness. Transitions between compartments are associated with parameters that determine the transition rates, e.g the level of infectivity, recovery rate and incubation time.

A number of parameters can be affected by policy makers through methods such as curfews and social distancing. Pharmaceutical advances can also have great effect, e.g recovery rate. The simplest SIR model is parametrized by two rates: the infection rate β, and the remove rate µ as shown in equation2.1and

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6 | Theory

2.2.

S + I −−→ 2I,^β (2.1)

I −−^µ→ R. (2.2)

Written in its differential equation form where S(t), I(t) and R(t) denotes the amount of people in respective compartment at time t, we have

S = −˙ β

NI(t)S(t), I =˙ β

NI(t)S(t) − µI(t), R = µI(t).˙

(2.3)

As individuals only transition between compartments the population size, N , is constant and given by N = S(t) + I(t) + R(t). Models can be made arbitrarily complex by adding more components, vital dynamics can also be included that introduces more parameters for birth and death rates. In this paper the discrete counterpart to the system of differential equations first introduced in [10] is used.

Figure 2.1: A subset of a population at a time t, depicting individual i with a contact rate c = 4 individuals per time step.

To motivate the system in2.3 we let c be the number of contact a individual encounter per time step as in Figure2.1. Assuming a homogeneous population such that an individual is as likely to have contact with any other, the probability of a contact being with an infectious individual is I(t)/N . As there are Let ν be the probability of transmission given contact between susceptible and infectious. The expected increase in the number of infectious individuals is therefore S(t)cνI(t)/N , causing a decrease by the same amount of susceptible individuals. By convention we let β = cν and we have the change in number

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Theory | 7

of susceptibles per time step

S(t + 1) − S(t) = −βS(t)I (t) /N . (2.4) To allow transitions to the removed compartment we let µ be the removed rate per time step for an infectious individual. The expected infectious period is therefore µ⁻¹ time steps and we obtain

I(t + 1) − I(t) = βS(t)I(t)/N − µI(t). (2.5) As the population size N is constant and the relation N = S(t) + I(t) + R(t) is true for all t, R(t) is derived implicitly. Albeit the models simplicity, many results are far from obvious due to its non-linearity. Following similar argu- ments as when derived Equation2.4and2.5we can add more compartments, and derive more complicated models, e.g the SEIR model.

An important metric is if there will be an outbreak if an infectious individual is introduced in to a fully susceptible population such that S = N − 1 ≈ N . Solving I(t + 1) − I(t) > 0 we arrive at the condition β/µ > 1. Here β/µ is the basic reproductive number R0, an important dimensionless number used to classify the potential danger of a disease. R0 can be interpreted as the number of secondary cases generated from the index case. See Appendix A for a method of computing R₀ for arbitrary compartmental models with n compartments.

2.1.1 Metapopulational models

The standard compartmental model’s homogeneous assumption, in which random mixing is assumed for the whole population can be relaxed [11]. By dividing the population into multiple sub-populations, each with the homogeneous assumption, we get a population of populations: a metapopulation. For instance, homogeneous mixing for a whole nation may be fallacious, whereas reasonable at the city scale. By introducing spatial heterogeneity, the model is now described by one set of differential equations per sub-population as seen in Figure2.2. The realism of the model is now dependent on inter sub-population interaction, such as commuting and also how the population is distributed.

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8 | Theory

Figure 2.2: Metapopulation where each sub-population is compartmentalized with an SEIR model.

2.2 GLEaM

In this section we describe GLEaM: a framework for creating arbitrary compartmental models in a metapopulational setting with both short and long distance mobility [8,12]. The metapopulation is defined by a voronoi tes- selation of the world with airports as seeds. This creates sub-populations with the property that the closest airport from anywhere in the region is the seed airport. The advantages of this approach is that administrative regions are often shaped by history and may not be the most natural area for the homogeneous assumption. This method is also generalizable to the whole world. The population size of each region is fitted with census data.

Each region’s disease dynamics is described by a set of difference equations.

Formally, we let C be a set of all compartments and T ⊆ C be compartments that allow for traveling. In each sub-population i, a compartment’s temporal evolution is defined by

X_i(t + 1) = X_i(t) + ∆X_i+ Ω_i(X) (2.6) where ∆Xi is net flux of individuals moving in and out of compartment X defined by transition probabilities. In general this is simulated as a random integer sampled from a multinomial distribution defined by respective transition probability [12]. The term, Ω_i(X) is the change due to air travel, see section2.2.1.

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Theory | 9

2.2.1 Long Distance Mobility

To simulate long distance mobility GLEaM approximates the exchange of people from different sub-populations in each compartment using real flight data between IATA airports [12]. The data includes passanger capacity ωij

on flights from airport i to j, as flights are not always full a stochasticity term is added. A scaling factor r_ω is also considered for periods with travel restrictions. On average each airport have an equal amount of travelers arriving and departing so no population drift will occur, i.e each sub-population size N is constant.

The probability p_ij that an individual travels from sub-population i with population size Ni to j is proportional to ωij/N_i. The number of individuals traveling to respective country therefore follows a multinomial distribution.

Using the travel probabiliteis, the net flux in compartment X_iin subpopulation i is

Ω_i(X) = (P

j(p_jiN_j− p_ijN_i) X ∈ T ,

0 otherwise. (2.7)

2.2.2 Short Distance Mobility

Let N_i be the original population in each subpopulation i. GLEaM follows the approach of Sattenspiel and Dietz [13] by separating the population into two classes: indigenous individuals Nii(t) and commuters originating from i residing in subpopulation j at time t, Nij(t). N_i can therefore be partition as

Ni = Nii(t) +X

j

Nij(t). (2.8)

A homogeneous commuting rate is assumed such that each individual in sub- population i commutes to sub-population j with probability σij and returns with universal rate τ . As seen in Figure 2.3 the change over time of Nii(t) is the net flux of commuters leaving i and returning to i per unit time. And similarly, ˙N_ij(t) is the net flux of commuters between i and j. By setting the rates ˙Niiand ˙Nij to 0 as in [13] we obtain the equilibrium quantities

Nii= N_i

1 + σ_i/τ and Nij = Ni

σ_ij/τ

1 + σ_i/τ (2.9)

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10 | Theory

where the total commuting rate for i, σ_i = P

jσ_ij. Balcan et al. shows in [12] that the equilibrium values are valid approximations given that the time in a disease compartment is greater than the duration a commuter resides in a neighboring population, τ⁻¹. Furthermore, Equation2.9can be generalized by assuming that individuals in compartments that allows commuting, commutes independent of compartment [12]. The number of individuals in each compartment Xi in sub-population i can therefore be partitioned as

X_ii= X_i

1 + σ_i/τ and X_ij = X_i σ_ij/τ

1 + σ_i/τ. (2.10) For compartments X ∈ C \ T , no commuting is allowed such that σ_ij = 0.

The equations then become, as expected Xjj = X_j and Xij = 0.

Figure 2.3: Population equilibrium with commuting between sub-populations i, j and k. Where each sub-population is divided into indigenous individuals and commuters currently in the sub-population.

From Figure 2.3 we note that each sub-population’s size is dependent on commuters from other sub-populations. The actual size of sub-population i, N_i^∗is calculated by summing over all individuals in respective sub-population as seen in Figure2.3, resulting in

N_i^∗ = N_ii+X

j

N_ji. (2.11)

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Theory | 11

The force of infection determines the rate that susceptible individuals become infected. In a metapopulational model the infection in a sub-population i occurs when susceptible individuals is in “contact” with infectious individuals in i or while visiting another sub-population j. For example, consider Stockholm as infectious, cities nearby would also be affected due to the bidirectional mobility of infectious individuals. Using the same argument again, the disease would continue to diffuse throughout space until all gradients vanish.

Setting the force of infection in each sub-population i to λ_i, it is dependent on infectious individuals in i, I_i. But individuals commuting from i to j will instead have contact with infectious individuals in j, Ij. λi is therefore split into λ_iiand λ_ij where the former is the force of infection for indigenous individuals and the latter is the force of infection for individuals that have commuted from i to j. The fraction of susceptible indigenous individuals is given by Sii/S_i and the fraction of commuters from i to j are Sij/S_i. By weighting λ_iiand λ_ij by their relative occurrence we obtain

λ_i = S_ii

S_iλ_ii+X

j

S_ij

S_i λ_ij. (2.12)

Using Equation2.10on Sii/Sj and Sij/Sj we have λ_i = λ_ii

1 + σ_i/τ +X

j

σ_ij/τ

1 + σ_i/τλ_ij. (2.13) Infectious individuals I_ithat λ_iidepends on are either from i and in i, or they are commuters from a neighbor j. In essence I_i = I_ii+P

jI_jiand λ_iican be expressed as

λ_ii = β

N_i^∗ I_ii+X

j

I_ji

!

. (2.14)

Note that β is time dependent in the case of seasonality. λ_ij can be derived as individuals in j are either from j and in j or from another sub-population k and currently in j. Continuing in the same manner as for λiiwe have

λ_ij = β

N_j^∗ I_jj+X

k

I_kj

!

. (2.15)

The realism of the commuting is determined by σ. Two well-known models for

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12 | Theory

commuting are the gravitation model and the radiation model [14]. The former can be seen in analogy to how newtons gravitational model attract masses.

The latter can be derived from first principles and is parameterless. It is also dependent on population densities which naturally creates asymmetries in the sigma matrix. Albeit the analogy’s limits, a fitted gravitational model has shown a high level of realism.

2.2.3 Seasonality

Sweden is located at a high latitude where seasonality strongly affects climate and social behavior. This has consequences on disease dynamics, which in cases such as influenza is evident. Looking at the data for COVID-19 both hospitalization and mortality rates where close to zero during Sweden’s warmer months [15]. The converse can be seen in Australia where the peak is in early August [1]. In GLEaM [12] seasonality is represented by scaling R₀ by a sinusoid

s(t) = 1 2

(αmax− αmin) cos 2π

365(t − tmax)

+ αmax+ αmin

(2.16) where α_max = 1.1 and α_min is the maximum and minimum scaling factor respectively. t_max = 15 such that the maximum scaling occurs on January 15.

2.2.4 Simulation Algorithm

Let G = (V, E) be a multigraph. Each element in V is a sub-population so V is the metapopulation. E represents the multiset of mobility patterns between sub-populations. We let E₁ ⊂ E be commuting connections and E₂ ⊂ E be flight travel connections such that E = E₁∪ E₂. Connections in E₁and E2are weighted by σij and pij respectively. Each sub-population in V

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Theory | 13

is compartmentalized with components X ∈ C.

Algorithm 1: GLEaM simulation overview [12]

foreach timestep t do

foreach sub-population i ∈ V do foreach compartment X do

Update Xiwith Ωi(X) using travel probabilities from E2, see Eq.2.7

end end

foreach sub-population i ∈ V do

Calculate the current sub-population size N_i^∗, see Eq.2.11 Calculate force of infection λ_iusing commuting probabilities

from E₁, see Eq.2.13 foreach compartment X do

Using the transition probabilities, stochastically move individuals between compartments

end end end

Note that flights are calculated at the start of each time step and that flight-time is zero, i.e all compartments are instantaneously updated.

2.3 Related Work

COVID-19 has been extensively studied at different scales and regions with a multitude of models. Kheirallah et al. use an SEIR model simulated with GLEAMviz to evaluate the strict non-pharmaceutical interventions (NPI) in Jordan [16]. They base their parameters on literature and simulate different scenarios ranging from no action to full lockdown.

They concluded that the NPIs introduced had successfully reduced the effec- tive reproductive number to less than one. In hindsight, strict NPIs are hard to enforce for too long and the situation in Jordan is now closer to their no action scenario that is on the same magnitude as Sweden [1]. Similar studies that assist policy makers has also been done with GLEAMviz in other countries [17,18].

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14 | Theory

In [19] the spatial effects is studied in England and Wales using a metapopulational SEIR model with a force of infection dependent on neighboring sub-populations. They show how spatial heterogeneity has large effects on spread and delays the time to peak by a factor of 4 compared to a standard mass-action model. They also conclude that peaks are hard to predict due to their great sensitivity to seasonality.

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Method | 15

Chapter 3 Method

In this chapter we construct a modified SEIR model and parametrize it accord- ing to literature. The data source for COVID-19 fatalities used as an evaluation metric is also presented.

3.1 Model Construction

The SEIR model has been extensively studied for close to a century and have been used to model a variety of diseases including influenza like illnesses [7, 8,12]. These are characterized by an incubation period and as COVID-19 follows a similar disease progression the SEIR model is frequently adopted [16–19].

The SEIR model used in this thesis is depicted in Figure 3.1. Individuals start as susceptibles except the initially infectious individuals who starts in either infectious compartment. Once infected by either an asymptomatic or an infectious individual with respective rates rβ and β the individual is moved to the exposed compartment. After the expected ⁻¹ days as exposed an individual becomes infectious with probability P_a of it being asymptomatic.

Once the expected infectious period of µ⁻¹ days the individual either transition to the death compartment with probability P_d, and otherwise to the removed compartment. The latter two compartments are absorbing states and individuals that have entered have no further effect on the disease dynamics.

This also means that reinfection is neglected which over our simulation period is a reasonable assumption [20]. As P_d is the infectious fatality rate, it

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16 | Method

also includes asymptomatic cases that may be hard to account for in reality.

However, this only affects the death magnitude by a constant factor 1 − Pa

and has no effect on the disease dynamics. To simulate the model we use the GLEAMviz client [6].

Figure 3.1: A modified SEIR model where Pa is the probability of asymptomatic infection and P_dis infectious fatality rate. r is the relative infectious- ness of an asymptomatic individual compared to a symptomatic.

As derived in AppendixA, the model’s basic reproductive number is given by R₀ = β

µ(rp_a+ 1 − p_a) . (3.1)

3.2 Model Parametrization

Parameters can be divides into two groups: COVID-19 intrinsic, such as incubation period, and COVID-19 extrinsic, such as commuting return rate.

We assume that intrinsic values are constant, e.g no mutations with noticeable effects, etc. Based on literature we set the intrinsic parameters as in Table3.1.

To initiate the pandemic we set five index cases in each of Sweden’s two largest cities, Stockholm and Gothenburg. These cities were chosen as the disease was imported and the major airports are located in these cities. Besides, these cities are most densely populated and thus, are the most influential in regard to contagion. The flight scaling rate, rω is set at 10 percent due to flight restrictions where the number of domestic passengers has decreased by 90 percent under simulation period [27]. Finally, we set the commuting return rate τ = 1/12.

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Method | 17

Table 3.1: Model parameters predominantly defined by COVID-19

Parameter Description Value Reference

β Transmission rate 0.28 Eq.3.1

Latent to infectious rate 1/4.5 [21]

µ Recovery rate 1/8 [22]

P_d Infectious fatality rate 0.003 [23]

P_a Probability of asymptomatic infection 0.3 [24]

R₀ Basic reproductive number 2.59 [25]

r_β Asymptomatic infectivity scaling factor 0.75 [26]

3.3 COVID-19 Fatality Data

The data we compare our results to is collected from the Public Health Agency of Sweden [15]. Deaths are reported as death-date, the only requirement for inclusion is that the deceased had COVID-19 confirmed by laboratory. Death data is well aligned with excess-mortality rates [28]. Delays in the data can occur up to a few days and also over weekends. There is also a negligible number of missing deaths where death date is unknown. Note that deaths in specific regions are reported as deaths per week instead of days.

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18 | Method

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Results and Discussion | 19

Chapter 4 Results and Discussion

In this chapter, we present the simulation results and discuss them.

4.1 Spread at a National Level

Figure 4.1: Simulation with α_min = 0.4. FOHM refers to the actual COVID- 19 data.

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20 | Results and Discussion

Figure 4.2: Comparing the maximum and minimum value in van Kampen’s [22] confidence interval for infectious period, µ⁻¹.

4.2 Spread in Regions

Figure 4.3: Regional results.

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Results and Discussion | 21

Figure 4.4: Comparison of Sweden’s northern region with a less extreme air- travel restrictions, r_ω = 0.5.

4.3 Discussion

Overall the model parametrized with basic disease data performs well. The main difference is the dramatic increase in deaths during the first wave. This may be explained by a high transient death rate where the most vulnerable are affected, e.g clusters in nursing or retirement homes. Super-spreading events may also be more common than the model suggests, as there is no specific control over the distribution of spread in the model. Furthermore, cases could be missed at the onset of the pandemic due to limited testing and when started, could result in a sudden jump. Figure 4.2suggests that the confidence interval’s lower-bound for infectious time is better with µ⁻¹ = 6 days. The difference is significant during the summer trough. This may be explained by studies that use virus shedding as a proxy for infectivity.

Although GLEaM’s metapopulation structure is not aligned with the administrative regions used in the collected data, GLEaM is more fine-grained with its 34 regions compared to the 21 official regions. This allows for comparing the three lands of sweden: Götaland, Svealand and Norrland with small errors in population size.

As seen in Figure 4.3the large-scale trends are captured in Sweden’s middle and southern regions while missing the first peak in the north, the mass is instead shifted onto the second wave. These are results of a 90 percent flight reduction as by [27]. Heterogeneity in flight reduction may explain this.

For example domestic flights to and from Northern Sweden may not have decreased as much as other destinations and using 50 percent instead results in high fidelity as seen in Figure4.4. Another thing to note is that the gravity

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22 | Results and Discussion

model for commuting was fitted with larger populations compared to Northern Sweden’s exceptionally low population density. There are also inconsistencies in literature regarding the phase shifts between urban and remote regions [29,30].

Spatial dimension can cause multiple peaks since it can move spatially. The COVID-19 pandemic shows the same pattern in every city indicating either seasonal effects or that smaller spatial scales must be considered.

To extend the simulation further in time, immunization effects would have to be considered as a significant fraction have been vaccinated. On the timescale of years, reinfection would also be highly relevant as it would determine if it would follow the recurring influenza pattern or be eradicated. Immunity duration and virus mutation are therefore crucial research questions.

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Conclusions and Future work | 23

Chapter 5 Conclusions and Future work

In this chapter we conclude and reflect on the thesis. Several ideas for future work is also presented.

5.1 Conclusion and Reflections

Epidemics, that quickly can become pandemics in this connected world are a profound threat to humanity and modelling them is essential for minimizing their impact. With increasing computational capacity, models utilizing real- world data show satisfactory spatiotemporal fidelity, albeit the multitude of simplifications imposed. They do this by capturing the large-scale multimodal property at a national and regional level. As much uncertainty exists at the onset of an epidemic, models should maximize their prior realism. The model’s spatial heterogeneity with mobility fluxes determined by real-world data is therefore important, as it can be determined beforehand. Although, considerable research have been done using both Apple’s and Google’s re- leased mobility data sets. Accurate mobility data from mobile phones have yet to be fully realized in epidemiology. However, privacy aspects has to be considered.

As acknowledged by Danon et al. [19] developing models and simulation software is time consuming. The GLEaM project amongst others are therefore valuable resources for future pandemics as they can be customized to respective disease. Their openness also allows any country to make data driven decisions. Models can also be used for other things such as vector borne

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24 | Conclusions and Future work

diseases, information spreading and even computer worms.

5.2 Future work

As this work concerns models, one can always add complexity. Age-structure is an interesting addition as COVID-19 shows great heterogeneity in regard to age, e.g probability of hospitalization. Another interesting aspect is to compare different transition distributions for the incubation and infectious period. A more thorough analysis in the spatial dimension by analyzing each city on is own could also bring new insights in the model’s fidelity. Finally, extending and evaluating the simulation world-wide may be insightful and as GLEaM has global data the only additional work would be the analysis part.

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REFERENCES | 25

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Appendix A: The next generation matrix | 31

Appendix A

The next generation matrix

In 1990 Diekmann et al. [31] defined R₀ as the dominant eigenvalue of the next generation linear operator. In 2002 Driessche et al. [32] outlines a method to calculate the linear operator for arbitrary compartmental models.

The operator is a matrix due to the compartmental model’s discreteness.

Following their method for the SEIIRD model constructed in Chapter 3 we let x be a vector of the disease-carriers. In our model these are defined as E, I and I_a. We decompose ^dx_dt = F − V where F is the number of new individuals transitioning into a disease carrier compartment per time. And V are the intra-disease-compartment transitions per time. For the SEIIRD model we have

F =





β^SI_N + rβ^SI_N^a 0 0



 (A.1)

and

V =





E

− (1 − p_a) + µI

−p_a+ µI_a



. (A.2)

The SEIIRD model has a disease-free-equilibrium (DFE) when the whole population is susceptible, i.e when S = N . Linearizing EquationA.1andA.2 around the DFE by forming Jacobian matrices F and V where (F )_ij = ^∂F_∂xⁱ

j

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32 | Appendix A: The next generation matrix

and (V )_ij = _∂x^∂Vⁱ

j we obtain

F =





0 β rβ

0 0 0



 (A.3)

and

V =





0 0

− (1 − p_a) µ 0

−pa 0 µ



. (A.4)

We can now calculate the next generation matrix as G = F V⁻¹ where each entry (G)_ij can be interpreted as the expected number of infections in compartment i due to an infectious individual in j. R₀is defined as the spectral radius of G. For our model, R0 = ρ (G) = ^β_µ(rpa+ 1 − pa).

Spatiotemporal Fidelity of a Metapopulational Model Evaluated on the COVID-19 Pandemic in Sweden