UNIVERSIDADE DE LISBOA
Faculdade de Medicina
Transmission Dynamics and the Impact of Non-Pharmaceutical Interventions in the COVID-19 Epidemic in Portugal:
A Modelling Study
Márcia Maria Galrão Luís
Supervisor: Prof. Dr. Ruy Miguel Sousa Soeiro de Figueiredo Ribeiro
Co-advisor: Prof. Dr. Paulo Jorge da Silva Nogueira
Dissertation to obtain the Master’s Degree in Epidemiology
2022
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UNIVERSIDADE DE LISBOA
Faculdade de Medicina
Transmission Dynamics and the Impact of Non-Pharmaceutical Interventions in the COVID-19 Epidemic in Portugal:
A Modelling Study
Márcia Maria Galrão Luís
Supervisor: Prof. Dr. Ruy Miguel Sousa Soeiro de Figueiredo Ribeiro
Co-advisor: Prof. Dr. Paulo Jorge da Silva Nogueira
Dissertation to obtain the Master’s Degree in Epidemiology
2022
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The printing of this dissertation was approved by the Scientific Council of the
Faculty of Medicine of Lisbon in a meeting of 28
thJune 2022.
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Acknowledgements
The accomplishment of this dissertation results not only from hard work and effort but also from the help, support, and guidance of many people over the past of this year.
I would first like to acknowledge that this work was produced with the support of Infraestrutura Nacional de Computação Distribuída (INCD) [National Infrastructure for Distributed Computation] funded by Fundação para a Ciência e Tecnologia (FCT) and FEDER under project 01/SAICT/2016 nº 022153.
I would like to extend my deepest gratitude to my supervisor, Professor Ruy Ribeiro, whose expertise and constant support and guidance since the beginning of this journey have unconditionally encouraged me in every step of this work. A sincere thank you for challenging me every week with further ideas and work, pushing me to sharpen my thinking and expanding my knowledge far beyond my expectations and beliefs.
A special thank you to my co-advisor, Professor Paulo Nogueira, for the precious insights and contribution during the development of this dissertation.
To my friends, for all the patience, understanding and missing events, for all your messages of kindness and encouragement, and for all much-needed distractions. To Adriana Ribeiro, for all support and friendship, even at a distance.
To Diogo Vargues, for all the love and kindness. For being my greatest supporter, encouraging and inspiring me every day to give the best version of myself even in the hardest times.
To my parents, Elsa and Jorge, for giving me the possibility of pursuing my academic research and supporting my ambitions. For teaching me to be resilient and persistent in every challenge and to strive for excellence in every step of my personal and academic life. My sincerest gratitude for all your patience throughout the development of this dissertation and in all other stages of my life.
Lastly, an unconditional thank you to those who directly or indirectly contributed to the completion of this dissertation.
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Resumo
Contexto: Em resposta à epidemia COVID-19, Portugal adoptou intervenções não farmacológicas (INFs), que incluíram o fecho das escolas, rastreamento de contactos e outras medidas de distanciamento social para conter a transmissão do SARS-CoV-2. Projetar a epidemia sob diferentes INFs e investigar a sua eficácia na reducão da transmissão têm auxiliado as decisões dos Governos. Esta dissertação propõe desenvolver um modelo matemático capaz de simular a epidemia da COVID-19 em Portugal entre Março 2020 e Fevereiro 2021 e avaliar o impacto de diferentes INFs no controlo da epidemia. Além de um cenário não mitigado, as INFs analisadas incluem diferentes intervenções de distanciamento físico e esforços adicionais no rastreamento/isolamento de contactos.
Métodos: Apresentamos um modelo de compartimentos estratificado por idades (uma extensão do modelo SEIR) para o SARS-CoV-2. Considerou-se ainda os diferentes estágios da infeção e da doença, distribuições realistas para os períodos de infeção, e as heterogeneidades da população portuguesa, incluindo a sua estrutura etária e padrões sociais de contacto. Para cada cenário estimou-se o número de infeções, mortes, hospitalizações e a proporção de infeções evitadas.
Resultados: Os nossos resultados sugerem que as medidas implementadas a 18 de Março 2020 possibilitaram o achatamento da curva epidémica e evitaram o colapso do Sistema Nacional de Saúde. As diferentes INFs investigadas permitiriam reduzir não só o número de infeções e mortes, mas também a pressão nos serviços de saúde. Também teria sido possível evitar o ressurgimento da terceira vaga da epidemia entre Dezembro 2020 e Janeiro 2021, como demonstrado pelo cenário de manter o ensino à distância ou manter maior restrições no Natal, embora estas intervenções possam ter outros custos sociais.
O impacto do rastreamento de contactos mostra depender do número de infeções ativas, cujos períodos de intensa transmissão exigem maiores recursos humanos.
Conclusão: O sucesso das INFs no controlo da transmissão do SARS-CoV-2 depende de uma resposta rápida e efetiva, essencial para determinar o crescimento da epidemia COVID-19.
Palavras-chave: Epidemia COVID-19, Modelação matemática, Intervenções não farmacológicas, Rastreamento de contactos, Portugal.
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Abstract
Background: In response to the COVID-19 epidemic, Portugal adopted non-pharmaceutical interventions (NPIs), including closing schools, contact tracing, and other social distancing measures, to contain the spread of SARS-CoV-2. Projecting the size of the epidemic under different NPIs and investigating their effectiveness to reduce spread have been essential to support governments’ response. This dissertation aims to develop a mathematical model to simulate the spread of SARS-CoV-2 in Portugal between March 2020 and February 2021 and explore the impact of different NPIs in controlling the epidemic. Besides an unmitigated scenario, the NPIs explored comprise different physical distancing interventions and additional efforts in tracing and isolating infected contacts.
Methods: We present an age-structured compartment model (an extension of the SEIR model) for SARS- CoV-2, accounting for the different stages of infection and disease, realistic distributions of infectious periods, and the Portuguese population’s heterogeneities, including age structure and social contact patterns. For each scenario, we estimated the burden of morbidity and mortality, the demand on healthcare facilities, and the fraction of infections averted.
Results: Our results suggest that the early control measures implemented on 18th March 2020 flattened the epidemic curve and avoided overwhelming the National Health System. The different NPIs could reduce the number of infections, deaths and pressure on the healthcare systems. Importantly, the third wave could have been avoided, as demonstrated by maintaining virtual learning for the 2020/2021 academic year or maintaining the 2020 Christmas under extreme restrictions, although these measures may have other social consequences. The impact of additional tracing efforts shows to be dependent on the number of active infectious cases, with periods of higher transmission implying more contact tracing effort and human resources.
Conclusion: The success of NPIs to curb SARS-CoV-2 transmission relies on a rapid and effective response which is crucial to determining the COVID-19 epidemic growth.
Keywords: COVID-19 epidemic, Mathematical modelling, Non-pharmaceutical interventions, Contact tracing, Portugal.
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Resumo Alargado
Contexto: No final de 2019, o vírus SARS-CoV-2 emergiu em Wuhan, na China, e propagou-se rapidamente pelo mundo, levando a Organização Mundial de Saúde a designar a COVID-19 como uma pandemia. Eventualmente esta transformou-se num dos maiores problemas de saúde de pública do século XXI. Até 1 de Janeiro de 2022, o vírus SARS-CoV-2 causou mais de 294 milhões de infeções e 5,5 milhões de mortes em todo o mundo [1]. Com o objectivo de reduzir a propagação do SARS-CoV-2, Portugal adoptou um conjunto intervenções não farmacológicas (INFs), incluindo o fecho das escolas, rastreamento de contactos e isolamento de casos infecciosos, entre outras medidas de distanciamento físico e social. O uso de modelos matemáticos têm sido uma ferramenta de apoio à decisão para Governos e Instituições de Saúde Pública com o propósito de avaliar as medidas de contenção e encontrar um equilíbrio entre maximizar a redução da transmissão e minimizar o impacto das restrições na sociedade e na economia.
Inúmeros estudos de modelação sobre o efeito das diferentes INFs têm sido publicados, no entanto apenas um número reduzido foca-se na epidemia em Portugal [2–4]. Entre esses, a maioria das análises usam modelos simplificados e são restritas até meados de 2020, não incluindo o maior surto de COVID-19 entre Dezembro 2020 e Janeiro 2021. A realização desta dissertação visa complementar os estudos existentes onde se pretende formular um modelo matemático mais realista capaz de simular a epidemia da COVID- 19 em Portugal entre Março 2020 e Fevereiro 2021, tendo em conta as características demográficas e sociais da população portuguesa, e ainda explorar a eficácia de diferentes INFs no controlo da transmissão da infeção.
Métodos: Nesta dissertação é desenvolvido um modelo epidemiológico de compartimentos (uma extensão do modelo SEIR – Susceptible, Exposed, Infectious, Recovered) onde são incluídas características biológicas e epidemiológicas da interação vírus-homem, distinguindo infeções assintomáticas e sintomáticas, diferentes estágios de severidade da doença, e a possibilidade do doente recuperar ou morrer pela doença COVID-19. O modelo também incorpora características demográficas e de contactos sociais da população portuguesa, incluindo a estrutura etária da população e matrizes de contactos que refletem os padrões sociais entre os diferentes grupos de idade e em diferentes locais (escola, trabalho, casa e outros).
Além disso, a diferentes grupos etários podem corresponder diferentes parâmetros subjacentes. Ao contrário da maioria dos modelos da epidemia existentes, cujos períodos de infeção são exponencialmente distribuídos, o nosso modelo utiliza uma duração mais realista desses períodos, baseada em dados epidemiólogicos empíricos, através do método dos estágios. O modelo é ajustado aos dados demográficos, sociais e epidemiológicos publicamente disponíveis para Portugal. Este modelo é posteriormente generalizado de modo a incluir novos compartimentos para avaliar o efeito do contact tracing (o rastreamento e isolamento de contactos infetados). Esta reformulação permite dar resposta ao último
x objectivo desta dissertação que consiste em estudar o impacto de diferentes intervenções na evolução da epidemia. Num primeiro cenário é explorado o efeito hipotético de uma epidemia sem a implementação de qualquer medida de controlo. Em seguida, são analisados dois principais grupos de INFs. O primeiro visa analisar o efeito de três medidas de distanciamento físico e social que poderiam ter sido implementados em Portugal, sendo estas (a) o efeito de prolongar o primeiro estado de emergência até ao final do Verão 2020, (b) manter o ensino à distância com as escolas encerradas para o ano letivo 2020/2021 e (c) manter o Natal 2020 sob apertadas restrições sem o alívio das mesmas. O segundo visa a rastrear e isolar mais 5% ou 10%
dos contactos infetados quando implementados em três momentos distintos da epidemia: (a) desde do levantamento das primeiras restrições a 4 de Maio 2020, (b) reabertura das escolas a 18 de Setembro 2020, e (c) durante a segunda vaga de COVID-19 a 17 de Novembro 2020. Nota-se que a implementação adicional de contact tracing nestes três momentos permite avaliar o seu efeito quando imposto em diferentes fases da atividade epidémica. Por fim, os resultados de cada cenário são comparados ao modelo base em termos da proporção de infeções evitadas, número de infeções e mortes (cumulativas e incidentes), e a carga hospitalar nos dias de pico, tendo em conta a capacidade do Serviço Nacional de Saúde (SNS) para doentes de COVID-19.
Resultados/Discussão: O modelo desenvolvido consegue descrever as diferentes fases da epidemia ocorridas em Portugal entre Março 2020 e Fevereiro 2021, incluindo períodos de crescimento exponencial (com índice de transmissibilidade Rt superior a 1) e períodos de decrescimento (com índice de transmissibilidade Rt inferior a 1). Os resultados sugerem ainda que as diferentes fases da epidemia seguem as tendências da mobilidade e dos padrões de contactos da população portuguesa. No entanto, ressalva-se que a mobilidade e os contactos são suscetíveis à adesão da população às medidas implementadas, o que pode comprometer a eficácia das intervenções no controlo da epidemia. Resultados da taxa de transmissão estratificada por grupos de idade sugerem ainda que as crianças e adolescentes (até aos 19 anos) possam ter desempenhado um papel importante na transmissão da infeção, nomeadamente desde do final do Verão 2020, coincidindo com a reabertura das escolas em Setembro 2020. Este resultado é justificável pela elevada taxa de contactos das crianças e adolescentes com outros indivíduos da mesma faixa etária, e quando infetados pelo SARS-CoV-2, estarem propensos a desenvolver uma infecção assintomática, sendo mais difícil de diagnosticar. Os nosso resultados são compatíveis com aproximadamente 45% das infeções não terem sido detetadas pelo sistema de vigilância Português, e este valor sobe para dois terços das infeções em crianças e adolescentes. Estes resultados estão de acordo com os estudos serológicos desenvolvidos em Portugal que apontam para uma subnotificação de casos após a primeira e segunda vaga da epidemia, mostrando ainda diferenças maiores entre os dados reportados e a seroprevalência em jovens [5]. Numa segunda parte, resultados do primeiro cenário hipotético sugerem que as medidas de controlo implementadas pelo Governo Português a 18 de Março de 2020 possibilitaram o achatamento da curva epidémica e evitaram o colapso do Sistema Nacional de Saúde na primeira vaga de COVID-19. Por outro
xi lado, o efeito das diferentes INFs investigadas nesta dissertação mostram que estas teriam um efeito notável na mitigação da transmissão, reduzindo o número de infeções e mortes, assim como a pressão nos hospitais e nas unidades de cuidados intensivos. No cenário mais optimista, também teria sido possível evitar o ressurgimento da terceira vaga da epidemia entre Dezembro 2020 e Janeiro 2021, como demonstrado pela projeção de manter o ensino à distância, ou manter restrições severas no Natal ou ainda, rastrear e isolar mais 10% dos contactos infetados numa fase inicial da epidemia. Ressalva-se que a implementação destas medidas poderia ter custos sociais e económicos, que não foram consideradas no model. Nota-se ainda que o sucesso do rastreamento e isolamento de contactos infetados depende amplamente no número de casos ativos, onde períodos de maior incidência implicam uma maior taxa de rastreamento, refletindo num maior esforço humano. É importante realçar que os cenários simulados, ainda que plausíveis, tiveram por base pressupostos optimistas. Nomeadamente, as projeções feitas têm por base o prolongamento e o reforço das INFs em Portugal, assumindo que a população mantém um nível de adesão às medidas implementadas semelhantes aos primeiros períodos da epidemia. Não obstante, é importante reconhecer o impacto socioeconómico e o peso da fatiga epidémica resultantes destas medidas (e projeções) que, em última instância, podem comprometer a eficácia dos cenários simulados no controlo da epidemia COVID-19 ou inviabilizar a projeção destes cenários.
Conclusão: As medidas de contenção implementadas em Portugal tiveram um papel crucial no combate à transmissão da doença e, em particular, no aliviar da pressão nos serviços de saúde. No entanto, devido ao carácter exponencial de uma epidemia, como a da COVID-19, o sucesso das INFs no controlo da transmissão pressupõem uma resposta rápida, proativa e dinâmica numa fase inicial de um novo surto, quando a incidência da infeção ainda é baixa. Sublinha-se ainda a premência destas respostas quando o vírus é dotado de uma elevada taxa de transmissibilidade e possível de ser transmitido por indivíduos não sintomáticos, que dificulta a sua deteção pelos serviços de vigilância epidemiológica, como é o caso do vírus SARS-CoV-2.
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Table of Contents
Acknowledgements ... v
Resumo ... vii
Abstract ... viii
Resumo Alargado ... ix
List of Figures ... xv
List of Tables ... xvii
List of Acronyms ... xviii
1. Introduction ... 1
1.1 The SARS-CoV-2 pandemic ... 1
1.2 Motivations and objectives ... 2
1.3 Dissertation outline ... 3
2 SARS-CoV-2 and COVID-19 Epidemiology ... 5
2.1 Coronaviruses and the emergence of SARS-CoV-2 ... 5
2.2 SARS-CoV-2 and COVID-19 disease overview ... 6
3 Mathematical Epidemiology ... 9
3.1 Deterministic compartmental models ... 9
3.2 The Threshold Theory and the Reproductive Numbers – R0 and Rt ... 12
3.3 Method of stages ... 14
3.4 Statistical tools ... 16
4 Modelling SARS-CoV-2 ... 19
4.1 Biological and epidemiological factors ... 19
4.2 Social and behavioural factors ... 20
4.3 The effect of realistic empirical distribution for the latent and infectious periods ... 21
5 Data and Methodology... 23
5.1 Datasets… ... 23
5.1.1 COVID-19 surveillance data ... 23
5.1.2 Demographic data ... 25
5.1.3 Contact matrices ... 26
5.2 Modelling contact patterns ... 27
5.3 Model development ... 28
xiv
5.3.1 Baseline Model: SARS-CoV-2 transmission model ... 28
5.3.2 Extension of the Baseline Model: Modelling Contact Tracing ... 31
5.4 Model fitting to the epidemic in Portugal ... 33
5.5 Scenarios ... 34
5.5.1 First Scenario: Unmitigated scenario ... 35
5.5.2 Second Scenario: Physical distancing interventions ... 35
5.5.3 Third Scenario: Contact tracing ... 37
5.6 Model implementation ... 38
6 Results ... 41
6.1 Model fitting ... 41
6.2 The spread of SARS-CoV-2 in Portugal ... 45
6.3 Impact of different intervention scenarios ... 47
6.3.1 First scenario: Unmitigated scenario ... 47
6.3.2 Second scenario: Physical distancing interventions ... 49
6.3.3 Third scenario: Contact tracing ... 52
7 Discussion... 59
7.1 Simulating the spread of SARS-CoV-2 in Portugal ... 59
7.2 Impact of different intervention scenarios ... 61
7.2.1 First scenario: Unmitigated scenario ... 61
7.2.2 Second scenario: Physical distancing interventions ... 62
7.2.3 Third scenario: Contact tracing ... 65
7.3 Limitations ... 67
8 Conclusion and Final Remarks ... 69
8.1 Conclusion and implications ... 69
8.2 Future work ... 71
References ... 73 Appendix ... I Appendix A – Details of SARS-CoV-2 transmission model ... III Appendix B – Model results by age group ... IX Appendix C – Summary of the scenarios simulated ... XV Appendix D – Sensitivity analyses ... XIX
xv
List of Figures
Figure 3.1. Flow diagrams for the SIR model (top) and SEIR model (bottom) with the transition rates presented above the arrows. ... 10 Figure 3.2. Comparison of the epidemic curves using SIR- and SEIR- type model.. ... 11 Figure 3.3. Probability density function of the infectious period using a non-realistic distribution (A) and empirically realistic distribution (B, C). ... 15 Figure 3.4. Diagram for the SEIR model accounting with the method of stages for the latent and infectious period with nE and nI stages, respectively. ... 15 Figure 4.1. Distribution of the incubation (left) and infectious period (right) for SARS-CoV-2. ... 22 Figure 5.1. Evolution of the epidemic in Portugal in terms of the number of new infections (a), new deaths (b), hospitalisations (c), and critical cases in ICU (d) between 3rd March 2020 and 10th February 2021. .. 24 Figure 5.2. Age distribution of the Portuguese population in 2020 among the eight age groups considered in this thesis... 26 Figure 5.3. Age-structured contact matrices at school (a), work (b), home (c), and others (d) stratified into 10-years age bands as used in this thesis. ... 27 Figure 5.4. SARS-CoV-2 transmission model and the transition rates between compartments. ... 29 Figure 5.5. Extended SARS-CoV-2 transmission model and the transition rates between compartments for traced and isolated contacts. ... 32 Figure 5.6. Daily infection incidence with the estimated breakpoints between 3rd March 2020 and 10th February 2021. ... 34 Figure 6.1. Model fitting to COVID-19 data from 3rd March 2020 to 10th February 2021 (a-f).. ... 41 Figure 6.2. Transmission rate 𝛿 per infective and per age group as a function of time from 3rd March 2020 to 10th February 2021. ... 43 Figure 6.3. The effective reproductive number between March 2020 and February 2021 using the baseline model. ... 44 Figure 6.4. Comparison of the total number of infections including only 25% (solid line) or 100% (dashed line) asymptomatic infections from 3rd March 2020 to 10th February 2021, depicted in a logarithmic scale (left) and linear scale (right)... 45
xvi Figure 6.5. Comparison of the total number of infections by age group including only 25% (solid line) or 100% (dashed line) of asymptomatic infections between 3rd March 2020 and 10th February 2021. ... 46 Figure 6.6. Effects of an unmitigated scenario on the evolution of the number of cases (a,b), healthcare demand (c,d), and deaths (e,f) compared to the baseline model from 3rd March 2020 to 10th February 2021.
... 48 Figure 6.7. Effects of different physical distancing interventions on the evolution of the number of cases (a,b), hospitalisations (c,d), and deaths (e,f) compared to the baseline model between 3rd March 2020 and 10th February 2021. ... 51 Figure 6.8. Fraction of cumulative infections averted % relative to the day upon each scenario is implemented. ... 52 Figure 6.9. Effective reproductive number Rt for the different physical distancing interventions (A, B, C) from March 2020 to February 2021. ... 52 Figure 6.10. Effect of implementing 5% and 10% additional contact tracing efforts from 4th May 2020 onwards on the evolution of the number of cases (a,b), hospitalisations (c,d), and deaths (e,f) compared to the baseline model. ... 55 Figure 6.11. Effect of implementing 5% and 10% additional contact tracing efforts from 18th September 2020 onwards on the evolution of the number of cases (a,b), hospitalisations (c,d), and deaths (e,f) compared to the baseline model.. ... 56 Figure 6.12. Effect of implementing 5% and 10% additional contact tracing efforts from 17th November 2020 onwards on the evolution of the number of cases (a,b), hospitalisations (c,d), and deaths (e,f) compared to the baseline model. ... 57 Figure 6.13. Fraction of infections averted under the different contact tracing scenarios relative to the day upon each measure is implemented. ... 58 Figure 6.14. Daily additional infected contacts traced under 5% and 10% additional contact tracing efforts from 3rd March 2020 to 10th February 2021. ... 58
xvii
List of Tables
Table 3.1. Notations of the SEIR model. ... 11 Table 6.1. Parameters result from model fitting. ... 42
xviii
List of Acronyms
ACE2 Angiotensin-Converting Enzyme 2
C The C Programming Language
CoV Coronavirus
COVID-19 Coronavirus Disease 2019
DGS Direção Geral de Saúde
DSSG Data Science for Social Good Portugal
GOF Goodness-of-Fit
HCoV Human Coronaviruses
ICTV International Committee on Taxonomy of Viruses
ICU Intensive Care Unit
IgG Immunoglobulin G
IgM Immunoglobulin M
iMM Instituto de Medicina Molecular
INE Instituto Nacional de Estatística
LVT Lisboa e Vale do Tejo
MERS-CoV Middle East Respiratory Syndrome Coronavirus
mRNA Messenger Ribonucleic Acid
NPI Non-Pharmaceutical Interventions
ODE Ordinary Differential Equation
PHEIC Public Health Emergency of International Concern
R R Foundation for Statistical Computing
RNA Ribonucleic Acid
rRT-PCR real-Time Reverse-Transcription Polymerase Chain Reaction SARS-CoV Severe Acute Respiratory Syndrome Coronavirus
SARS-CoV-2 Severe Acute Respiratory Syndrome Coronavirus 2
SSE Sum of Square Error
VOC Variant of Concern
WHO World Health Organization
1
1. Introduction
1.1 The SARS-CoV-2 pandemic
The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) pandemic has become the major global public health concern of the 21st century, challenging multiple nations since its emergence in Wuhan, China, in late 2019. By 1st January 2022, SARS-CoV-2 has already caused nearly 294 million cases and 5.5 million deceased individuals worldwide [1].
The vast and rapid spread of this virus all over the world is primarily due to its inherent nature as a new infectious and contagious pathogen with a high transmissibility rate from person to person, able to cause multiple secondary infections within a short period of time, but also due to the minimal pre-existing population immunity against the virus. In the absence of antiviral drugs and vaccines, especially at the beginning of the pandemic, several countries adopted non-pharmaceutical interventions (NPIs), such as school closures, social distancing, quarantine and isolation of high-risk contacts and infectious cases, to slow down the spread of the infection. In a similar manner, the Portuguese government quickly took multiple mitigation measures, including a nationwide lockdown, decreed on 18th March 2020 until 2nd May 2020 [6] to control the spread of the infection after the first positive SARS-CoV-2 cases were reported on 2nd March 2020.
According to Google and Apple mobility data, the Portuguese population seemed to adhere to the government’s mitigation measures by changing their mobility patterns in different settings, with the highest reduction in retail and leisure (-83%), in contrast to grocery and pharmacy (-59%) and work (-53%) [7].
According to Ricoca et al. [8], containment measures employed in Portugal in mid-March 2020 were effective at reducing morbidity and mortality due to COVID-19’s first wave, estimating 25% fewer deaths, 69% fewer admissions to intensive care units (ICUs), 28% fewer hospitalisations, and 23% fewer cases than would have been expected without any intervention. Nonetheless, as in other European countries, subsequent COVID-19 waves emerged in Portugal, varying in magnitude and time, following mainly two additional major outbreaks (i) between November 2020 and December 2020 and (ii) between mid- December 2020 and January 2021, leading to the implementation of more lockdowns and other extreme containment measures. Although effective, the extreme restrictions had enormous consequences in society and the economy, leading, for example, to an increase in lay-offs, the progression of other diseases not timely diagnosed and the postponement of surgeries, and a remarkable increase in mental health disorders, especially among young adults and women [9]. The lack of reliable data for SARS-CoV-2, along with other social, geopolitics, and economic factors, have challenged nations to design a prompt and timely response to combat the disease. As an example, besides the World Health Organization (WHO) guidelines, the gauge
2 of testing policies and case and deaths definitions were generally established according to each country’s limited testing capacity and medical resources [10]. Part of these challenges has been bypassed by the role of mathematical transmission models, which have supported policymakers in designing control programmes to suppress the spread of SARS-CoV-2. Since the beginning of the epidemic, those models were at the basis of multiple forecasts, predictions, and scenarios analyses, which have contributed to a better understanding of the dynamics of SARS-CoV-2 and have provided notable insights related to the evolution of the epidemic and the effectiveness of different NPIs to curb the transmission [11].
1.2 Motivations and objectives
Although two years of the pandemic have already passed and knowledge has continuously evolved as time goes on, several questions remain to date. Such knowledge has been partially obtained by modelling studies that were developed for several countries and many of them with different objectives, which have included, for example, the estimation of key epidemiological distributions and epidemiological measures. However, a primordial concern has been understanding the dynamics of the virus and assessing the effect of NPIs to control the disease spread more effectively, avoiding massive disruptions of the healthcare systems and diminishing the impact at an economic and social level [12-14]. Nevertheless, despite the large numbers of modelling studies that have been developed for SARS-CoV-2, only a few studies were found for the Portuguese case [2–4]. In particular, Teles [2]adoptedan extension of the SIR model to predict the evolution of COVID-19 in Portugal and assess the effectiveness of different mitigation measures in the number of cases. Pais & Taveira [3] employed a SIR model to predict the evolution of the epidemic under different levels of compliance of the Portuguese population to the control measures implemented by the Portuguese government. Both studies describe the first months of the epidemic in Portugal, excluding the major outbreak between mid-December 2020 and January 2021. In Viana et al. [4], the authors used an age- structured transmission model to evaluate relaxation intervention scenarios as mass vaccination rolls out in Portugal. Also, at the time of writing, we are aware of other work ongoing by the Instituto Nacional de Saúde Doutor Ricardo Jorge (the National Health Institute) developing a transmission model to assess the impact of the contact tracing policy.
The lack of a general overview of the dynamics of SARS-CoV-2 and the scarce scientific evidence regarding the effectiveness of NPIs in controlling the transmission in Portugal considering demographic and social features of the Portuguese population motivated us to expand the current knowledge in both those directions. For this reason, the objectives of this dissertation aim to develop a mathematical model able to simulate the spread of SARS-CoV-2 in Portugal during one year of the epidemic and explore the effect of different non-pharmaceutical interventions (NPIs) in the evolution of the epidemic.
3 With that purpose, we propose an age-structured transmission model (an extension of the SEIR model) to characterise the evolution of the epidemic in Portugal from March 2020 to February 2021, encompassing the three major COVID-19 outbreaks. The model accounts for biological and epidemiological features of COVID-19 disease, distinguishing subclinical and clinical infections and outlining the progression of more severe disease with the possibility of death induced by COVID-19 disease. We also introduce demographic and social features of the Portuguese population by incorporating the population's age structure and social contact patterns within and across the age groups and at different locations (school, work, home, and others).
We also allow different age groups to differ in their underlying parameters, specifying age-specific rates for subclinical infections, hospitalisations and ICU admission, and deaths by the disease. Also, in contrast to the previous studies [2,3] that adopted a SIR or SEIR compartmental model with an exponential latent and recovery period, our model accounts for realistic, empirically-based distribution for both of these periods. The model is then calibrated to demographic, social, and epidemiological data available for Portugal, including the number of cases and deaths by age group and the number of patients in hospitals and ICUs.
With this baseline framework, we then propose an extension of our model to include new compartments for tracing and isolating infected contacts. This approach enables us to achieve the last objective of this dissertation: analysing the effect of different NPIs in terms of the number of cases, deaths, and the demand on healthcare services. Besides the projection of a hypothetical unmitigated scenario in which no measures are taken, we explore mainly two different NPIs: (i) physical/social distancing interventions and (ii) additional efforts to trace and isolate infected contacts (usually referred to as contact tracing strategy).
Among physical distancing interventions, we explore the effect of (a) maintaining the 2020 summer break under extreme restrictions, (b) maintaining virtual learning for the 2020/2021 academic year, and (c) maintaining the 2020 Christmas under extreme restrictions. Among the contact tracing scenarios, we explore the effect of implementing 5% and 10% additional tracing at three different moments of the epidemic, since (a) the lifting of the first phase of interventions on 4th May 2020, (b) schools’ reopening on 18th September 2020, and (c) during the second wave of the epidemic on 17th November 2020.
1.3 Dissertation outline
This dissertation is divided into eight main chapters, with the introduction being the first one.
Chapter 2 introduces the epidemiology of SARS-CoV-2 as it pertains to our work here. It starts by presenting the main key events related to the emergence of SARS-CoV-2 and the COVID-19 outbreak. We also present an up-to-date description of the current knowledge of SARS-CoV-2 infection and COVID-19
4 disease, including the main clinical and epidemiological features, as well as the progress of potential treatments and vaccines against SARS-CoV-2 infection.
Chapter 3 provides a basic introduction to Mathematical Modelling of Infectious Diseases needed to understand the model developed in this dissertation. We begin by describing the simplest SIR- and SEIR- type compartmental models. Then, we outline basic concepts and terminology frequently used in mathematical modelling, such as the threshold theory and the basic and effective reproductive numbers (R0
and Rt). We also present mathematical and statistical tools which will be used in this dissertation, including the method of stages (usually used to provide a more realistic infectious period distribution) and the residual sum of squares method (commonly employed to calibrate mathematical models to epidemiological data).
Chapter 4 highlights the importance of introducing biological, epidemiological, social, and behavioural factors into epidemiological models. Also, we report which features have been most used for modelling SARS-CoV-2 by providing examples from the literature and which ones will be considered in our model.
Chapter 5 presents the datasets and the methodologies required to develop the baseline model and its extension accounting for contact tracing policy. Subsequently, we introduce the model developed, the assumptions assumed in its formulation, and the process used to calibrate the model to data. We also describe the scenarios explored in this dissertation and how to implement them in order to achieve our second objective. Lastly, we outline some remarks and computational techniques used to implement the model.
Chapter 6 and Chapter 7 present the numerical results and discuss the findings obtained in this work, respectively. We start by presenting the results of the calibration process, showing the fitting to the Portuguese COVID-19 data. Subsequently, we present the results for the different scenarios explored in this dissertation which are then compared to the baseline model regarding the number of cases, hospitalisations, critical cases in ICU, deaths, and the number of infections averted for each situation.
Finally, Chapter 8 presents final remarks and exposes future perspectives and work.
5
2 SARS-CoV-2 and COVID-19 Epidemiology
2.1 Coronaviruses and the emergence of SARS-CoV-2
Coronaviruses (CoVs) are a family of RNA viruses recognised by their crown-like spikes on their viral surfaces. CoVs are divided into four genera: Alphacoronavirus, Betacoronavirus, Gammacoronavirus, and Deltacoronavirus, of which only the first two groups can infect people [14,15]. Human coronaviruses (HCoVs) were first described in the mid-1960s and were related to affecting the respiratory and gastrointestinal tract [14,15]. Most HCoVs became endemic diseases, re-emerging especially during the winter in temperate-climate regions [15]. Additionally, evidence suggests that HCoVs are transmitted via direct physical contact, fomites, as well as through respiratory droplets and aerosols of infectious individuals [16]. In 2002 and 2012, two novels of betacoronaviruses with a zoonotic origin1 emerged in China and Saudi Arabia, respectively – the severe acute respiratory syndrome coronavirus (SARS-CoV) and the middle east respiratory syndrome coronavirus (MERS-CoV) [15]. Both viruses rapidly spread across the globe and were responsible for severe respiratory disease, causing atypical pneumonia outbreaks.
In late December 2019, clusters of patients with pneumonia of unknown cause were reported by the local health authorities in Wuhan, Hubei Province, China [17]. Based on the first surveillance data in Wuhan, most of the cases were epidemiologically linked to the Huanan Seafood Wholesale Market – where seafood and other live animals like poultry and wildlife were sold – and subsequent confirmed cases showed the possibility of human-to-human transmission [18]. On 9th January 2020, results from RNA sequencing and virus isolation from patient samples with severe pneumonia demonstrated a novel betacoronavirus as the causative agent for the ongoing epidemic [19]. On 11th February 2020, The International Committee on Taxonomy of Viruses (ICTV) classified the novel virus as the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) [20], and subsequently, the WHO named the disease coronavirus disease 2019 (COVID-19) [21]. Due to the increasing number of newly reported COVID-19 cases in other provinces of China, on 23rd January 2020, Wuhan city imposed a lockdown, and mainland China implemented public health measures to control the spread of the disease [22]. With several outbreaks starting to emerge in Europe and other regions like South Korea and Japan, on 30th January 2020, the WHO declared COVID-19 a Public Health Emergency of International Concern (PHEIC) [23]. Later, with the rapid distribution of the virus geographically and subsequent outbreaks occurring worldwide, the WHO declared the SARS-CoV-2 outbreak a pandemic on 11th March 2020 [23].
1 Zoonoses are defined as infections/diseases that might be transmitted between vertebrate animals and man. Previous studies have suggested that the origin of SARS-CoV and MERS-CoV emerged from bats and camels, respectively [15].
6
2.2 SARS-CoV-2 and COVID-19 disease overview
Early phylogenetic analyses demonstrated that SARS-CoV-2 might be a zoonotic virus closely related to (SARS)-like coronaviruses with bat origin [19,24]. However, its origin in humans – when and where it emerged – remains uncertain. In the study by Lu et al. [19], it is suggested that bats may act as a reservoir of SARS-CoV-2, and another animal may act as an intermediate host between bats and humans; however, these assumptions have not been yet clearly proven. Lab-leak theories have also been suggested as a possible hypothesis for the disease’s origin [25].
Similar to SARS-CoV, the spike (S) protein present on the surface of the SARS-CoV-2 recognises and binds to a specific human cell-surface protein called the angiotensin-converting enzyme 2 (ACE2), which is mainly expressed in the respiratory tract (e.g., lungs and bronchial epithelial cells). When the S protein attaches to the ACE2 human receptor, the virus enters into the cell, and subsequently, it can replicate and initiate the infection within the human host [19,26].
Like other viral infections, SARS-CoV-2 infection may be sub-divided into three distinct stages in the process of infection: latent, infectious, and recovery or death stages. Firstly, there is a latent period during which the host is infected but not yet infectious, which for SARS-CoV-2 may last on average 3-4 days [27].
After that period, the patient becomes infectious. At the infectious stage, individuals shed viral particles to the environment and transmit the infection to others. Multiple evidence suggests that the infectious period may last on average 4-5 days [27]. The time between infection and clinical symptom onset is called the incubation period, which lasts a median of 5-6 days [27]. Following the infectious period comes the recovery period when the viral load declines and antibodies against SARS-CoV-2 start rising. Alternatively, it may follow a death induced by COVID-19 disease. Estimating these time intervals for SARS-CoV-2 has been addressed by several methods, such as data from infector-infectee pairs, fitting epidemiological compartmental models, and observational data from exposure time for reported cases.
According to He et al. [28], there is also a presymptomatic period when individuals are infectious but have not developed clinical symptoms yet, starting at least two days before symptoms onset. These findings suggest that transmission may occur before symptomatic illness, and the beginning of the state of infectiousness and the onset of disease symptoms might not be synchronous. Infectiousness seems to peak two days before to one day after symptom onset. According to that study, presymptomatic cases are also important drivers of infection, and they are responsible for 44% of secondary infections.
Besides presymptomatic carriers, case reports have also confirmed infectious individuals who never developed symptoms throughout the course of infection, the so-called asymptomatic cases [29]. The prevalence of asymptomatic infections and their attack rate have been hard to quantify due to a wide range
7 of biases usually present in such studies. Examples of such biases have included, for example, selection bias (e.g., non-representative sample or small sample sizes) and measurement bias (e.g., short follow-up duration, incomplete or unclear case definition) [30]. Nevertheless, according to a systematic review [31]
based on 13 studies with low risk-of-bias, the fraction of asymptomatic infections varied from 4% to 41%, and the result of the combined estimate was 17% (95% CI 14 - 20%). However, one should note that the prevalence of asymptomatic infections may vary substantially across countries, depending, for example, on their testing policies and population structure (e.g., younger populations may have a higher rate of asymptomatic cases than countries with older populations). Also, based on five of those 13 studies, the relative risk of asymptomatic transmission was 42% lower than symptomatic transmission.
SARS-CoV-2 infection presents a broad spectrum of manifestations, encompassing no clinical symptoms, mild respiratory symptoms, and severe illness, with patients requiring medical intervention, such as hospitalisation, intensive care, or mechanical ventilation. Although the most common symptoms reported are fever, dry cough, fatigue, sore throat, the most specific ones are the loss of the senses of smell and/or taste [22,32]. In some instances, patients may progress to severe viral pneumonia with respiratory failure and develop other complications, including acute respiratory distress syndrome (ARDS), septic shock, or multiple organ failures like heart and kidney. Such complications have also been associated with the cause of death by COVID-19 [33]. The duration of COVID-19 disease depends on the severity of the disease, lasting a median of nearly two weeks for mild cases and between three-six weeks for severe cases [22].
Evidence suggests that the severity of the illness differs with age. In particular, children and young adults are prone to become asymptomatic or develop mild symptoms, whereas older patients with underlying conditions (e.g., hypertension, cardiovascular disease, and immune suppression) have a higher risk of developing more severe illness and death [22].
As the epidemic evolved and data emerged, results from immunoglobulin M (IgM) and immunoglobulin G (IgG) antibody test for COVID-19 suggest that patients start developing IgM and IgG antibodies against SARS-CoV-2 as early as the fourth day after symptom onset; however, higher levels seem to occur solely from the second week onwards [34]. Serological studies have been a valuable tool to quantify the prevalence of antibodies within the community and to understand the longevity of natural and induced immunity. A population-based longitudinal study estimated that the seroprevalence of SARS-CoV-2 antibodies in Wuhan after the first-wave epidemic was 6.92% (95% CI, 6.41-7.43) and 82.1% of those were from asymptomatic individuals [35]. According to The COVID-19 Serological Panel developed by the Institute of Molecular Medicine (iMM) between 1st –17th March 2021, the seropositivity resulting from natural infection in Portugal after the second and third wave epidemic was 13% (when accounted with induced immunity from vaccination, the overall estimation was 17%) [36]. The study also reported that the level of antibodies from natural infection remains stable after six months of infection [36]. Similarly, evidence from
8 a population-level observational study conducted in Denmark showed that after six months, natural infection with SARS-CoV-2 was able to protect against reinfection of approximately 80% in people younger than 65 and nearly 47% in patients over 65 years old [37].
To date, several treatments against SARS-CoV-2 have been proposed. Although clinical cases have shown remarkable benefits using some treatments, mainly in patients with severe COVID-19 disease, their effectiveness remains controversial because of the lack of clinical safety and efficacy data. Examples of such treatments include antiviral drugs, immunosuppressant drugs, and immunoglobulin therapy [38,39].
Antiviral drugs (e.g., remdesivir, ribavirin, and lopinavir) have been shown to inhibit virus replication and to shorten the period of hospitalisation [38,39]. Immunosuppressant drugs (e.g., dexamethasone and tocilizumab) have been shown to reduce the inflammatory response among severely ill patients and, in some cases, reduce mortality among hospitalised patients who require respiratory support [38,39].
Immunoglobulin therapy (e.g., convalescent plasma and monoclonal antibodies) has shown an effect in neutralising the virus and diminishing the risk of hospitalisation among vulnerable individuals of developing severe disease [38,39].
Besides those therapies, in late 2020, several National Health Authorities authorised the first COVID-19 vaccines developed by Pfizer-BioNTech, Moderna-NIAID, Oxford & AstraZeneca, and Sinopharm, with many others in development at the time [40]. Different vaccines have used different strategies to prime the immune system, encompassing novel technologies with mRNA in lipid nanoparticles and traditional methods, such as live attenuated viruses and inactivated viruses [40]. These vaccines showed high effectiveness in preventing disease and reducing clinical symptoms, and real-world evidence from Israel [41] and England [42] also suggest that vaccines are effective at preventing infection. Additionally, some vaccines have also shown remarkable protection against new variants of SARS-CoV-2 with higher transmissibility and infectious rate, including some Variants of Concern (VOCs), such as the Delta (B.1.617.2) variant [43]. Such variants have added further challenges to fight the pandemic, requiring more efforts in promptly identifying new mutations of SARS-CoV-2. Despite all knowledge and progress regarding new therapies and vaccines to combat the virus, several unknown questions remain concerning the longevity of our immune response against SARS-CoV-2 and how the virus will evolve and escape antibodies.
9
3 Mathematical Epidemiology
Mathematical epidemiology of infectious diseases aims to use mathematical tools to understand epidemiological processes. It consists of developing mathematical models to understand the dynamics of an infectious pathogen in the real world at an individual level or across the community. Also, it has been a remarkable tool to predict the course of an epidemic under different public health policies. Until the 18th century, many mathematicians had already developed mathematical methods to study the evolution of infectious diseases. However, it was only in 1911 that Ronald Ross formulated the first framework of infectious disease modelling to describe malaria transmission dynamics between mosquitoes and humans [44]. Following Ross’ work, in 1927, Kermack and McKendrick developed the first deterministic compartmental model, the so-called SIR model and postulated the “threshold theory” in which the initial fraction of susceptibles in the community must exceed a certain critical threshold for an epidemic outbreak to occur [45, p.7]. Although stochasticity also has an important role in the spread of an infection, especially at the beginning of an outbreak and in modelling small populations, we shall focus on deterministic compartmental models in this dissertation.
3.1 Deterministic compartmental models
Compartmental models are epidemiological models that divide the population into several compartments according to their infection status, i.e., their ability to spread the infection. Despite the vast amount of different host-pathogen interactions, most compartmental models are variations of two basic types: SIR and SEIR models. The SIR-type model divides the population into susceptibles S (if the host has not previously been exposed to the pathogen and therefore has nonspecific immunity); infectious I (if the host can transmit the infection), and removed/recovered R (if the individual is no longer infectious and remains immune to further re-infections). In the case that the pathogen has a latent period before infectiousness onset, then an additional compartment E is introduced in the SIR model, and the population is modelled as an SEIR-type model (Susceptible-Exposed-Infectious-Recovered). Figure 3.1 presents flow diagrams for both SIR and SEIR models.
10
Figure 3.1. Flow diagrams for the SIR model (top) and SEIR model (bottom) with the transition rates presented above the arrows.
Considering the SEIR model diagram in Figure 3.1, 𝜀 is the rate at which an exposed individual becomes infectious (i.e., the latent rate), and 𝛾 is the rate at which infectious individuals recover from the infection (i.e., the recovery rate). Susceptible individuals become infected and move to compartment E at a time- dependent rate, the so-called force of infection usually denoted by 𝜆(𝑡). The force of infection measures the per capita rate that a susceptible individual will acquire the infection [45, p.63] and is defined as:
𝜆(𝑡) = 𝛽𝐼(𝑡) 𝑁
(3.1)
where N is the total population size, such that N = S + E + I + R, and 𝛽 denotes the transmission rate (the rate at which infection occurs due to the contact between a susceptible and an infectious individual [45, p.14]).
By contrast to 𝜀 and 𝛾, which are possible to measure from epidemiological data, 𝛽 is highly affected by biological, environmental, and social factors, being unsuitable for direct measurement [45, p.63]. The average durations of stay in the latent and infectious states are given by 1/𝜀 and 1/γ, respectively. 𝜆 and 𝛾 parameters have the same meaning for the SIR model. In fact, the addition of the latent period in the SEIR model will mainly cause a delay at the start of the individual's infectivity, resulting in the slower initial growth of an outbreak compared to the SIR model, as illustrated in Figure 3.2. Note that in the SEIR model, infected individuals comprise all individuals in both latent and infectious states, whereas infectious individuals comprise only individuals in the infectious period. Table 3.1 summarises the notation used in the SEIR model.
11
Table 3.1. Notations of the SEIR model.
Symbol Description
S Susceptible individuals
E Exposed individuals
I Infectious individuals
R Recovered individuals
𝛽 Transmission rate
𝜀 Latent rate
𝛾 Recovery rate
1/𝜀 Average latent period 1/γ Average infectious period
N Total population size
Figure 3.2. Comparison of the epidemic curves using SIR- and SEIR- type model. We assumed that the population size is 1,000,000. Recovery rates are equal for both models, that is 1
𝛾𝑆𝐼𝑅= 1
𝜀𝑆𝐸𝐼𝑅+ 1
𝛾𝑆𝐸𝐼𝑅. In the SIR model, we fixed 𝛾SIR = 1/4. In the SEIR model, we fixed ε = 1/2, 𝛾SEIR =1/2. Adapted from [46, Fig.3.14].
The number of individuals who move from one compartment to the next is determined mathematically by the derivative with respect to time of the number of individuals who leave a compartment [44]. Thus, in the SEIR model, (i) 𝜆(𝑡)𝑆 represents the number of susceptibles who become infected per unit of time (also referred to as the infection incidence), (ii) 𝜀𝐸 is the number of exposed individuals who become infectious per unit of time, and (iii) γI is the number of infected individuals who recover per unit of time. The number of derivatives is equivalent to the number of compartments, and the overall model is formulated as a system of differential equations2. Of note, modelling epidemics using differential equations implies that we implicitly assume that the process is deterministic; that is, models are only affected by the assumptions
2 In general, when modelling constant population sizes, the derivative for the removed/recovered state (𝑑𝑅/𝑑𝑡) might be ignored in the SEIR-type model if S, E and I are known. Similar considerations might be used for the SIR-type model [44].
12 considered, ignoring stochastic effects [44]. Hence, the SEIR model is mathematically translated by the following system of ordinary differential equations (ODEs):
𝑑𝑆/𝑑𝑡 = −𝜆𝑆
(3.2) 𝑑𝐸/𝑑𝑡 = 𝜆𝑆 − 𝜀𝐸
𝑑𝐼/𝑑𝑡 = 𝜀𝐸 − 𝛾𝐼 𝑑𝑅/𝑑𝑡 = 𝛾𝐼 with non-negative initial conditions:
𝑆(0) ≥ 0 , 𝐸(0) ≥ 0, 𝐼(0) ≥ 0, 𝑅(0) ≥ 0 (3.3)
The ODE system is then solved using numerical methods in programming languages like R or Matlab. In general, those methods require as inputs: (i) the values of the initial conditions, (ii) the timestep at which the output is required, (iii) the model function with the ODE system, and (iv) the parameters' values. The output (the solution of the ODEs system) is the number of individuals in each compartment for each timestep.
With the simplest SIR and SEIR models (as depicted in Figure 3.1), death induced by disease and demographic effects such as births, deaths, and migration are not considered. Consequently, the number of individuals in the population remains constant over time. This assumption is suitable for short-term outbreaks in which the timescale is faster than the replenishment of susceptibles in the population [45, pp.81-82]. Nonetheless, the biology of most pathogens and the epidemiological features of host-pathogen interaction usually require more complex models with more compartments and parameters. The SIR and SEIR models can easily be extended to include more realistic characteristics of the disease, heterogeneities of the host and the pathogen, as well as control measures to analyse the dynamics of the pathogen under different mitigation scenarios.
3.2 The Threshold Theory and the Reproductive Numbers – R
0and R
t According to Kermack and McKendrick’s threshold theory, an epidemic outbreak occurs if the initial fraction of susceptible individuals is above a certain critical value. With the SIR model (and also with the SEIR model when demographic changes are ignored [47]), it is easily demonstrated that such critical value is expressed by γ/β, usually referred to as the relative removal. Thus, when the initial fraction of susceptibles exceeds this value, it follows that 𝑑𝐼𝑑𝑡> 0, and consequently, the number of infections will increase.
Otherwise, the epidemic cannot invade. In some situations (especially at the beginning of an epidemic)
13 when the whole population might be susceptible to the infection, the relative removal can alternatively be represented by its inverse value, which results in the so-called basic reproduction number R0 [45, pp.19- 20], [46, pp. 19-21], expressed as:
R0 = 𝛽
𝛾 (3.4)
where β is the transmission rate, and 1/γ is the average infectious period, as seen in the previous subsection.
In those situations, it is also easily shown that the threshold theory can be re-defined using R0,and the epidemic outbreak only occurs if R0 > 1. In contrast to R0,which refers to the pathogen's ability to establish or die out in the population at the beginning of an epidemic, we use the effective reproductive number Rt 3
to analyse the evolution of the epidemic at any time. As the epidemic evolves, Rt tends to be lower than R0
due to the effect of control interventions and the decreased number of susceptible individuals, who acquire immunity, which reduces the probability of transmission between susceptible and infectious individuals. Rt
can then be defined as the average number of secondary infections generated by an infectious individual over its entire infectious period in a partially susceptible population [44].
Several methods have been proposed to calculate Rt over the course of an epidemic, including, for example, the methods of Cori et al. [48], Bettencourt & Ribeiro [49], and Walling & Teunis [50]. Specifically, the Cori et al. method [48] uses past daily infection cases to infer the number of secondary infections that an infectious individual would infect if the conditions are maintained. The Bettencourt & Ribeiro method [49]
uses an epidemiological model to infer the number of cases and employs a Bayesian approach to estimate the probability distribution of the effective reproduction number Rt. The Walling and Teunis method [50]
uses past and future daily case infections based on a specific cohort of individualsto determine the number of secondary infections that an infectious individual infects.
R0 and Rt represent some of the most valuable key quantities in epidemiology to characterise the potential of a pathogen to spread in the community and predict the course of the epidemic. Additionally, using a proper transmission model, we can estimate R0 and Rt values under different intervention scenarios to compare interventions' effectiveness and provide insights about the magnitude of the effort needed to diminish those values below one. For example, the magnitude of R0 provides information on the minimum vaccination coverage to reach the so-called herd immunity threshold c (i.e., the degree of population immunity [45, p.31]) at the beginning of the epidemic. Considering a population homogeneously mixed, then we can determine c as:
c ≥ 1 - 1
𝑅0 (3.5)
3Sometimes it can also be denoted as Re.
14 In this case, if R0 = 4, then at least 75% of the population should be vaccinated to achieve the herd immunity threshold. It is worth noting that the value of R0 is affected by notification biases and the implementation of early measures at the beginning of the epidemic, which hamper a correct estimation of its value. Also, the estimation of R0 and Rt is sensitive to demographics, environmental, and population characteristics (e.g., level of immunity in the community, mobility, contacts patterns, etc.) [46, p.20], which consequently may lead to variations of epidemic phases across different regions of a country.
3.3 Method of stages
With the simplest SIR and SEIR models, as described previously in this section, the latent and infectious periods are exponentially distributed. Biologically, this means that for most people the latent period (from infection to the onset of infectiousness) is very short and they quickly transition from compartment E to compartment I. This phenomenon is easily observed by the shape of the exponential distribution depicted in A, Figure 3.3. However, this assumption is biologically unrealistic. In a more realistic scenario, most people would transition through the exposed period and become infectious with a characteristic time, with few people becoming infectious quickly or very late, that is, assuming a bell-shaped distribution similar to the ones represented in B and C, Figure 3.3. Statistically, these central distributions might be captured by a Gamma, Erlang, or Weibull distribution. Although we discuss the biological effect of modelling the latent period using a non-realistic versus realistic distribution, analogous considerations may be applied to the other state transition, such as the infectious period of SARS-CoV-2 [51,52], as discussed in the following chapters.
Several mathematical methods enable capturing a central tendency distribution for the latent and infectious periods. Examples of such methods include the method of stages, integro-differential equations, and partial differential equations. In this dissertation, we shall focus on the method of stages adapted from the work of Lloyd [53].
Considering the SEIR model and including a realistic empirical distribution in the latent and infectious periods, the method of stages consists of replacing the single latent and infectious compartment by a series of nE and nI successive sub-compartments (or stages), respectively. Figure 3.4 shows the modified SEIR flux diagram with the method of stages for both of these periods.
15
Figure 3.3. Probability density function of the infectious period using a non-realistic distribution (A) and empirically realistic distribution (B, C). The figure represents the distribution of how long a person remains infectious, and shows that for an exponential distribution (A) most people would leave the infectious compartment almost instantaneously (i.e., at t = 0). For the Gamma distributions (B and C) most people stay in the infectious compartment over a few days. Here the Gamma distributions have shape parameter n, corresponding to the number of stages assumed in the method of stages, and the rate was fixed at 1/5 in all situations A, B, C (that is, the average time spent in the infectious period by one person is 5 days).
Figure 3.4. Diagram for the SEIR model accounting with the method of stages for the latent and infectious periods with nE and nI stages, respectively. Here, ε represents the exit from the latent rate, γ is the recovery rate, and 𝜆 is the force of infection expressed by 𝜆 = 𝛽𝐼(𝑡)
𝑁, where I is the sum of all infectious individuals in all nI
stages, i.e., 𝐼 = ∑𝑛𝐼 𝑖=1𝐼𝑖.
Equations (3.6) present the modified ODEs system for the flux diagram shown in Figure 3.4, accounting for nE stages inthe latent period and nI stages in the infectious period.
𝑑𝑆/𝑑𝑡 = – 𝜆𝑆
(3.6) dE1/𝑑𝑡 = 𝜆𝑆 – nE𝜀E1
dEk/𝑑𝑡 = nE𝜀EK-1 – nE𝜀𝐸k 2 ≤ k ≤ nE
dI1/𝑑𝑡 = nE𝜀EnE– nI𝛾I1
dIk/𝑑𝑡 = nI 𝛾Ik-1 – nI𝛾Ik, 2 ≤ k ≤ nI
dR/𝑑𝑡 = nI 𝛾InI
