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Stochastic epidemic models with contact tracing
Stockholm University, Faculty of Science, Department of Mathematics.ORCID iD: 0000-0001-6365-5491
2024 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The COVID-19 pandemic has significantly heightened research interest in infectious disease modelling. Specifically, the pandemic has underscored the critical role of epidemic models in understanding and predicting epidemic dynamics as well as evaluating the impact of public health interventions. This thesis delves into stochastic epidemic modelling, concentrating especially on the effectiveness of contact tracing to prevent large outbreaks.

In Paper I, we consider a Markovian SIR epidemic model in a homogeneous community with a rate of diagnosis (testing). This model is then incorporated with traditional manual contact tracing: once an infectious individual tests positive, it is immediately isolated. Each of its contacts is traced and tested independently with some fixed probability. Using large population approximations, we analyzed the early stage of the epidemic when the process of “to-be-traced components” behaves like a branching process. Based on these approximations, the analytic expressions for the reproduction numbers (for the components and individuals), the probability of a major outbreak, and the final fraction getting infected are derived and numerically evaluated. For the main stage of the epidemic, the process of to-be-traced components converges to a deterministic process (defined by a system of differential equations). Our numerical results suggest that the manual tracing probability is more effective in reducing the reproduction number than the testing fraction.

Paper II concerns a similar epidemic model as in Paper I but focuses on digital contact tracing (based on a contact tracing app) and a hybrid approach combining both manual and digital tracing. We assume manual as well as digital contact tracing occurs instantaneously and recursively for mathematical tractability. This model is then analyzed using a two-type branching process relying on a large community, where one type of “individuals” are “app-using components” and the other is non-app-users. Further, we investigate the combined preventive effect of two tracing methods. This combined model is analyzed by a different two-type branching process, with both types being the “to-be-traced components” but starting with different “roots”. The corresponding reproduction numbers R are derived. It is proven that the critical fraction of app-users for which digital contact tracing to R=1 is larger than the critical manual reporting probability to reach R=1.

Paper III presents an SEIR epidemic model allowing for network and random contacts, incorporating manual and digital (app-based) contact tracing. Manual tracing is only allowed to happen on the network and has random tracing delays. In contrast, instantaneous digital tracing occurs among network and random contacts (both need to be app-users). Both manual and digital tracing are assumed to be forward only and non-iterative. We show that the initial phase of the epidemic with both manual and/or digital tracing can be approximated by different multi-type branching processes, leading to the derivation of three respective reproduction numbers. This paper sets a lower bound on the effectiveness of contact tracing through its conservative assumptions, while Paper II offers an upper bound by presenting an optimistic scenario. The actual effectiveness of contact tracing in real-world situations is expected to fall somewhere in between.

Paper IV explores an epidemic model with sideward contact tracing. Individuals are involved in mixing events that occur at some rate. Infection is driven by interaction with at least one infective at the mixing event rather than pairwise interaction outlined in Papers I, II and III. Once an infective is diagnosed, each individual who got infected at the same event as the diagnosed individual is traced with a certain probability. Assuming few initial infectives within a large population, the initial stage of the epidemic is approximated by a limiting process where individuals can be traced by their siblings (individuals who become infected at the same event), thus creating dependencies. Treating sibling groups as macro-individuals allows for a macro branching process interpretation, from which we derive the reproduction number. Finally, we present some numerical results showing how the reproduction number varies with the size of the mixing event, the fraction of diagnosis and the tracing probability.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2024. , p. 43
Keywords [en]
epidemic model, contact tracing, branching process, reproduction number
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
ISBN: 978-91-8014-789-7 (print)ISBN: 978-91-8014-790-3 (electronic)OAI: oai:DiVA.org:su-228477DiVA, id: diva2:1853031
Public defence
2024-06-12, ALB Auditorium 4, 2nd floor, house 2, Campus Albano, Albanovägen 18, Stockholm, 13:00 (English)
Funder
Swedish Research Council, 2020-04744Available from: 2024-05-20 Created: 2024-04-20 Last updated: 2024-05-08Bibliographically approved
List of papers
1. Analysing the Effect of Test-and-Trace Strategy in an SIR Epidemic Model
Open this publication in new window or tab >>Analysing the Effect of Test-and-Trace Strategy in an SIR Epidemic Model
2022 (English)In: Bulletin of Mathematical Biology, ISSN 0092-8240, E-ISSN 1522-9602, Vol. 84, no 10, article id 105Article in journal (Refereed) Published
Abstract [en]

Consider a Markovian SIR epidemic model in a homogeneous community. To this model we add a rate at which individuals are tested, and once an infectious individual tests positive it is isolated and each of their contacts are traced and tested independently with some fixed probability. If such a traced individual tests positive it is isolated, and the contact tracing is iterated. This model is analysed using large population approximations, both for the early stage of the epidemic when the “to-be-traced components” of the epidemic behaves like a branching process, and for the main stage of the epidemic where the process of to-be-traced components converges to a deterministic process defined by a system of differential equations. These approximations are used to quantify the effect of testing and of contact tracing on the effective reproduction numbers (for the components as well as for the individuals), the probability of a major outbreak, and the final fraction getting infected. Using numerical illustrations when rates of infection and natural recovery are fixed, it is shown that Test-and-Trace strategy is effective in reducing the reproduction number. Surprisingly, the reproduction number for the branching process of components is not monotonically decreasing in the tracing probability, but the individual reproduction number is conjectured to be monotonic as expected. Further, in the situation where individuals also self-report for testing, the tracing probability is more influential than the screening rate (measured by the fraction infected being screened).

Keywords
Epidemic model, Contact tracing, Branching process, Testing, Reproduction number
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-208211 (URN)10.1007/s11538-022-01065-9 (DOI)000844190400001 ()36001175 (PubMedID)2-s2.0-85137034886 (Scopus ID)
Available from: 2022-08-24 Created: 2022-08-24 Last updated: 2024-04-20Bibliographically approved
2. Epidemic models with digital and manual contact tracing
Open this publication in new window or tab >>Epidemic models with digital and manual contact tracing
Abstract [en]

We consider the Markovian SIR epidemic model, but where individuals either recover naturally or are diagnosed, the latter implying isolation and subject to being contact traced. We focus on the novel digital contact tracing based on individuals using a tracing app and consider both the situation when this is the only form of contact tracing and when combined with manual contact tracing. It is proven that the epidemic process converges to a limiting process as the population size $n$ grows large. The limiting process is not a branching process (as is common for epidemic processes) because infectees may be contact traced "from" their infectors, thus creating dependencies. Instead, treating to-be-traced components as macro-individuals allows a multi-type branching process interpretation, thus giving an expression for the reproduction number $R$. This reproduction number (for the digital-only case) is then converted to an individual reproduction number $R^{(ind)}$ being easier to interpret. It is proven that contrary to $R$, $R^{(ind)}$ decays monotonically with $\pi$ (the fraction of app-users), and the two reproduction numbers are proven to have the same threshold at 1. The digital (only) contact tracing is then compared with earlier results for manual (only) contact tracing, where a fraction $p$ of all contacts are traced. It is proven that the critical fraction app-users $\pi_c$ for which digital contact tracing is reduced to $R=1$, and the critical fraction manually contact traced $p_c$ to reach $R=1$, satisfy $\pi_c>p_c$.

Keywords
epidemic model, contact tracing, branching process, reproduction number
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-228480 (URN)
Funder
Swedish Research Council, 2020-04744 Available from: 2024-04-19 Created: 2024-04-19 Last updated: 2024-04-20
3. A network SEIR epidemic model with manual and digital contact tracing allowing delays
Open this publication in new window or tab >>A network SEIR epidemic model with manual and digital contact tracing allowing delays
Abstract [en]

We consider an SEIR epidemic model on a network also allowing random contacts, where recovered individuals could either recover naturally or be diagnosed. Upon diagnosis, manual contact tracing is triggered so that some network contacts are reported and isolated after a random delay. Additionally, digital tracing is based on a tracing app, if the diagnosed index case is an app-user, all of the app-using contacts are immediately notified and isolated. The early phases of the epidemic with manual and/or digital tracing are approximated by different multi-type branching processes, and three respective reproduction numbers are derived. The effectiveness of contact tracing is numerically quantified through the reduction of reproduction numbers. It suggests that app-using fraction plays an essential role in the overall effectiveness of contact tracing. The effectiveness of manual tracing increases if more transmission occurs on the network and the tracing delay is shortened. While manual tracing on its own is insufficient to control the epidemic, it still requires a substantially large proportion of app-users when combined with digital tracing.

Keywords
epidemic model, contact tracing, branching process, reproduction number
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-228481 (URN)
Funder
Swedish Research Council, 2020-04744 Available from: 2024-04-19 Created: 2024-04-19 Last updated: 2024-04-20
4. Sideward contact tracing in an epidemic model with mixing groups
Open this publication in new window or tab >>Sideward contact tracing in an epidemic model with mixing groups
Abstract [en]

We consider a stochastic epidemic model with sideward contact tracing. We assume that infection is driven by interactions within mixing events (gatherings of two or more individuals). Once an infective is diagnosed, each individual who was infected at the same event is contact traced with some probability. Assuming few initial infectives in a large population, the early phase of the epidemic is approximated by a branching process with sibling dependencies. To address the challenges given by the dependencies, we consider sibling groups (individuals who become infected at the same event) as macro-individuals and define a macro-branching process. This allows us to derive an expression for the effective macro-reproduction number, which corresponds to the effective individual reproduction number and represents a threshold for the behaviour of the epidemic. Through a numerical illustration, we show how the reproduction number varies with the mean size of mixing events, the rate of diagnosis and the tracing probability.

Keywords
epidemic model, contact tracing, branching process, reproduction number
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:su:diva-228482 (URN)
Funder
Swedish Research Council, 2020-04744Knut and Alice Wallenberg Foundation, 2020.072 Available from: 2024-04-19 Created: 2024-04-19 Last updated: 2024-04-20

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