http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

A precise definition of the basic reproduction number, Ro, is presented for a general compartmental disease transmission model based on a system of ordinary dierential equations. It is shown that, if Ro 1, then it is unstable. Thus,Ro is a threshold parameter for the model. An analysis of the local centre manifold yields a simple criterion for the existence and stability of super- and sub-threshold endemic equilibria for Ro near one. This criterion, together with the definition of Ro, is illustrated by treatment, multigroup, staged progression, multistrain and vectorhost models and can be applied to more complex models. The results are significant for disease control.

/pdf/reproduction-numbers-and-sub-threshold-endemic-equilibria-4b4f32pm0g.pdf

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

Many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. Threshold theorems involving the basic reproduction number $R_{0}$, the contact number $\sigma$, and the replacement number $R$ are reviewed for the classic SIR epidemic and endemic models. Similar results with new expressions for $R_{0}$ are obtained for MSEIR and SEIR endemic models with either continuous age or age groups. Values of $R_{0}$ and $\sigma$ are estimated for various diseases including measles in Niger and pertussis in the United States. Previous models with age structure, heterogeneity, and spatial structure are surveyed.

https://epubs.siam.org/doi/pdf/10.1137/S0036144500371907

The Mathematics of Infectious Diseases

Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study.

Estimation of the prevalence and contagiousness of undocumented novel coronavirus [severe acute respiratory syndrome–coronavirus 2 (SARS-CoV-2)] infections is critical for understanding the overall prevalence and pandemic potential of this disease. Here, we use observations of reported infection within China, in conjunction with mobility data, a networked dynamic metapopulation model, and Bayesian inference, to infer critical epidemiological characteristics associated with SARS-CoV-2, including the fraction of undocumented infections and their contagiousness. We estimate that 86% of all infections were undocumented [95% credible interval (CI): 82–90%] before the 23 January 2020 travel restrictions. The transmission rate of undocumented infections per person was 55% the transmission rate of documented infections (95% CI: 46–62%), yet, because of their greater numbers, undocumented infections were the source of 79% of the documented cases. These findings explain the rapid geographic spread of SARS-CoV-2 and indicate that containment of this virus will be particularly challenging.

/pdf/substantial-undocumented-infection-facilitates-the-rapid-20nusdrfp7.pdf

Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV-2)

The expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population is mathematically defined as the dominant eigenvalue of a positive linear operator. It is shown that in certain special cases one can easily compute or estimate this eigenvalue. Several examples involving various structuring variables like age, sexual disposition and activity are presented.

/pdf/on-the-definition-and-the-computation-of-the-basic-1422gpmabf.pdf

On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations

On the definition and the computation of the basic reproduction ratio : $R_ 0$ in models for infectious diseases in heterogeneous populations

Preface, Creed and Apology. What it is All About and What Not. The Top Down Approach. A Workbook. Portrait of the Reader as a Young Person. A Brief Outline of the Book. Acknowledgments. And What About Reality? I: THE BARE BONES: BASIC ISSUES EXPLAINED IN THE SIMPLEST CONTEXT. The Epidemic in a Closed Population. Heterogeneity: The Art of Averaging. Dynamics at the Demographic Time Scale. II: STRUCTURED POPULATIONS. The Concept of State. The Basic Reproduction Ratio. And Everything Else... Age Structure. Spatial Spread. Macroparasites. What is Contact? III: THE HARD PART: ELABORATIONS (ALMOST) ALL EXERCISES. Elaborations for Part I. Elaborations for Part II. Appendix A: Stochastic Basic of the Kermack-McKendrick ODE Model. Appendix B: Bibliographic Skeleton. Index.

Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation

The basic reproduction number ℛ0 is arguably the most important quantity in infectious disease epidemiology. The next-generation matrix (NGM) is the natural basis for the definition and calculation...

/pdf/the-construction-of-next-generation-matrices-for-108c1k29w4.pdf

The construction of next-generation matrices for compartmental epidemic models.

Increasing evidence that the strengths of interactions among populations in biological communities form patterns that are crucial for system stability requires clarification of the precise form of these patterns, how they come about, and why they influence stability. We show that in real food webs, interaction strengths are organized in trophic loops in such a way that long loops contain relatively many weak links. We show and explain mathematically that this patterning enhances stability, because it reduces maximum "loop weight" and thus reduces the amount of intraspecific interaction needed for matrix stability. The patterns are brought about by biomass pyramids, a feature common to most ecosystems. Incorporation of biomass pyramids in 104 food-web descriptions reveals that the low weight of the long loops stabilizes complex food webs. Loop-weight analysis could be a useful tool for exploring the structure and organization of complex communities.

/pdf/stability-in-real-food-webs-weak-links-in-long-loops-4p3tntdtch.pdf

J.A.P. Heesterbeek

Papers

On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations

On the definition and the computation of the basic reproduction ratio : $R_ 0$ in models for infectious diseases in heterogeneous populations

Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation

The construction of next-generation matrices for compartmental epidemic models.

Stability in real food webs: weak links in long loops.