scholarly journals The Probability of Containment for Multitype Branching Process Models for Emerging Epidemics

2011 ◽  
Vol 48 (1) ◽  
pp. 173-188
Author(s):  
Simon E. F. Spencer ◽  
Philip D. O‘Neill
Keyword(s):  
Infectious Disease ◽  
Multiple Solutions ◽  
Branching Process ◽  
Specific Model ◽  
Process Models ◽  
Multitype Branching Process ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Disease Epidemics ◽  
Special Case

This paper is concerned with the definition and calculation of containment probabilities for emerging disease epidemics. A general multitype branching process is used to model an emerging infectious disease in a population of households. It is shown that the containment probability satisfies a certain fixed point equation which has a unique solution under certain conditions; the case of multiple solutions is also described. The extinction probability of the branching process is shown to be a special case of the containment probability. It is shown that Laplace transform ordering of the severity distributions of households in different epidemics yields an ordering on the containment probabilities. The results are illustrated with both standard epidemic models and a specific model for an emerging strain of influenza.

scholarly journals The Probability of Containment for Multitype Branching Process Models for Emerging Epidemics

2011 ◽  
Vol 48 (01) ◽  
pp. 173-188
Author(s):  
Simon E. F. Spencer ◽  
Philip D. O‘Neill
Keyword(s):  
Infectious Disease ◽  
Multiple Solutions ◽  
Branching Process ◽  
Specific Model ◽  
Process Models ◽  
Multitype Branching Process ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Disease Epidemics ◽  
Special Case

This paper is concerned with the definition and calculation of containment probabilities for emerging disease epidemics. A general multitype branching process is used to model an emerging infectious disease in a population of households. It is shown that the containment probability satisfies a certain fixed point equation which has a unique solution under certain conditions; the case of multiple solutions is also described. The extinction probability of the branching process is shown to be a special case of the containment probability. It is shown that Laplace transform ordering of the severity distributions of households in different epidemics yields an ordering on the containment probabilities. The results are illustrated with both standard epidemic models and a specific model for an emerging strain of influenza.

scholarly journals Unified model of short- and long-term HIV viral rebound for clinical trial planning

2021 ◽  
Vol 18 (177) ◽  
Author(s):  
Jessica M. Conway ◽  
Paige Meily ◽  
Jonathan Z. Li ◽  
Alan S. Perelson
Keyword(s):  
Branching Process ◽  
Treatment Interruption ◽  
Process Models ◽  
Viral Rebound ◽  
Multitype Branching Process ◽  
Viral Loads ◽  
Biological Interpretation ◽  
Short And Long Term ◽  
Study Participants

Antiretroviral therapy (ART) effectively controls HIV infection, suppressing HIV viral loads. Typically suspension of therapy is rapidly followed by rebound of viral loads to high, pre-therapy levels. Indeed, a recent study showed that approximately 90% of treatment interruption study participants show viral rebound within at most a few months of therapy suspension, but the remaining 10%, showed viral rebound some months, or years, after ART suspension. Some may even never rebound. We investigate and compare branching process models aimed at gaining insight into these viral dynamics. Specifically, we provide a theory that explains both short- and long-term viral rebounds, and post-treatment control, via a multitype branching process with time-inhomogeneous rates, validated with data from Li et al. (Li et al. 2016 AIDS 30 , 343–353. ( doi:10.1097/QAD.0000000000000953 )). We discuss the associated biological interpretation and implications of our best-fit model. To test the effectiveness of an experimental intervention in delaying or preventing rebound, the standard practice is to suspend therapy and monitor the study participants for rebound. We close with a discussion of an important application of our modelling in the design of such clinical trials.

scholarly journals Interventions targeting nonsymptomatic cases can be important to prevent local outbreaks: COVID-19 as a case-study

2020 ◽  
Author(s):  
F.A. Lovell-Read ◽  
S. Funk ◽  
U. Obolski ◽  
C.A. Donnelly ◽  
R.N. Thompson
Keyword(s):  
Infectious Disease ◽  
Optimal Strategy ◽  
Process Model ◽  
Branching Process ◽  
Host Population ◽  
Imported Cases ◽  
New Locations ◽  
Disease Epidemics ◽  
And Control

ABSTRACTBackgroundDuring infectious disease epidemics, a key question is whether cases travelling to new locations will trigger local outbreaks. The risk of this occurring depends on a range of factors, such as the transmissibility of the pathogen, the susceptibility of the host population and, crucially, the effectiveness of local surveillance in detecting cases and preventing onward spread. For many pathogens, presymptomatic and/or asymptomatic (together referred to here as nonsymptomatic) transmission can occur, making effective surveillance challenging. In this study, using COVID-19 as a case-study, we show how the risk of local outbreaks can be assessed when nonsymptomatic transmission can occur.MethodsWe construct a branching process model that includes nonsymptomatic transmission, and explore the effects of interventions targeting nonsymptomatic or symptomatic hosts when surveillance resources are limited. Specifically, we consider whether the greatest reductions in local outbreak risks are achieved by increasing surveillance and control targeting nonsymptomatic or symptomatic cases, or a combination of both.FindingsSeeking to increase surveillance of symptomatic hosts alone is typically not the optimal strategy for reducing outbreak risks. Adopting a strategy that combines an enhancement of surveillance of symptomatic cases with efforts to find and isolate nonsymptomatic hosts leads to the largest reduction in the probability that imported cases will initiate a local outbreak.InterpretationDuring epidemics of COVID-19 and other infectious diseases, effective surveillance for nonsymptomatic hosts can be crucial to prevent local outbreaks.

scholarly journals Single-seed cascades on clustered networks

2020 ◽  
pp. 1-14
Author(s):  
John K. McSweeney
Keyword(s):  
Branching Process ◽  
Dynamic Network ◽  
Average Degree ◽  
Random Networks ◽  
Threshold Condition ◽  
Cascade Process ◽  
Active Node ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Clustered Networks

Abstract We consider a dynamic network cascade process developed by Duncan Watts applied to a class of random networks, developed independently by Newman and Miller, which allows a specified amount of clustering (short loops). We adapt existing methods for locally tree-like networks to formulate an appropriate two-type branching process to describe the spread of a cascade started with a single active node and obtain a fixed-point equation to implicitly express the extinction probability of such a cascade. In so doing, we also recover a formula that has appeared in various forms in works by Hackett et al. and Miller which provides a threshold condition for certain extinction of the cascade. We find that clustering impedes cascade propagation for networks of low average degree by reducing the connectivity of the network, but for networks with high average degree, the presence of small cycles makes cascades more likely.

scholarly journals Interventions targeting non-symptomatic cases can be important to prevent local outbreaks: SARS-CoV-2 as a case study

2021 ◽  
Vol 18 (178) ◽  
pp. 20201014
Author(s):  
Francesca A. Lovell-Read ◽  
Sebastian Funk ◽  
Uri Obolski ◽  
Christl A. Donnelly ◽  
Robin N. Thompson
Keyword(s):  
Infectious Disease ◽  
Optimal Strategy ◽  
Process Model ◽  
Branching Process ◽  
Host Population ◽  
Imported Cases ◽  
New Locations ◽  
Disease Epidemics ◽  
And Control

During infectious disease epidemics, an important question is whether cases travelling to new locations will trigger local outbreaks. The risk of this occurring depends on the transmissibility of the pathogen, the susceptibility of the host population and, crucially, the effectiveness of surveillance in detecting cases and preventing onward spread. For many pathogens, transmission from pre-symptomatic and/or asymptomatic (together referred to as non-symptomatic) infectious hosts can occur, making effective surveillance challenging. Here, by using SARS-CoV-2 as a case study, we show how the risk of local outbreaks can be assessed when non-symptomatic transmission can occur. We construct a branching process model that includes non-symptomatic transmission and explore the effects of interventions targeting non-symptomatic or symptomatic hosts when surveillance resources are limited. We consider whether the greatest reductions in local outbreak risks are achieved by increasing surveillance and control targeting non-symptomatic or symptomatic cases, or a combination of both. We find that seeking to increase surveillance of symptomatic hosts alone is typically not the optimal strategy for reducing outbreak risks. Adopting a strategy that combines an enhancement of surveillance of symptomatic cases with efforts to find and isolate non-symptomatic infected hosts leads to the largest reduction in the probability that imported cases will initiate a local outbreak.

scholarly journals Demographic Dynamics in Multitype Populations with Migrations

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 246
Author(s):  
Manuel Molina-Fernández ◽  
Manuel Mota-Medina
Keyword(s):  
Mathematical Modeling ◽  
Population Dynamics ◽  
Branching Process ◽  
Probability Model ◽  
Biological Systems ◽  
Research Work ◽  
General Setting ◽  
Process Theory ◽  
Multitype Branching Process ◽  
Demographic Dynamics

This research work deals with mathematical modeling in complex biological systems in which several types of individuals coexist in various populations. Migratory phenomena among the populations are allowed. We propose a class of mathematical models to describe the demographic dynamics of these type of complex systems. The probability model is defined through a sequence of random matrices in which rows and columns represent the various populations and the several types of individuals, respectively. We prove that this stochastic sequence can be studied under the general setting provided by the multitype branching process theory. Probabilistic properties and limiting results are then established. As application, we present an illustrative example about the population dynamics of biological systems formed by long-lived raptor colonies.

Branching processes and functional differential equations determining steady-size distributions in cell populations

1995 ◽  
Vol 32 (01) ◽  
pp. 1-10
Author(s):  
Ziad Taib
Keyword(s):  
Differential Equation ◽  
Branching Process ◽  
Branching Processes ◽  
Functional Differential Equations ◽  
Cell Populations ◽  
Size Distributions ◽  
Multitype Branching Process ◽  
Functional Differential ◽  
Intuitive Interpretation ◽  
Infinite Sum

The functional differential equation y′(x) = ay(λx) + by(x) arises in many different situations. The purpose of this note is to show how it arises in some multitype branching process cell population models. We also show how its solution can be given an intuitive interpretation as the probability density function of an infinite sum of independent but not identically distributed random variables.

scholarly journals Bounding the Size and Probability of Epidemics on Networks

2008 ◽  
Vol 45 (2) ◽  
pp. 498-512 ◽  
Author(s):  
Joel C. Miller
Keyword(s):  
Infectious Disease ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Marginal Probability ◽  
Transmission Properties ◽  
Disease Spreading ◽  
Special Case

We consider an infectious disease spreading along the edges of a network which may have significant clustering. The individuals in the population have heterogeneous infectiousness and/or susceptibility. We define the out-transmissibility of a node to be the marginal probability that it would infect a randomly chosen neighbor given its infectiousness and the distribution of susceptibility. For a given distribution of out-transmissibility, we find the conditions which give the upper (or lower) bounds on the size and probability of an epidemic, under weak assumptions on the transmission properties, but very general assumptions on the network. We find similar bounds for a given distribution of in-transmissibility (the marginal probability of being infected by a neighbor). We also find conditions giving global upper bounds on the size and probability. The distributions leading to these bounds are network independent. In the special case of networks with high girth (locally tree-like), we are able to prove stronger results. In general, the probability and size of epidemics are maximal when the population is homogeneous and minimal when the variance of in- or out-transmissibility is maximal.

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