Perturbation analysis in finite LD‐QBD processes and applications to epidemic models

Gómez‐Corral, Antonio; López‐García, Martín

doi:10.1002/nla.2160

Cited by 22 publications

(26 citation statements)

References 43 publications

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“…This section is devoted to numerical experiments to implement the analytical results we obtained in Section as well as to provide evidences that these results are likely to hold in realistic situations. Specifically, experiments in Figures are related to the compartmental model of Lipsitch et al, which addresses antibiotic resistance in hospitals; for related work, see the papers by Cen et al and Gómez‐Corral and López‐García ,. Section 3.3…”

Section: Numerical Experiments and Discussionmentioning

confidence: 99%

“…From the above, Lipsitch et al, page 1939 derive a set of ordinary differential equations for the fractions f AS ( t ) and f AR ( t ) of patients who are colonized with either sensitive bacteria or resistant bacteria, respectively, and the fraction f F ( t ) of patients who are free of bacteria at time t . Since f AS ( t ) + f AR ( t ) + f F ( t ) = 1 at any time t , it is summarized by the pair

\begin{matrix} \frac{d f_{A S} false(t false)}{d t} & = ((), λ_{A S} + β f_{A S} false(t false)) false(1 - f_{A S} false(t false) - f_{A R} false(t false) false) - ((), τ_{A} + τ_{B} + γ + μ) f_{A S} false(t false), \\ \frac{d f_{A R} false(t false)}{d t} & = ((), λ_{A R} + false(1 - c false) β f_{A R} false(t false)) false(1 - f_{A S} false(t false) - f_{A R} false(t false) false) - ((), τ_{B} + γ + μ) f_{A R} false(t false), \end{matrix}

of differential equations and is thus a particular instance of a multitype SIS model; see Gómez‐Corral and López‐García, Sections 3.2‐3.3 and Saunders …”

Section: Numerical Experiments and Discussionmentioning

confidence: 99%

“…The focus has been on the two‐strain SIS model with partial cross‐immunity, where neither coinfection nor mutation is possible, and it is assumed that infective individuals become susceptible individuals to both strains of bacteria after recovering. This work is part of an ongoing study on how heterogeneous infectiousness and/or susceptibility may influence the dynamics of epidemic models, including SIS models with heterogeneous contacts, multitype SIS models, and SIR models with two strains and total cross‐immunity …”

Section: Discussionmentioning

confidence: 99%

“…Specifically, the dynamics of the population are driven by the non‐null infinitesimal rates

q_{false(i, j false), false(i^{'}, j^{'} false)} = ({, \begin{matrix} left left \\ array (N - i) (λ_{1} + (i - j) β_{1}), & array if i^{'} = i + 1, j^{'} = j, \\ array (N - i) (λ_{2} + j β_{2}), & array if i^{'} = i + 1, j^{'} = j + 1, \\ array (i - j) γ_{1}, & array if i^{'} = i - 1, j^{'} = j, \\ array j γ_{2}, & array if i^{'} = i - 1, j^{'} = j - 1, \end{matrix})

and q ( i , j ),( i , j ) = − q ( i , j ) with

q_{(i, j)} = (N - i) (λ_{1} + λ_{2} + (i - j) β_{1} + j β_{2}) + (i - j) γ_{1} + j γ_{2},

for

false(i, j false) \in scriptS

. By conveniently labeling states, (1) results in distinct level‐dependent quasi‐birth‐death (LD‐QBD) processes, which are used by Gómez‐Corral and López‐García(, Section 3.2) to provide the sensitivities and elasticities of first‐passage times and hitting probabilities, extreme values, and stationary measures of either an outbreak of the disease or the outbreak of the type‐ k strain, for k ∈ {1,2}.…”

Section: The Sis Model With Two Strains and Partial Cross‐immunitymentioning

confidence: 99%

“…The aim of this paper is to complement the treatment of the two‐strain SIS model with partial cross‐immunity we started in Gómez‐Corral and López‐García, Section 3.2 —which can be seen as an IDS model with

p_{M}^{false(M false)} = p_{S}^{false(S false)} = 1

– by focusing here on the numbers M 1 and M 2 of type‐1 and type‐2 infections suffered by a focal individual during an outbreak of the disease. To begin with, we define in Section the underlying Markov chain model.…”

Section: Introductionmentioning

confidence: 99%

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Number of infections suffered by a focal individual in a two‐strain SIS model with partial cross‐immunity

Almaraz

Gómez‐Corral

2019

Math Methods in App Sciences

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We study a stochastic model for the spread of two pathogen strains—termed type 1 and type 2—among a homogeneously mixing community consisting of a finite number of individuals. In the model, we assume partial cross‐immunity, exogenous streams of infection, and that the degree of severity of a newly infective individual depends on who this infective individual was infected by. The aim is to characterize the joint probability distribution of the numbers M1 and M2 of type‐1 and type‐2 infections suffered by a focal individual during an outbreak of the disease. We present iterative procedures for computing the probability mass function of (M1,M2) under the assumption that the initial state of the focal individual is known, and a numerical study of the model is performed to investigate the influence of certain key parameters on the spread of resistant bacteria in hospitals.

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Section: Numerical Experiments and Discussionmentioning

confidence: 99%

\begin{matrix} \frac{d f_{A S} false(t false)}{d t} & = ((), λ_{A S} + β f_{A S} false(t false)) false(1 - f_{A S} false(t false) - f_{A R} false(t false) false) - ((), τ_{A} + τ_{B} + γ + μ) f_{A S} false(t false), \\ \frac{d f_{A R} false(t false)}{d t} & = ((), λ_{A R} + false(1 - c false) β f_{A R} false(t false)) false(1 - f_{A S} false(t false) - f_{A R} false(t false) false) - ((), τ_{B} + γ + μ) f_{A R} false(t false), \end{matrix}

of differential equations and is thus a particular instance of a multitype SIS model; see Gómez‐Corral and López‐García, Sections 3.2‐3.3 and Saunders …”

Section: Numerical Experiments and Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

“…Specifically, the dynamics of the population are driven by the non‐null infinitesimal rates

q_{false(i, j false), false(i^{'}, j^{'} false)} = ({, \begin{matrix} left left \\ array (N - i) (λ_{1} + (i - j) β_{1}), & array if i^{'} = i + 1, j^{'} = j, \\ array (N - i) (λ_{2} + j β_{2}), & array if i^{'} = i + 1, j^{'} = j + 1, \\ array (i - j) γ_{1}, & array if i^{'} = i - 1, j^{'} = j, \\ array j γ_{2}, & array if i^{'} = i - 1, j^{'} = j - 1, \end{matrix})

and q ( i , j ),( i , j ) = − q ( i , j ) with

q_{(i, j)} = (N - i) (λ_{1} + λ_{2} + (i - j) β_{1} + j β_{2}) + (i - j) γ_{1} + j γ_{2},

for

false(i, j false) \in scriptS

Section: The Sis Model With Two Strains and Partial Cross‐immunitymentioning

confidence: 99%

p_{M}^{false(M false)} = p_{S}^{false(S false)} = 1

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Number of infections suffered by a focal individual in a two‐strain SIS model with partial cross‐immunity

Almaraz

Gómez‐Corral

2019

Math Methods in App Sciences

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Sensitivity Analysis of Continuous Markov Chains

Caswell

2019

Sensitivity Analysis: Matrix Methods in Demography and Ecology

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When Markov chains are used as mathematical models of natural or social phenomena, the transition intensities or probabilities are usually defined in terms of parameters that are relevant to the scientific question at hand. Sensitivity analysis of such models is important because it quantifies the dependence of the model behavior on the parameters. This chapter presents sensitivity results for finite-state, continuous-time absorbing Markov chains, paralleling the approach for discretetime chains in Chap. 11. In absorbing chains, interest focuses on behavior prior to absorption (time spent in transient states and time to absorption) and on the probabilities of absorption in each absorbing state. Here we will derive formulae for the sensitivity and the elasticity (i.e., proportional sensitivity) of the moments of the time to absorption, the time spent in each transient state, and the number of visits to each transient state. The most basic difference between discrete-time and continuous-time Markov chains is that the former are defined by transition probabilities, while the latter are defined by transition rates. This leads to differences in the structure of the matrices, but there is a nice parallelism in the results. Perturbation analysis of Markov chains has a long history (Schweitzer 1968; Meyer 1975). Most of the literature, however, is devoted to discrete-time chains, and most of that focuses on ergodic chains and the perturbation analysis of the stationary distribution; e.g. Funderlic and Meyer (1986), Golub and Meyer (1986), Hunter (2005), Cho and Meyer (2000), and Seneta (1993). Much less attention has been paid to continuous-time chains. Perturbation expansions have been developed Chapter 12 is modified, by permission of John Wiley and Sons, from: Caswell, H. 2012. Perturbation analysis of continuous-time absorbing Markov chains. Numerical Linear Algebra with Applications 18:901-917. ©John Wiley and Sons.

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Sensitivity Analysis of Markovian Exact Reproduction Numbers

Gamboa

Lopez-Herrero

2023

Lecture Notes in Computer Science

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Perturbation analysis in finite LD‐QBD processes and applications to epidemic models

Cited by 22 publications

References 43 publications

Number of infections suffered by a focal individual in a two‐strain SIS model with partial cross‐immunity

Number of infections suffered by a focal individual in a two‐strain SIS model with partial cross‐immunity

Sensitivity Analysis of Continuous Markov Chains

Sensitivity Analysis of Markovian Exact Reproduction Numbers

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