Solvability of implicit final size equations for SIR epidemic models

Bidari, Subekshya; Chen, Xinying; Peters, Daniel M.; Pittman, Dylanger; Šimon, Peter

doi:10.1016/j.mbs.2016.10.012

Cited by 16 publications

(9 citation statements)

References 26 publications

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“…The solution is denoted here by b † c , as the b c to be used is slightly different, as it is shown below. The non-principal Lambert Appendix D. Proofs of Equations (102) and (103) With the approximate expression (95), that is useful up to order O(η 2 ), one obtains for corresponding cumulative number fraction, with the substitution ξ = e −bτ /b:…”

Section: Data Availability Statementmentioning

confidence: 99%

“…There are several approaches that investigate the optimal control of the model with time-dependent or nonlinear functions [49,50,55,59,60,62,[67][68][69][70]75,[77][78][79][80][81][82][83][84][85][86][87][88][89][90][91][92][93] and the SIR model had also been applied to networks [34,[94][95][96]. While it is comparably easy to set up a network or agentbased [97][98][99] model or to solve a nonlinear problem numerically, pure analytic works are relatively scarce [100][101][102][103][104][105][106][107][108][109][110][111].…”

Section: Introductionmentioning

confidence: 99%

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Analytical Modeling of the Temporal Evolution of Epidemics Outbreaks Accounting for Vaccinations

Schlickeiser

Kröger

2021

Physics

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With the vaccination against Covid-19 now available, how vaccination campaigns influence the mathematical modeling of epidemics is quantitatively explored. In this paper, the standard susceptible-infectious-recovered/removed (SIR) epidemic model is extended to a fourth compartment, V, of vaccinated persons. This extension involves the time t-dependent effective vaccination rate, v(t), that regulates the relationship between susceptible and vaccinated persons. The rate v(t) competes with the usual infection, a(t), and recovery, μ(t), rates in determining the time evolution of epidemics. The occurrence of a pandemic outburst with rising rates of new infections requires k+b<1−2η, where k=μ(0)/a(0) and b=v(0)/a(0) denote the initial values for the ratios of the three rates, respectively, and η≪1 is the initial fraction of infected persons. Exact analytical inverse solutions t(Q) for all relevant quantities Q=[S,I,R,V] of the resulting SIRV model in terms of Lambert functions are derived for the semi-time case with time-independent ratios k and b between the recovery and vaccination rates to the infection rate, respectively. These inverse solutions can be approximated with high accuracy, yielding the explicit time-dependences Q(t) by inverting the Lambert functions. The values of the three parameters k, b and η completely determine the reduced time evolution of the SIRV-quantities Q(τ). The influence of vaccinations on the total cumulative number and the maximum rate of new infections in different countries is calculated by comparing with monitored real time Covid-19 data. The reduction in the final cumulative fraction of infected persons and in the maximum daily rate of new infections is quantitatively determined by using the actual pandemic parameters in different countries. Moreover, a new criterion is developed that decides on the occurrence of future Covid-19 waves in these countries. Apart from in Israel, this can happen in all countries considered.

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Section: Data Availability Statementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analytical Modeling of the Temporal Evolution of Epidemics Outbreaks Accounting for Vaccinations

Schlickeiser

Kröger

2021

Physics

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“…To our knowledge, there are relatively few articles in which GSPT has been applied rigorously to epidemics models [33,17,14,42,3,39]; however, for most infectious diseases, the presence of different time scales is natural. Moreover, though a SIR model on networks has been studied with moment closure already [1,20], the SIRS extension has not. Likewise, a thorough bifurcation analysis on compartment models such as the one we analyse in this paper is not present in the literature.…”

Section: Introductionmentioning

confidence: 99%

A geometric analysis of the SIR, SIRS and SIRWS epidemiological models

Jardón-Kojakhmetov

Kuehn

Sensi

2021

Nonlinear Analysis: Real World Applications

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We study a fast-slow version of an SIRS epidemiological model on homogeneous graphs, obtained through the application of the moment closure method. We use GSPT to study the model, taking into account that the infection period is much shorter than the average duration of immunity. We show that the dynamics occurs through a sequence of fast and slow flows, that can be described through 2-dimensional maps that, under some assumptions, can be approximated as 1-dimensional maps. Using this method, together with numerical bifurcation tools, we show that the model can give rise to periodic solutions, differently from the corresponding model based on homogeneous mixing.

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“…In Bidari et al. ( 2016 ), Bidari et al studied the solvability of implicit equations for the final epidemic size such as the classical SIR model, the pairwise SIR model and the heterogeneous mean-field SIR model, and proved the convergence of the solution for a suitable fixed point equation. Wang and Cao ( 2021 ) studied the final size of various network SIR models, and showed that how the final epidemic size is related to the basic reproduction number for different degree distributions.…”

Section: Introductionmentioning

confidence: 99%

Mathematical Analysis of Epidemic Models with Treatment in Heterogeneous Networks

et al. 2023

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In this paper, we formulate two different network-based epidemic models to investigate the effect of partly effective treatment on disease dynamics. The first network model represents the individuals with heterogeneous number of contacts in a population as choosing a new partner at each moment, whereas the second one assumes the individuals have fixed or stable neighbors. The basic reproduction number is computed for each model, using the next generation matrix method. In particular, the critical treatment rate is defined for the model, above which the disease can be eliminated through the treatment. The final epidemic size relations are derived, and the solvability of these implicit equations is studied. In particular, a unique solution of the implicit equation for the final epidemic size is determined, and by rewriting the implicit equation as a suitable fixed point problem, it is proved that the iteration of the fixed point problem converges to the unique solution. Stochastic simulations and numerical simulations, including in comparison with the model outputs and the joint influence of network topology and treatment on the final epidemic size, are conducted to illustrate the theoretical results.

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Solvability of implicit final size equations for SIR epidemic models

Cited by 16 publications

References 26 publications

Analytical Modeling of the Temporal Evolution of Epidemics Outbreaks Accounting for Vaccinations

Analytical Modeling of the Temporal Evolution of Epidemics Outbreaks Accounting for Vaccinations

A geometric analysis of the SIR, SIRS and SIRWS epidemiological models

Mathematical Analysis of Epidemic Models with Treatment in Heterogeneous Networks

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