Journal articles: 'Subcritical epidemics'

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Frazer, L. Neil, Alexandra Morton, and Martin Krkošek. "Critical thresholds in sea lice epidemics: evidence, sensitivity and subcritical estimation." Proceedings of the Royal Society B: Biological Sciences 279, no. 1735 (January 4, 2012): 1950–58. http://dx.doi.org/10.1098/rspb.2011.2210.

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Host density thresholds are a fundamental component of the population dynamics of pathogens, but empirical evidence and estimates are lacking. We studied host density thresholds in the dynamics of ectoparasitic sea lice ( Lepeophtheirus salmonis ) on salmon farms. Empirical examples include a 1994 epidemic in Atlantic Canada and a 2001 epidemic in Pacific Canada. A mathematical model suggests dynamics of lice are governed by a stable endemic equilibrium until the critical host density threshold drops owing to environmental change, or is exceeded by stocking, causing epidemics that require rapid harvest or treatment. Sensitivity analysis of the critical threshold suggests variation in dependence on biotic parameters and high sensitivity to temperature and salinity. We provide a method for estimating the critical threshold from parasite abundances at subcritical host densities and estimate the critical threshold and transmission coefficient for the two epidemics. Host density thresholds may be a fundamental component of disease dynamics in coastal seas where salmon farming occurs.

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Neal, Peter. "Endemic Behaviour of SIS Epidemics with General Infectious Period Distributions." Advances in Applied Probability 46, no. 01 (March 2014): 241–55. http://dx.doi.org/10.1017/s0001867800007023.

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We study the endemic behaviour of a homogeneously mixing SIS epidemic in a population of size N with a general infectious period, Q, by introducing a novel subcritical branching process with immigration approximation. This provides a simple but useful approximation of the quasistationary distribution of the SIS epidemic for finite N and the asymptotic Gaussian limit for the endemic equilibrium as N → ∞. A surprising observation is that the quasistationary distribution of the SIS epidemic model depends on Q only through

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Neal, Peter. "Endemic Behaviour of SIS Epidemics with General Infectious Period Distributions." Advances in Applied Probability 46, no. 1 (March 2014): 241–55. http://dx.doi.org/10.1239/aap/1396360112.

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We study the endemic behaviour of a homogeneously mixing SIS epidemic in a population of size N with a general infectious period, Q, by introducing a novel subcritical branching process with immigration approximation. This provides a simple but useful approximation of the quasistationary distribution of the SIS epidemic for finite N and the asymptotic Gaussian limit for the endemic equilibrium as N → ∞. A surprising observation is that the quasistationary distribution of the SIS epidemic model depends on Q only through

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Tiwari, Pankaj Kumar, Rajanish Kumar Rai, Arvind Kumar Misra, and Joydev Chattopadhyay. "Dynamics of Infectious Diseases: Local Versus Global Awareness." International Journal of Bifurcation and Chaos 31, no. 07 (June 15, 2021): 2150102. http://dx.doi.org/10.1142/s0218127421501029.

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Public awareness programs may deeply influence the epidemic pattern of a contagious disease by altering people’s perception of risk and individuals behavior during the course of the epidemic outbreak. Regardless of the veracity, social media advertisements are expected to execute an increasingly prominent role in the field of infectious disease modeling. In this paper, we propose a model which portrays the interplay between dissemination of awareness at local and global levels, and prevalence of disease. Our sensitivity results determine the correlations between some epidemiologically important parameters and disease prevalence. The growth rate of broadcasting information through social media is found to destabilize the system through limit cycle oscillations whereas the baseline number of social media advertisements stabilize the system by terminating persistent oscillations. The system first undergoes supercritical Hopf-bifurcation and then subcritical Hopf-bifurcation on gradual increase in dissemination rate of awareness at local/global level. Moreover, the disease is eradicated if the dissemination rates of awareness and baseline number of social media advertisements are too large. We also study the effect of seasonal variation of the growth rate of social media advertisements. Our nonautonomous system generates globally attractive positive periodic solution if the growth rate of social media advertisements lies between certain ranges. However, the global attractivity is affected on enhancement in growth rate of social media advertisements and three-periodic solution is observed. Our findings show that baseline number of social media advertisements and dissemination of awareness at individual as well as community levels play a tremendous role in eliminating the burden of disease. Furthermore, a comparison of the effects of local and global awareness reveals that the latter is more effective in curtailing the disease. We believe these findings may be beneficial to understand the contagion characteristics of real epidemics and help to adopt suitable precautionary measures in the form of nonpharmaceutical interventions.

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Hueter, Irene. "Branching processes in generalized autoregressive conditional environments." Advances in Applied Probability 48, no. 4 (December 2016): 1211–34. http://dx.doi.org/10.1017/apr.2016.71.

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AbstractBranching processes in random environments have been widely studied and applied to population growth systems to model the spread of epidemics, infectious diseases, cancerous tumor growth, and social network traffic. However, Ebola virus, tuberculosis infections, and avian flu grow or change at rates that vary with time—at peak rates during pandemic time periods, while at low rates when near extinction. The branching processes in generalized autoregressive conditional environments we propose provide a novel approach to branching processes that allows for such time-varying random environments and instances of peak growth and near extinction-type rates. Offspring distributions we consider to illustrate the model include the generalized Poisson, binomial, and negative binomial integer-valued GARCH models. We establish conditions on the environmental process that guarantee stationarity and ergodicity of the mean offspring number and environmental processes and provide equations from which their variances, autocorrelation, and cross-correlation functions can be deduced. Furthermore, we present results on fundamental questions of importance to these processes—the survival-extinction dichotomy, growth behavior, necessary and sufficient conditions for noncertain extinction, characterization of the phase transition between the subcritical and supercritical regimes, and survival behavior in each phase and at criticality.

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Worden, Lee, Ira B. Schwartz, Simone Bianco, Sarah F. Ackley, Thomas M. Lietman, and Travis C. Porco. "Hamiltonian Analysis of Subcritical Stochastic Epidemic Dynamics." Computational and Mathematical Methods in Medicine 2017 (2017): 1–11. http://dx.doi.org/10.1155/2017/4253167.

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We extend a technique of approximation of the long-term behavior of a supercritical stochastic epidemic model, using the WKB approximation and a Hamiltonian phase space, to the subcritical case. The limiting behavior of the model and approximation are qualitatively different in the subcritical case, requiring a novel analysis of the limiting behavior of the Hamiltonian system away from its deterministic subsystem. This yields a novel, general technique of approximation of the quasistationary distribution of stochastic epidemic and birth-death models and may lead to techniques for analysis of these models beyond the quasistationary distribution. For a classic SIS model, the approximation found for the quasistationary distribution is very similar to published approximations but not identical. For a birth-death process without depletion of susceptibles, the approximation is exact. Dynamics on the phase plane similar to those predicted by the Hamiltonian analysis are demonstrated in cross-sectional data from trachoma treatment trials in Ethiopia, in which declining prevalences are consistent with subcritical epidemic dynamics.

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Brightwell, Graham, Thomas House, and Malwina Luczak. "Extinction times in the subcritical stochastic SIS logistic epidemic." Journal of Mathematical Biology 77, no. 2 (January 31, 2018): 455–93. http://dx.doi.org/10.1007/s00285-018-1210-5.

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Helmstetter, Agnès, and Didier Sornette. "Subcritical and supercritical regimes in epidemic models of earthquake aftershocks." Journal of Geophysical Research: Solid Earth 107, B10 (October 2002): ESE 10–1—ESE 10–21. http://dx.doi.org/10.1029/2001jb001580.

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9

Windridge, Peter. "The extinction time of a subcritical branching process related to the SIR epidemic on a random graph." Journal of Applied Probability 52, no. 04 (December 2015): 1195–201. http://dx.doi.org/10.1017/s002190020011318x.

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We give an exponential tail approximation for the extinction time of a subcritical multitype branching process arising from the SIR epidemic model on a random graph with given degrees, where the type corresponds to the vertex degree. As a corollary we obtain a Gumbel limit law for the extinction time, when beginning with a large population. Our contribution is to allow countably many types (this corresponds to unbounded degrees in the random graph epidemic model, as the number of vertices tends to∞). We only require a second moment for the offspring-type distribution featuring in our model.

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Windridge, Peter. "The extinction time of a subcritical branching process related to the SIR epidemic on a random graph." Journal of Applied Probability 52, no. 4 (December 2015): 1195–201. http://dx.doi.org/10.1239/jap/1450802763.

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We give an exponential tail approximation for the extinction time of a subcritical multitype branching process arising from the SIR epidemic model on a random graph with given degrees, where the type corresponds to the vertex degree. As a corollary we obtain a Gumbel limit law for the extinction time, when beginning with a large population. Our contribution is to allow countably many types (this corresponds to unbounded degrees in the random graph epidemic model, as the number of vertices tends to∞). We only require a second moment for the offspring-type distribution featuring in our model.

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11

Heinzmann, Dominik. "Extinction Times in Multitype Markov Branching Processes." Journal of Applied Probability 46, no. 01 (March 2009): 296–307. http://dx.doi.org/10.1017/s0021900200005374.

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In this paper, a distributional approximation to the time to extinction in a subcritical continuous-time Markov branching process is derived. A limit theorem for this distribution is established and the error in the approximation is quantified. The accuracy of the approximation is illustrated in an epidemiological example. Since Markov branching processes serve as approximations to nonlinear epidemic processes in the initial and final stages, our results can also be used to describe the time to extinction for such processes.

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Heinzmann, Dominik. "Extinction Times in Multitype Markov Branching Processes." Journal of Applied Probability 46, no. 1 (March 2009): 296–307. http://dx.doi.org/10.1239/jap/1238592131.

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In this paper, a distributional approximation to the time to extinction in a subcritical continuous-time Markov branching process is derived. A limit theorem for this distribution is established and the error in the approximation is quantified. The accuracy of the approximation is illustrated in an epidemiological example. Since Markov branching processes serve as approximations to nonlinear epidemic processes in the initial and final stages, our results can also be used to describe the time to extinction for such processes.

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13

Xiao, Yanju, Weipeng Zhang, Guifeng Deng, and Zhehua Liu. "Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function." Mathematical Problems in Engineering 2015 (2015): 1–14. http://dx.doi.org/10.1155/2015/745732.

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This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.

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14

Campi, Gaetano, Antonio Valletta, Andrea Perali, Augusto Marcelli, and Antonio Bianconi. "Epidemic spreading in an expanded parameter space: the supercritical scaling laws and subcritical metastable phases." Physical Biology 18, no. 4 (June 21, 2021): 045005. http://dx.doi.org/10.1088/1478-3975/ac059d.

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15

Lemonnier, Rémi, Kevin Scaman, and Nicolas Vayatis. "Spectral bounds in random graphs applied to spreading phenomena and percolation." Advances in Applied Probability 50, no. 2 (June 2018): 480–503. http://dx.doi.org/10.1017/apr.2018.22.

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Abstract In this paper we derive nonasymptotic upper bounds for the size of reachable sets in random graphs. These bounds are subject to a phase transition phenomenon triggered by the spectral radius of the hazard matrix, a reweighted version of the adjacency matrix. Such bounds are valid for a large class of random graphs, called local positive correlation (LPC) random graphs, displaying local positive correlation. In particular, in our main result we state that the size of reachable sets in the subcritical regime for LPC random graphs is at most of order O(√n), where n is the size of the network, and of order O(n2/3) in the critical regime, where the epidemic thresholds are driven by the size of the spectral radius of the hazard matrix with respect to 1. As a corollary, we also show that such bounds hold for the size of the giant component in inhomogeneous percolation, the SIR model in epidemiology, as well as for the long-term influence of a node in the independent cascade model.

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Ren, Xiao-Long, Niels Gleinig, Dirk Helbing, and Nino Antulov-Fantulin. "Generalized network dismantling." Proceedings of the National Academy of Sciences 116, no. 14 (March 15, 2019): 6554–59. http://dx.doi.org/10.1073/pnas.1806108116.

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Finding an optimal subset of nodes in a network that is able to efficiently disrupt the functioning of a corrupt or criminal organization or contain an epidemic or the spread of misinformation is a highly relevant problem of network science. In this paper, we address the generalized network-dismantling problem, which aims at finding a set of nodes whose removal from the network results in the fragmentation of the network into subcritical network components at minimal overall cost. Compared with previous formulations, we allow the costs of node removals to take arbitrary nonnegative real values, which may depend on topological properties such as node centrality or on nontopological features such as the price or protection level of a node. Interestingly, we show that nonunit costs imply a significantly different dismantling strategy. To solve this optimization problem, we propose a method which is based on the spectral properties of a node-weighted Laplacian operator and combine it with a fine-tuning mechanism related to the weighted vertex cover problem. The proposed method is applicable to large-scale networks with millions of nodes. It outperforms current state-of-the-art methods and opens more directions for understanding the vulnerability and robustness of complex systems.

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17

Vere-Jones, David. "A limit theorem with application to Båth's law in seismology." Advances in Applied Probability 40, no. 03 (September 2008): 882–96. http://dx.doi.org/10.1017/s0001867800002834.

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In this paper a limit theorem is proved that establishes conditions under which the distribution of the difference between the size of the initial event in a random sequence, modeled as a finite point process, and the largest subsequent event approaches a limiting form independent of the size of the initial event. The underlying assumptions are that the sizes of the individual events follow an exponential distribution, that the expected total number of events increases exponentially with the size of the initial event, and that the structure of the sequence approximates that of a Poisson process. Particular cases to which the results apply include sequences of independent and identically distributed exponential variables, and the epidemic-type aftershock (ETAS) branching process model in the subcritical case. In all these cases the form of the limit distribution is shown to be that of a double exponential (type-I extreme value distribution). In sampling from a family of aftershock sequences, with possibly different underlying parameters, the limit distribution is a mixture of such double exponential distributions. The conditions for the simple limit to exist relate to the approximation of the distribution of the number of events by a Poisson distribution. One such condition requires the coefficient of variation (ratio of standard deviation to mean) of the number of events to converge to 0 as the mean increases. The results provide a statistical background to Båth's law in seismology, which asserts that in an aftershock sequence the magnitude of the main shock is commonly around 1.2 units higher than the magnitude of the largest aftershock.

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Vere-Jones, David. "A limit theorem with application to Båth's law in seismology." Advances in Applied Probability 40, no. 3 (September 2008): 882–96. http://dx.doi.org/10.1239/aap/1222868190.

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In this paper a limit theorem is proved that establishes conditions under which the distribution of the difference between the size of the initial event in a random sequence, modeled as a finite point process, and the largest subsequent event approaches a limiting form independent of the size of the initial event. The underlying assumptions are that the sizes of the individual events follow an exponential distribution, that the expected total number of events increases exponentially with the size of the initial event, and that the structure of the sequence approximates that of a Poisson process. Particular cases to which the results apply include sequences of independent and identically distributed exponential variables, and the epidemic-type aftershock (ETAS) branching process model in the subcritical case. In all these cases the form of the limit distribution is shown to be that of a double exponential (type-I extreme value distribution). In sampling from a family of aftershock sequences, with possibly different underlying parameters, the limit distribution is a mixture of such double exponential distributions. The conditions for the simple limit to exist relate to the approximation of the distribution of the number of events by a Poisson distribution. One such condition requires the coefficient of variation (ratio of standard deviation to mean) of the number of events to converge to 0 as the mean increases. The results provide a statistical background to Båth's law in seismology, which asserts that in an aftershock sequence the magnitude of the main shock is commonly around 1.2 units higher than the magnitude of the largest aftershock.

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Aguiar, Maíra, Joseba Bidaurrazaga Van-Dierdonck, Javier Mar, Nicole Cusimano, Damián Knopoff, Vizda Anam, and Nico Stollenwerk. "Critical fluctuations in epidemic models explain COVID-19 post-lockdown dynamics." Scientific Reports 11, no. 1 (July 5, 2021). http://dx.doi.org/10.1038/s41598-021-93366-7.

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AbstractAs the COVID-19 pandemic progressed, research on mathematical modeling became imperative and very influential to understand the epidemiological dynamics of disease spreading. The momentary reproduction ratio r(t) of an epidemic is used as a public health guiding tool to evaluate the course of the epidemic, with the evolution of r(t) being the reasoning behind tightening and relaxing control measures over time. Here we investigate critical fluctuations around the epidemiological threshold, resembling new waves, even when the community disease transmission rate $$\beta$$ β is not significantly changing. Without loss of generality, we use simple models that can be treated analytically and results are applied to more complex models describing COVID-19 epidemics. Our analysis shows that, rather than the supercritical regime (infectivity larger than a critical value, $$\beta > \beta _c$$ β > β c ) leading to new exponential growth of infection, the subcritical regime (infectivity smaller than a critical value, $$\beta < \beta _c$$ β < β c ) with small import is able to explain the dynamic behaviour of COVID-19 spreading after a lockdown lifting, with $$r(t) \approx 1$$ r ( t ) ≈ 1 hovering around its threshold value.

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Campi, Gaetano, Maria Vittoria Mazziotti, Antonio Valletta, Giampietro Ravagnan, Augusto Marcelli, Andrea Perali, and Antonio Bianconi. "Metastable states in plateaus and multi-wave epidemic dynamics of Covid-19 spreading in Italy." Scientific Reports 11, no. 1 (June 14, 2021). http://dx.doi.org/10.1038/s41598-021-91950-5.

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AbstractThe control of Covid 19 epidemics by public health policy in Italy during the first and the second epidemic waves has been driven by using reproductive number Rt(t) to identify the supercritical (percolative), the subcritical (arrested), separated by the critical regime. Here we show that to quantify the Covid-19 spreading rate with containment measures there is a need of a 3D expanded parameter space phase diagram built by the combination of Rt(t) and doubling time Td(t). In this space we identify the Covid-19 dynamics in Italy and its administrative Regions. The supercritical regime is mathematically characterized by (i) the power law of Td vs. [Rt(t) − 1] and (ii) the exponential behaviour of Td vs. time, either in the first and in the second wave. The novel 3D phase diagram shows clearly metastable states appearing before and after the second wave critical regime. for loosening quarantine and tracing of actives cases. The metastable states are precursors of the abrupt onset of a next nascent wave supercritical regime. This dynamic description allows epidemics predictions needed by policymakers interested to point to the target "zero infections" with the elimination of SARS-CoV-2, using the Finding mobile Tracing policy joint with vaccination-campaign, in order to avoid the emergence of recurrent new variants of SARS-CoV-2 virus, accompined by recurrent long lockdowns, with large economical losses, and large number of fatalities.

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21

Jain, Shikha, and Sachin Kumar. "Chaos detection in SIR model with modified Beddington–De Angelis type incidence rate and saturated treatment." International Journal of Modeling, Simulation, and Scientific Computing, May 25, 2021, 2150049. http://dx.doi.org/10.1142/s1793962321500495.

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In this paper, we construct an SIR epidemic model with a modified Beddington–De Angelis type incidence rate and saturated treatment rate. We modify the incidence rate to incorporate the isolation of infected individuals after detection, and separation of some susceptible individuals from the rest to avoid the infection, without an increase in the number of classes. We find that the system has a unique disease-free equilibrium (DFE) which is locally asymptotically stable when the reproduction number is less than unity. The multiple endemic equilibria may exist irrespective of the basic reproduction number. The existence of bistability is encountered. Supercritical transcritical (forward), as well as subcritical transcritical (backward) bifurcation, may occur at [Formula: see text] where contact rate, [Formula: see text] acts as the bifurcation parameter. Therefore, DFE need not be globally stable. The conditions for the existence of Andronov–Hopf bifurcation are deduced with maximum treatment capacity, [Formula: see text] as the bifurcation parameter. The impacts of isolation of confirmed infected cases and separation of some susceptible from rest are studied numerically as well as the effect of saturation in treatment. The existence of chaotic behavior is deduced by showing the maximum Lyapunov exponent to be positive as well as the sensitivity to initial conditions. The computation of the Kalpan–Yorke dimension to be fractional confirms the existence of fractal-type strange attractor. The positive Kolmogorov–Sinai entropy further strengthens the claim of the existence of chaos.

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22

Pan, Qin, Jicai Huang, and Qihua Huang. "Global dynamics and bifurcations in a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate." Discrete & Continuous Dynamical Systems - B, 2021, 0. http://dx.doi.org/10.3934/dcdsb.2021195.

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In this paper, we analyze a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate. The nonmonotone incidence rate describes the "psychological effect": when the number of the infected individuals (denoted by <inline-formula><tex-math id="M1">\begin{document}$ I $\end{document}</tex-math></inline-formula>) exceeds a certain level, the incidence rate is a decreasing function with respect to <inline-formula><tex-math id="M2">\begin{document}$ I $\end{document}</tex-math></inline-formula>. The piecewise-smooth treatment rate describes the situation where the community has limited medical resources, treatment rises linearly with <inline-formula><tex-math id="M3">\begin{document}$ I $\end{document}</tex-math></inline-formula> until the treatment capacity is reached, after which constant treatment (i.e., the maximum treatment) is taken.Our analysis indicates that there exists a critical value <inline-formula><tex-math id="M4">\begin{document}$ \widetilde{I_0} $\end{document}</tex-math></inline-formula> <inline-formula><tex-math id="M5">\begin{document}$ ( = \frac{b}{d}) $\end{document}</tex-math></inline-formula> for the infective level <inline-formula><tex-math id="M6">\begin{document}$ I_0 $\end{document}</tex-math></inline-formula> at which the health care system reaches its capacity such that:(i) When <inline-formula><tex-math id="M7">\begin{document}$ I_0 \geq \widetilde{I_0} $\end{document}</tex-math></inline-formula>, the transmission dynamics of the model is determined by the basic reproduction number <inline-formula><tex-math id="M8">\begin{document}$ R_0 $\end{document}</tex-math></inline-formula>: <inline-formula><tex-math id="M9">\begin{document}$ R_0 = 1 $\end{document}</tex-math></inline-formula> separates disease persistence from disease eradication. (ii) When <inline-formula><tex-math id="M10">\begin{document}$ I_0 < \widetilde{I_0} $\end{document}</tex-math></inline-formula>, the model exhibits very rich dynamics and bifurcations, such as multiple endemic equilibria, periodic orbits, homoclinic orbits, Bogdanov-Takens bifurcations, and subcritical Hopf bifurcation.

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Journal articles on the topic 'Subcritical epidemics'

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