Convergence and Monotonicity for a Model of Spontaneous Infection and Transmission

A version of the contact process (effectively an SIS model) on a finite set of sites is considered in which there is the possibility of spontaneous infection. A companion process is also considered in which spontaneous infection does not occur from the disease-free state. Monotonicity with respect to parameters and initial data is established, and conditions for irreducibility and exponential convergence of the processes are given. For the spontaneous process, a set of approximating equations is derived, and its properties investigated.

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A non-standard discretized SIS model of epidemics

Mathematical Biosciences and Engineering ◽

10.3934/mbe.2022006 ◽

2021 ◽

Vol 19 (1) ◽

pp. 115-133

Author(s):

Marcin Choiński ◽

◽

Mariusz Bodzioch ◽

Urszula Foryś ◽

◽

...

Keyword(s):

Global Stability ◽

Stationary State ◽

Continuous System ◽

Continuous Model ◽

Free State ◽

Discrete Version ◽

Step Size ◽

Sis Model ◽

Homogeneous Population ◽

Disease Free

<abstract><p>In this paper we introduce and analyze a non-standard discretized SIS epidemic model for a homogeneous population. The presented model is a discrete version of the continuous model known from literature and used by us for building a model for a heterogeneous population. Firstly, we discuss basic properties of the discrete system. In particular, boundedness of variables and positivity of solutions of the system are investigated. Then we focus on stability of stationary states. Results for the disease-free stationary state are depicted with the use of a basic reproduction number computed for the system. For this state we also manage to prove its global stability for a given condition. It transpires that the behavior of the disease-free state is the same as its behavior in the analogous continuous system. In case of the endemic stationary state, however, the results are presented with respect to a step size of discretization. Local stability of this state is guaranteed for a sufficiently small critical value of the step size. We also conduct numerical simulations confirming theoretical results about boundedness of variables and global stability of the disease-free state of the analyzed system. Furthermore, the simulations ascertain a possibility of appearance of Neimark-Sacker bifurcation for the endemic state. As a bifurcation parameter the step size of discretization is chosen. The simulations suggest the appearance of a supercritical bifurcation.</p></abstract>

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Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function

Mathematical Problems in Engineering ◽

10.1155/2015/745732 ◽

2015 ◽

Vol 2015 ◽

pp. 1-14 ◽

Cited By ~ 1

Author(s):

Yanju Xiao ◽

Weipeng Zhang ◽

Guifeng Deng ◽

Zhehua Liu

Keyword(s):

Sufficient Conditions ◽

Global Dynamics ◽

Sis Model ◽

Delayed Treatment ◽

Sis Epidemic Model ◽

Treatment Function ◽

Lyapunov Coefficient ◽

Saturated Treatment ◽

Theoretical Results ◽

Disease Free

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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Network inference from population-level observation of epidemics

Scientific Reports ◽

10.1038/s41598-020-75558-9 ◽

2020 ◽

Vol 10 (1) ◽

Cited By ~ 1

Author(s):

F. Di Lauro ◽

J.-C. Croix ◽

M. Dashti ◽

L. Berthouze ◽

I. Z. Kiss

Keyword(s):

Real World ◽

Continuous Time ◽

Network Inference ◽

Population Level ◽

Death Process ◽

Disease Dynamics ◽

Birth And Death Process ◽

Sis Model ◽

Underlying Network ◽

Finite Set

Abstract Using the continuous-time susceptible-infected-susceptible (SIS) model on networks, we investigate the problem of inferring the class of the underlying network when epidemic data is only available at population-level (i.e., the number of infected individuals at a finite set of discrete times of a single realisation of the epidemic), the only information likely to be available in real world settings. To tackle this, epidemics on networks are approximated by a Birth-and-Death process which keeps track of the number of infected nodes at population level. The rates of this surrogate model encode both the structure of the underlying network and disease dynamics. We use extensive simulations over Regular, Erdős–Rényi and Barabási–Albert networks to build network class-specific priors for these rates. We then use Bayesian model selection to recover the most likely underlying network class, based only on a single realisation of the epidemic. We show that the proposed methodology yields good results on both synthetic and real-world networks.

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Laser myringotomy for otitis media with effusion in nasopharyngeal carcinoma patients

Otolaryngology ◽

10.1016/j.otohns.2005.01.042 ◽

2005 ◽

Vol 132 (6) ◽

pp. 924-927 ◽

Cited By ~ 3

Author(s):

Chung-Feng Hwang ◽

Chih-Yen Chien ◽

Hsin-Ching Lin ◽

Jyh-Ping Peng ◽

Hsueh-Wen Chang ◽

...

Keyword(s):

Nasopharyngeal Carcinoma ◽

Clinical Outcome ◽

Otitis Media ◽

Study Design ◽

Otitis Media With Effusion ◽

Tumor Stage ◽

Free State ◽

History Of ◽

Determining Factor ◽

Disease Free

OBJECTIVE: To evaluate the effectiveness of laser myringotomy in otitis media with effusion (OME) in patients with nasopharyngeal carcinoma (NPC). STUDY DESIGN AND SETTING: Laser myringotomy was performed with a CO2 laser otoscope in 53 NPC patients (68 ears) and 32 non-NPC patients (37 ears). All patients had a history of OME. RESULTS: In the NPC group, 40% of ears developed persistent perforation, 37% had recurrent OME, and 23% were free of disease. In the non-NPC group, only 3% of ears developed persistent perforation, and 43% were free of disease. Significant differences in perforation and disease-free state were evident between the 2 groups. Tumor stage was found to be a significant determining factor for clinical outcome in NPC group. CONCLUSION: Laser myringotomy for NPC group carries a significantly higher risk of persistent perforation and lower rate of cure. This procedure may be considered for early staging NPC patients with serious OME symptoms.

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Regularity and large time behaviour of solutions of a conservation law without convexity

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500014256 ◽

1985 ◽

Vol 99 (3-4) ◽

pp. 201-239 ◽

Cited By ~ 52

Author(s):

C. M. Dafermos

Keyword(s):

Initial Data ◽

Conservation Law ◽

Large Time ◽

Decay Rates ◽

Time Behaviour ◽

Large Time Behaviour ◽

Jump Discontinuities ◽

Contact Discontinuities ◽

Finite Set ◽

Line Characteristic

SynopsisUsing the method of generalized characteristics, we discuss the regularity and large time behaviour of admissible weak solutions of a single conservation law, in one space variable, with one inflection point.We show that when the initial data are C∞ then, generically, the solution is C∞ except: (a) on a finite set of C∞ arcs across which it experiences jump discontinuities (genuine shocks or left contact discontinuities); (b) on a finite set of straight line characteristic segments across which its derivatives of order m, m = 1, 2,…, experience jump discontinuities (weak waves of order m); and (c) on the finite set of points of interaction of shocks and weak waves. Weak waves of order 1 are triggered by rays grazing upon contact discontinuities. Weak waves of order m, m ≥ 2, are generated by the collision of a weak wave of order m − 1 with a left contact discontinuity.We establish sharp decay rates for solutions with initial data of the following types: (a) with bounded primitive; (b) with primitive having sublinear growth; (c) in L1; (d) of compact support; and (e) periodic.

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SPREADING DYNAMICS OF A DISEASE-AWARENESS SIS MODEL ON COMPLEX NETWORKS

International Journal of Biomathematics ◽

10.1142/s1793524513500253 ◽

2013 ◽

Vol 06 (04) ◽

pp. 1350025 ◽

Cited By ~ 6

Author(s):

BEI LIN ◽

ZHENLONG SUN ◽

XINCHU FU ◽

GUANGHU ZHU

Keyword(s):

Theoretical Analysis ◽

Network Structure ◽

Control Variable ◽

Model Parameters ◽

Spread Rate ◽

Sis Model ◽

Disease Awareness ◽

Spreading Dynamics ◽

The Impact ◽

Disease Free

In this paper, to better understand the impact of awareness and the network structure on epidemic transmission, we divide the population into four subpopulations corresponding to different physical states and conscious states, and we first propose a modified disease-awareness model, then verify the global stability of the disease-free and endemic equilibria, and finally present numerical simulations to demonstrate the theoretical analysis. By examining the spreading influences of model parameters, we find that the outbreak scale can be effectively controlled through increasing the spread rate of awareness or reducing the rate of awareness loss. That is to say, all sorts of media publicity are meaningful. Meanwhile, we find that infection will be affected by consciousness through the control variable.

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