Global stability of an SIR epidemic model with constant infectious period

Zhang, Fengpan; Li, Zizhen; Zhang, Feng

doi:10.1016/j.amc.2007.09.053

Cited by 67 publications

(49 citation statements)

References 7 publications

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“…Equilibrium stability analyses have been conducted on 'unforced' models that assume constant contact rates [6,7,[29][30][31][32], and bifurcation analyses have been conducted on 'forced' models in which contact rates vary seasonally [6][7][8][9][10][11][12]33]. Lloyd [7] found that the biennial pattern observed in the SI 1 R model is reproduced by the SI n R model but with much weaker seasonality.…”

Section: Dynamics Of Epidemic Models With Erlang-distributed Stage Dumentioning

confidence: 99%

Effects of the infectious period distribution on predicted transitions in childhood disease dynamics

Krylova

Earn

2013

J. R. Soc. Interface.

106

140

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The population dynamics of infectious diseases occasionally undergo rapid qualitative changes, such as transitions from annual to biennial cycles or to irregular dynamics. Previous work, based on the standard seasonally forced 'susceptible-exposed-infectious -removed' (SEIR) model has found that transitions in the dynamics of many childhood diseases result from bifurcations induced by slow changes in birth and vaccination rates. However, the standard SEIR formulation assumes that the stage durations (latent and infectious periods) are exponentially distributed, whereas real distributions are narrower and centred around the mean. Much recent work has indicated that realistically distributed stage durations strongly affect the dynamical structure of seasonally forced epidemic models. We investigate whether inferences drawn from previous analyses of transitions in patterns of measles dynamics are robust to the shapes of the stage duration distributions. As an illustrative example, we analyse measles dynamics in New York City from 1928 to 1972. We find that with a fixed mean infectious period in the susceptible -infectious -removed (SIR) model, the dynamical structure and predicted transitions vary substantially as a function of the shape of the infectious period distribution. By contrast, with fixed mean latent and infectious periods in the SEIR model, the shapes of the stage duration distributions have a less dramatic effect on model dynamical structure and predicted transitions. All these results can be understood more easily by considering the distribution of the disease generation time as opposed to the distributions of individual disease stages. Numerical bifurcation analysis reveals that for a given mean generation time the dynamics of the SIR and SEIR models for measles are nearly equivalent and are insensitive to the shapes of the disease stage distributions.

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Section: Dynamics Of Epidemic Models With Erlang-distributed Stage Dumentioning

confidence: 99%

Effects of the infectious period distribution on predicted transitions in childhood disease dynamics

Krylova

Earn

2013

J. R. Soc. Interface.

106

140

View full text Add to dashboard Cite

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“…The proportion of children that develop active TB fast can thus be assumed reasonably to be higher than 10% • Active TB has always been taken as a continued development of the primary infection first acquired or due to endogenous reactivation or exogenous reinfection with a second or the same strain. Separation and quantification of these mechanisms especially (endogenous reactivation and exogenous re-infection) requires technology that can differentiate between strains (Zhang et al, 2007). We crudely lump the two processes (Jung et al, 2002) in this study to represent a progression to active TB, denoted by r a and r c • Recovery, may be due to treatment or natural recovery In which case individuals revert back to the exposed class at rates σ c and σ a .…”

Section: Numerical Resultsmentioning

confidence: 99%

Modeling the Dynamics of Tuberculosis Transmission in Children and Adults

Nyabadza¹,

Kgosimore²

2012

Journal of Mathematics and Statistics

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Problem statement:The fight Against Tuberculosis (TB) has mainly focused on the adult population because children were perceived to pose a very low risk in TB transmission. This assumption ignores the potential risk children had as reservoirs of latent infections from which future cases evolve when they become adults. It was therefore important to investigate the dynamics of TB taking into consideration, children. Approach: We formulated a compartmental model for TB with two age classes, children and adults. Qualitative analysis of the model was done to investigate the stability of the model equilibria in terms of the model reproduction number R 0 Numerical simulations were also done to investigate the role played by some key epidemiological parameters in the dynamics of the disease. Results: The model had two equilibria: The disease free equilibrium which was globally stable for R 0 <1 and the endemic equilibrium which was locally asymptotically stable for R 0 >1, for R 0 near 1. The study showed increased latent infections in the adult population as a result of increased latently infected children who mature to adulthood with latent infections. Conclusion/Recommendations: Progression to active TB among adults is epidemiologically significant and interventions should focus on the adult population. Anti-tuberculosis, treatment of adults is crucial in controlling the epidemic and should interventions be proposed, they should target progression to active TB for those latently infected. The fight against TB should also take into consideration tuberculosis among children.

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“…> σ n ≥ 0 and σ i ∈ R. This form of DDE allows models to describe events having a fixed duration. They have been used to describe biological systems in which events have a non-negligible duration [3,15,10] or in which a sequence of simple events is abstracted as a single complex event associated with a duration [14,7]. In what follows we recall an example of DDE model of a biological system that we shall use to compare delay stochastic simulation approaches.…”

Section: Delay Differential Equations (Ddes)mentioning

confidence: 99%

“…In [3,15] an epidemiological model is defined that computes the theoretical number of people infected with a contagious illness in a closed population over time; in the model a delay is used to model the length of the infectious period. In [10] a simple predator-prey model with harvesting and time delays is presented; in the model a constant delay is used based on the assumption that the change rate of predators depends on the number of prey and predators at some previous time.…”

Section: Introductionmentioning

confidence: 99%

Delay Stochastic Simulation of Biological Systems: A Purely Delayed Approach

Barbuti

Caravagna

Maggiolo-Schettini

et al. 2011

Lecture Notes in Computer Science

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Abstract. Delays in biological systems may be used to model events for which the underlying dynamics cannot be precisely observed. Mathematical modeling of biological systems with delays is usually based on Delay Differential Equations (DDEs), a kind of differential equations in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. In the literature, delay stochastic simulation algorithms have been proposed. These algorithms follow a "delay as duration" approach, which is not suitable for biological systems in which species involved in a delayed interaction can be involved at the same time in other interactions. We show on a DDE model of tumor growth that the delay as duration approach for stochastic simulation is not precise, and we propose a simulation algorithm based on a "purely delayed" interpretation of delays which provides better results on the considered model. Moreover, we give a formal definition of a stochastic simulation algorithm which combines both the delay as duration approach and the purely delayed approach.

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Global stability of an SIR epidemic model with constant infectious period

Cited by 67 publications

References 7 publications

Effects of the infectious period distribution on predicted transitions in childhood disease dynamics

Effects of the infectious period distribution on predicted transitions in childhood disease dynamics

Modeling the Dynamics of Tuberculosis Transmission in Children and Adults

Delay Stochastic Simulation of Biological Systems: A Purely Delayed Approach

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