Epidemic Thresholds in SIR and SIIR Models Applying an Algorithmic Method

Hincapié, P. Doracelly; Ospina, G. Juan; Afuwape, Anthony Uyi; Gómez, DivierAntonio Agudelo

doi:10.1007/978-3-540-89746-0_12

Cited by 2 publications

(1 citation statement)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Most of the previous works are concerned with the numerical estimation of the probability of occurrence of epidemics, but the authors know less symbolic solutions such as are given in [5]. For some complex deterministic models with the inclusion of two states of infection, treatment modality, the basic reproduction number was obtained using computer algebra [6,7,8,9]. In such works the symbolic expression of the epidemic threshold was derived, identifying the relationship between the parameters for the occurrence of an epidemic, that is, when R 0 > 1 or when it is controlled, with R 0 <1.…”

Section: Introductionmentioning

confidence: 99%

Solving stochastic epidemiological models using computer algebra

Hincapié

Ospina

2011

SPIE Proceedings

View full text Add to dashboard Cite

Mathematical modeling in Epidemiology is an important tool to understand the ways under which the diseases are transmitted and controlled. The mathematical modeling can be implemented via deterministic or stochastic models. Deterministic models are based on short systems of non-linear ordinary differential equations and the stochastic models are based on very large systems of linear differential equations. Deterministic models admit complete, rigorous and automatic analysis of stability both local and global from which is possible to derive the algebraic expressions for the basic reproductive number and the corresponding epidemic thresholds using computer algebra software. Stochastic models are more difficult to treat and the analysis of their properties requires complicated considerations in statistical mathematics. In this work we propose to use computer algebra software with the aim to solve epidemic stochastic models such as the SIR model and the carrier-borne model. Specifically we use Maple to solve these stochastic models in the case of small groups and we obtain results that do not appear in standard textbooks or in the books updated on stochastic models in epidemiology. From our results we derive expressions which coincide with those obtained in the classical texts using advanced procedures in mathematical statistics. Our algorithms can be extended for other stochastic models in epidemiology and this shows the power of computer algebra software not only for analysis of deterministic models but also for the analysis of stochastic models. We also perform numerical simulations with our algebraic results and we made estimations for the basic parameters as the basic reproductive rate and the stochastic threshold theorem. We claim that our algorithms and results are important tools to control the diseases in a globalized world.

show abstract

Section: Introductionmentioning

confidence: 99%