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Modeling Highly Random Dynamical Infectious Systems

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Applied Mathematical Analysis: Theory, Methods, and Applications

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 177))

Abstract

Random dynamical processes are ubiquitous in all areas of life:- in the arts, in the sciences, in the social sciences and engineering systems etc. The rates of various types of processes in life are subject to random fluctuations leading to variability in the systems. The variabilities give rise to white noise which lead to unpredictability about the processes in the systems. This chapter exhibits compartmental random dynamical models involving stochastic systems of differential equations, Markov processes, and random walk processes etc. to investigate random dynamical processes of infectious systems such as infectious diseases of humans or animals, the spread of rumours in social networks, and the spread of malicious signals on wireless sensory networks etc. A step-to-step approach to identify, and represent the various constituents of random dynamic processes in infectious systems is presented. In particular, a method to derive two independent environmental white noise processes, general nonlinear incidence rates, and multiple random delays in infectious systems is presented. A unique aspect of this chapter is that the ideas, mathematical modeling techniques and analysis, and the examples are delivered through original research on the modeling of vector-borne diseases of human beings or other species. A unique method to investigate the impacts of the strengths of the noises on the overall outcome of the infectious system is presented. Numerical simulation results are presented to validate the results of the chapter.

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Notes

  1. 1.

    A sub-life cycle may refer to a phase where the organism exhibits different behavioral characteristics.

  2. 2.

    A half-life is a series of intermediary developmental stages of the organism between the egg age and adult or sexual age of the organism.

  3. 3.

    The intensity levels of the white noise processes in the system are described as infinitesimally small, significant in magnitude and small but not infinitesimally small, big in size and finite, and sufficiently large.

  4. 4.

    A seed is set on the random number generator to reproduce the same sequence of random numbers for the Brownian motion in order to generate reliable graphs for the trajectories of the system under different intensity values for the white noise processes, so that comparison can be made to identify differences that reflect the effect of intensity values.

  5. 5.

    That is, \(\sigma _{\beta }=O(1)\).

  6. 6.

    That is \(\sigma _{S}=\theta (\frac{1}{\epsilon })\).

  7. 7.

    That is, \(\sigma _{i}=\theta (\frac{1}{\epsilon }), i= S, E, I, R\).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Georgia Southern University, 65 Georgia Ave, Room 3042, Statesboro, GA, 30460, USA

    Divine Wanduku

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  1. Divine Wanduku
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Correspondence to Divine Wanduku .

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Editors and Affiliations

  1. Department of Mathematics, Gauhati University, Guwahati, India

    Hemen Dutta

  2. Computational Intelligence Laboratory, Department of Electrical & Computer Engineering, University of Manitoba, Winnipeg, MB, Canada

    James F. Peters

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Wanduku, D. (2020). Modeling Highly Random Dynamical Infectious Systems. In: Dutta, H., Peters, J. (eds) Applied Mathematical Analysis: Theory, Methods, and Applications. Studies in Systems, Decision and Control, vol 177. Springer, Cham. https://doi.org/10.1007/978-3-319-99918-0_17

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