Optimal control of the treatment frequency in a stochastic model of tuberculosis
DOI:
https://doi.org/10.11145/j.biomath.2017.05.077Abstract
This paper presents a stochastic model of the Tuberculosis (TB) infection with treatment in a population composed of four individuals compartments: susceptible individuals, latent infected individuals, active infected individuals and recovered individuals after the therapy. The mathematical model of TB infection includes in addition to the deterministic term, a stochastic term that translates the random noise. The random nature of this model is due to the fact that the contraction of the Mycobacterium Tuberculosis, the vector agent of the TB infection and his transmission within the population are done in a random manner according to the variable efficiency of control of the immune system of the individuals. While supposing that only the active infected individuals transmit the infection, their survey in order to observe the rules of hygiene, to adopt a positive behavior with respect to the susceptible individuals (who must also take some precautions) to follow the treatment up to finish, constitute measures of adequate control. A preliminary survey of the model is made before approaching the crucial left of the topic. The main objective of this paper is to control the frequency of treatment in a stochastic model of the TB infection while minimizing the cost of the measures. We formulate an optimal control problem that consists in minimizing the relative cost to the dynamics of the model in order to reduce the prevalence and the mortality due to this infection. The optimal control problem is solved by applying the Projection Stochastic Gradient Method in order to find the optimal numerical solution. Finally, we provide some numerical simulations of the controlled model.Downloads
Published
Issue
Section
License
The journal Biomath is an open access journal. All published articles are immeditely available online and the respective DOI link activated. All articles can be access for free and no reader registration of any sort is required. No fees are charged to authors for article submission or processing. Online publications are funded through volunteer work, donations and grants.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).