Deterministic and Stochastic Schistosomiasis Models with General Incidence

Abstract

In this paper, deterministic and stochastic models for schistosomiasis involving four sub-populations are developed. Conditions are given under which system exhibits thresholds behavior. The disease-free equilibrium is globally asymptotically stable if and the unique endemic equilibrium is globally asymptotically stable when . The populations are computationally simulated under various conditions. Comparisons are made between the deterministic and the stochastic model.

Optimal control of the treatment frequency in a stochastic model of tuberculosis

“…Under a given condition, the model (1) admits a random endemic equilibrium exponentially p-stable (p ≥ 2) and globally stable. The introduction of a treatment control function in model (1) gives an optimal control problem governed by model (22). The Projection Gradient method permits to determine numerically the optimal control as well as the cost function corresponding to this problem.…”

“…The Projection Gradient Method applied to the stochastic model of TB with control, consist therefore in considering the system (30) of two equations (22) and (28) in order to solve it numerically,…”

“…The diagram of the stochastic model of the TB infection is given by Fig 1. The mathematical stochastic model of TB infection in the presence of treatment is written under the compact form by the following equation (1) (its formulation uses [1], [5], [6], [7], and [22])…”

“…In epidemiology and others domain as biology, demography, economy..., many stochastic models deriving from their deterministic formulation. The reference of the literature for a variety of wellknown stochastic models deriving from their deterministic counterparts include the books [1], [5], [6], [7], and [22]. Our contribution is first in Sub Section II.A, the formulation of a stochastic model of TB with treatment from a deterministic model of TB-only (Sharomi [18]) which is formulated along the lines of the model in Feng and al.…”