This article explains a nonlinear mathematical model of forestry biomass depletion due to human population, population pressure and industrialization. The model also considers crowding by industrialization. Analysis of model shows that equilibrium level the biomass density of forestry resources decreases as the equilibrium level in density of human population, population pressure and industrialization increase. It is found that if the crowding by industrialization increases, then biomass density of forestry resources decreases. Therefore, it is necessary control industrialization to protect the forestry resources stability.

In this article, the homotopy analysis method is applied to obtain approximate analytical solution for a single species population model with viral infection in polluted environment. The resulting solutions are compared with the numerical method. The comparison reveals that our approximate solutions are in very good agreement with the solutions by numerical method. Moreover, The result show that the homotopy analysis method is very e¤ective and simple for solving the nonlinear ordinary di¤erential equation systems.

Improving services through goods pickups and deliveries to expand the market is currently being carried out by many companies. This effort needs another step of optimization to avoid losses due to additional resources, such as time, which must be held by the company. In the case of simultaneous goods pickups and deliveries using a vehicle with single transport access, the policy of arranging goods in the vehicle can affect the length of time for service. Single transport access causes goods that want to be unloaded obstructed by the other goods, thus requires additional time for handling. In this paper, the problem of a vehicle’s route and handling time in picking and delivering goods with time windows is modeled as the Traveling Salesman Problem. The objective function of this problem is to minimize the total travel time and handling time during the picking and delivering goods. The model is implemented in the case of bicycle pickup and delivery services involving 15 bicycle shops. The capacity of the vehicle is 15 units of bicycle. In this case, a solution consisting the pickup and delivery route and optimal goods order is obtained with an objective function value of 334 minutes.

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In this paper, we study an SVEIR disease model of tuberculosis transmission dynamics in which the infected population is divided into two groups, namely infectious infected and noninfectious infected population. The equilibrium points and the basic reproduction number R 0 are determined. The stability analysis of the model was conducted by considering the basic reproduction number R 0. We show that if R 0 < 1, then the disease-free equilibrium is locally asymptotically stable. If R 0 > 1, then a unique endemic equilibrium is locally asymptotically stable.

This paper considers a deterministic model for the dynamics of measles transmission in a population divided into six classes with respect to the disease states: susceptible, vaccinated, exposed, infected, treated, and recovered. First, we investigate the dynamical properties of the SVEITR model such as its equilibrium points, their stability, and parameter sensitivity by applying constant controls. Criteria for determining the stability of disease-free and endemic equilibrium points are provided in terms of basic reproduction number. The model is then extended by incorporating vaccination, therapy, and treatment rates as time-dependent control variables representing the level of coverages. Application of Pontryagin’s maximum principle provides the necessary conditions that must be satisfied for the existence of optimal controls aiming at minimization of the number of exposed and infected individuals simultaneously with the control effort. Numerical simulations that were carried out using the backward sweep method and Runge–Kutta scheme suggest that optimal controls under moderate and high scenarios can effectively reduce the cases of measles. In particular, the moderate scenario that utilizes the existing coverage level of 86% for MCV1 and 69% for MCV2 can degrade the cost functional by 47% of the low scenario. Meanwhile, high scenario that takes the 2020 target of 96% as coverage only makes a slight difference in reducing the number of exposed and infected individuals.

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