A Review of Matrix SIR Arino Epidemic Models

Many of the models used nowadays in mathematical epidemiology, in particular in COVID-19 research, belong to a certain subclass of compartmental models whose classes may be divided into three “(x,y,z)” groups, which we will call respectively “susceptible/entrance, diseased, and output” (in the classic SIR case, there is only one class of each type). Roughly, the ODE dynamics of these models contains only linear terms, with the exception of products between x and y terms. It has long been noticed that the reproduction number R has a very simple Formula in terms of the matrices which define the model, and an explicit first integral Formula is also available. These results can be traced back at least to Arino, Brauer, van den Driessche, Watmough, and Wu (2007) and to Feng (2007), respectively, and may be viewed as the “basic laws of SIR-type epidemics”. However, many papers continue to reprove them in particular instances. This motivated us to redraw attention to these basic laws and provide a self-contained reference of related formulas for (x,y,z) models. For the case of one susceptible class, we propose to use the name SIR-PH, due to a simple probabilistic interpretation as SIR models where the exponential infection time has been replaced by a PH-type distribution. Note that to each SIR-PH model, one may associate a scalar quantity Y(t) which satisfies “classic SIR relations”, which may be useful to obtain approximate control policies.

Characterizing Forest Dynamics with Landsat-Derived Phenology Curves

Landsat is among the most popular satellites used for forest change assessments. Traditionally, Landsat data users relied on annual or biennial images to measure forest recovery after disturbance, a process that is difficult to monitor at broad scales. With the availability of free Landsat data, intra-annual change analyses are now possible. Phenology, the timing of cyclical vegetation events, can be estimated using indices derived from intra-annual remote sensing data and used to classify different vegetation types after a disturbance. We used a smoothed harmonic modelling approach to estimate NDVI and NBR phenology patterns in pre- and post-fire Landsat sample pixels for two forest groups in South Carolina, using nearby unburned samples as an approximate control group. These methods take advantage of all available images collected by Landsat 5, 7, and 8 for the study area. We found that within burned samples, there were differences in phenology for the two forest groups, while the unburned samples showed no forest group differences. Phenology patterns also differed based on fire severity. These methods take advantage of the freely available Landsat archive and can be used to characterize intra-annual fluctuations in vegetation following a variety of disturbances in the southeastern U.S. and other regions. Our approach builds on other harmonic approaches that use the Landsat archive to detect forest change, such as the Continuous Change Detection and Classification (CCDC) algorithm, and provides a tool to describe post-disturbance forest change.

ON APPROXIMATE OPTIMAL CONTROL FOR THE REACTION-DIFFUSION PROCESS IN MICROINHOMOGENEOUS MEDIUM

The problem of constructing an approximate optimal control for controlled processes of chemical kinetics in microinhomogeneous medium is considered. Such processes are described by semilinear parabolic equations of the reaction-diffusion type with coefficients of the form . The preference of an approximate control as the optimal control in the problem with averaged coefficients is justified. An example of the construction of such a control is discussed and its efficiency is demonstrated.

Automatic wireless Monitoring and Controlling of Greenhouse using multiple sensors

Many parameters like temperature, soil moisture, light intensity, Humidity, Carbon dioxide (CO2) leads to the healthy growth of plants in greenhouse environment. Observing only few of those leads to improper growth of plants and minimize the yields. Every grower cannot visit the field and observe the parameters continuously. In order to monitor the parameters and give the approximate control to the greenhouse, we proposed this system. This system continuously monitors the plants and communicate the information to the grower through wireless Sensor Network (WSN), thus reducing the risk of staying at the field. The proposed system has three stations - Transmitter Station (TS), Control Station (CS), and Communication Station (CMS). The ZigBee plays a major role by enabling communication between the three stations. This implementation supports the farmers to simplify the management and to increase the crop production. The overall system has shown the benefits in price, volume, and strength.

Adaptive control parameterization method by density functions for optimal control problems

This paper proposes an efficient adaptive control parameterization method for solving optimal control problems. In this method, mesh density functions are used to generate mesh points. In the first step, the problem is solved by control parameterization on uniform mesh points. Then at each step, the approximate control obtained from the previous step is applied to construct a mesh density function, and consequently a new adapted set of mesh points. Several numerical examples are included to demonstrate that the adaptive control parameterization method is more accurate than a uniform control parameterization one.