Sampling Distribution: Impact of the Population Reliability on the Sample Size Determination

One of the spiniest problems in the elementary theory of the sampling is the determination of the size of the sample. Several authors tried to answer this question. In this paper, we present the methods of determination of the sample size in order to estimate to a given precision the   population mean, the variance of the population being known or unknown. In the same way, we present the methods of determination of the size of the sample in order to estimate to a given precision the frequency of a character in the population at different level of its reliability denoted here by.the method uses the sampling error of the distribution of the specified statistic.  Notice that the bigger the population reliability (or 100%), the smaller is the necessary sample size to get the best estimation of the parameter. The representativeness of the population doesn't essentially depend on the size of the sample; the size of the sample influences the precision of the measures; to achieve a highest accuracy, it is necessary to increase the sample size. The results of this research will help in determining the sample size to be drawn from different populations (finite or finite).

Solving an optimal control problem of hepatitis B virus dynamics: Efficacy of fuzzy logic strategy

This work aims at using fuzzy logic strategy to solve a hepatitis B virus (HBV) optimal control problem. To test the efficacy of this numerical method, we compare numerical results with those obtained using direct method. We consider a patient under treatment during 12 months where the two drugs are taken as controls. The results are rather satisfactory. In particular, the reaction of HBV to drugs can be modeled and a feedback can be approximated by the solution of a linear quadratic problem. The drugs reduce the risk of HBV. Furthermore, results of both numerical methods are in good agreement with experimental data and this justifies the efficacy of fuzzy logic strategy in solving optimal control problems

Mathematical modelling of the effects of peer-educators’ campaign on the dynamics of HIV/AIDS in Rwanda

In this paper, we analyse the effects of peer-educator’s campaign on the dynamics of HIV. We present a sex-structured model for heterosexual transmission of HIV/AIDS in a community. The model is formulated using integro-differential equations, which help to account for a time delay due to incubation period of infective before developing AIDS. The sex-structured HIV/AIDS model divides the population into two subpopulations, namely; females and males. Both disease Free equilibrium and the endemic equilibrium points for the model are determined and their stability are examined. The model is extended to assess the effect of peer- educational campaigns in slowing or eradicating the epidemic. The exposure risk of infection after each intervention is obtained. Basic reproductive numbers for these models are computed and compared to assess the effectiveness of each intervention in a community. The models are numerically analyzed to assess the effectiveness of the treatment free measure, namely; peer educational campaign on the transmission dynamics of HIV/AIDS using demographic and epidemiological parameters of Rwanda. The study demonstrates the use of sex-structured HIV/AIDS models in assessing the effectiveness of educational campaigns as a preventive strategy in a heterosexually active populationMathematics Subject classifications (MSC 2010): 34D20, 34K60, 92D25, 34K25, 34K28Keywords: Population dynamics, Stability, Basic reproductive numbers, Equilibriu

Morphometric characteristics of four important fish species from the Rwandan side of Lake Kivu

Inland fisheries are important socio-economic resources for rural communities in Africa. In Rwanda, about two million people directly depend on capture fisheries from Lake Kivu. Sustainable fisheries management requires monitoring of population dynamics that help devise effective intervention strategies. However, the relevant data such as length-length and length-weight relationships do not exist for most of the key species in Lake Kivu. The present study generated morphometric and body weight data for Limnothrissa miodon (n=82), Lamprichthys tanganicanus (n=55), Haplochromis vittatus (n=51) and Haplochromis graueri (n=34) from Lake Kivu. Length-weight relationships (W=aLb) were highly significant for all species (p < 0.0001), high adjusted R2 = 78 to 98% of the variance in weight explained by total length. Negative allometric length-weight relationships were found with regression slope (b) values of 2.7 for Limnothrissa miodon, 2.8 for H. graueri, and 2.8 for L. tanganicanus, and positive (b=3.4) for Haplochromis vittatus. Length-length relationships were highly significant for all species (p ≤ 0.0001), except for the relationship between total and caudal length for L. miodon, and total and caudal peduncle length for L. tanganicanus, both not significant at p > 0.05. The highest R2 was for the relationships between total and standard length for H. vittatus, and total and fork length for L. tanganicanus. No description of length-length relationships exists for these species in literature or in FishBase database. The data generated in the current study will be used to assess changes in fish populations over time

Mathematical Modelling of Glucose-Insulin System and Test of Abnormalities of Type 2 Diabetic Patients

This paper presents a mathematical model of glucose-insulin dynamics which is specific for type 2 diabetic patients. The general modelling is obtained by simplification of a global compartmental model by John Thomas Sorensen. The model parameters are estimated using nonlinear optimization and data collected in Rwanda for type 2 diabetic patients. In order to identify and evaluate possible abnormalities of type 2 diabetic patients, the Sampling Importance Resampling (SIR) particle filtering algorithm is used and implemented through discretization of the developed mathematical model. This process is done by clamping insulin and glucose concentrations at around clinical trial values as proposed by Defronzo. Furthermore, for detecting potential abnormalities in type 2 diabetic patients, we compare our results with results obtained from a simulation of the mathematical model for healthy subjects. The proposed mathematical model allows further investigation of the dynamic behavior of glucose, insulin, glucagon, stored insulin, and labile insulin in different organs for type 2 diabetic patients