Modeling and Analysis of Cholera Dynamics with Vaccination

Cholera is a disease as a kind of acute diarrhea caused by bacteria V. Cholerae. The spread of cholera can be modeled in the form of nonlinear differential equation systems with 4 variables SIRB. This paper aims to research to analyze the local stability of the equilibrium point of the dynamical population in the spread of cholera by the Routh-Hurwitz stability criterion and bifurcation method. Next Generation Matrix (NGM) method is used to get the basic reproductive numbers ( ℜ0 ) to find the local stability at the equilibrium point of the. The disease-free equilibrium point is locally asymptotically stable if ℜ0 < 1, while the endemic equilibrium point is locally asymptotically stable if ℜ0 > 1 the results of numerical simulations obtained ℜ0 = 0 0.87 indicated that the disease-free equilibrium point is locally asymptotically stable. In endemic condition ( ℜ0 > 1 ) show that increasing the rate of vaccination and disinfection can reduce the population of susceptible, infected and bacteria of V. Cholerae.

In this paper, we propose a COVID-19 epidemic model with quarantine class. The model contains 6 sub-populations, namely the susceptible (S), exposed (E), infected (I), quarantined (Q), recovered (R), and death (D) sub-populations. For the proposed model, we show the existence, uniqueness, non-negativity, and boundedness of solution. We obtain two equilibrium points, namely the disease-free equilibrium (DFE) point and the endemic equilibrium (EE) point. Applying the next generation matrix, we get the basic reproduction number (R0). It is found that R0 is inversely proportional to the quarantine rate as well as to the recovery rate of infected subpopulation. The DFE point always exists and if R0 < 1 then the DFE point is asymptotically stable, both locally and globally. On the other hand, if R0 > 1 then there exists an EE point, which is globally asymptotically stable. Here, there occurs a forward bifurcation driven by R0. The dynamical properties of the proposed model have been verified our numerical simulations.

This paper presents a new mathematical model of a tuberculosis transmission dynamics incorporating first and second line treatment. We calculated a control reproduction number which plays a vital role in biomathematics. The model consists of two equilibrium points namely disease free equilibrium and endemic equilibrium point, it has been shown that the disease free equilibrium point was locally asymptotically stable if thecontrol reproduction number is less than one and also the endemic equilibrium point was locally asymptotically stable if the control reproduction number is greater than one. Numerical simulation was carried out which supported the analytical results.&#x0D; Keywords: Mathematical Model, Biomathematics, Reproduction Number, Disease Free Equilibrium, Endemic Equilibrium Point

A four (4) compartmental model of (S, E, I , I ) were presented to have better understanding of parameters that influence the dynamical spread of Ebola in a population. The model is analyzed for all the parameters responsible for the dynamical spread of the disease in order to find the most sensitive parameters that need to be given attention. &#x0D; The stability of the model was analyzed for the existence of disease free and endemic equilibrium points. Basic Reproduction Number ( ) was obtained using next generation matrix method (NGM), and it is shown that the disease free equilibrium point is locally asymptotically stable whenever the basic reproduction number is less than unity i.e ( ) and unstable whenever the basic reproduction number is greater than unity ( ).The relative sensitivity indices of the model with respect to each parameter in the basic reproduction number is calculated in order to find the most sensitive parameter which the medical practitioners and policy health makers should work on in order to reduce the spread of Ebola in the population. The result shows that effective contact rate and fraction of individuals with low immunity are the most sensitive parameters in the reproduction number. Therefore, effort should be put in place so that the basic reproduction number should not be greater unity so as to prevent the endemic situation.

We present in this research work, mathematical modeling of the transmission dynamics of measles using treatment as a control measure. We determined the Disease Free Equilibrium (DFE) point of the model after which we obtained the Basic Reproduction Number ( R0 ) of the model using the next generation approach. The model Endemic Equilibrium (EE) point was also determined after which we performed Local Stability Analysis(LAS) of the Disease Free Equilibrium point and result shows that the Disease Free Equilibrium point of the model would be stable if ( R0 &lt;1). Global Stability Analysis (GAS) result shows that, ( R0 ≤ 1) remains the necessary and sufficient condition for the infection to go into extinction from a population. We carried out Sensitivity Analysis of the model using the Basic Reproduction Number and we discovered that ( δ , μ, ν , θ ) are sensitive parameters that should be targeted towards control intervention strategy as an increase in these values can reduce the value of ( R0 ) to a value less than unity and such can reduce the spread of measles in a population. Model simulation was carried out using mat lab software to support our analytical results.

In this paper, we formulated a new nine (9) compartmental mathematical model to have better understanding of parameters that influence the dynamical spread of Human immunodeficiency virus (HIV) interacting with Tuberculosis (TB) in a population. The model is analyzed for all the parameters responsible for the disease spread in order to find the most sensitive parameters out of all. Sub models of HIV and TB only were considered first, followed by the full HIV-TB co-infection model. Stability of HIV model only, TB model only and full model of HIV-TB co-infection were analyzed for the existence of the disease free and endemic equilibrium points. Basic Reproduction Number (R0) was obtained using next generation matrix method (NGM), and it has been shown that the disease free equilibrium point is locally asymptotically stable whenever R0 &gt; 1and unstable whenever this threshold exceeds unity. i.e.. R0 &gt; 1 The relative sensitivity solutions of the model with respect to each of the parameters is calculated, Parameters are grouped into two categories: sensitive parameters and insensitive parameters. Numerical simulation was carried out by maple software using Runge-kunta method, to show the effect of each parameter on the dynamical spread of HIV-TB co-infection, i.e. detection of infected undetected individuals plays a vital role, it decreases infected undetected individuals. Also, increased in effective contact rate has a pronounced effect on the total population; it decreases susceptible individuals and increases the infected individuals. However, effective contact rate needs to be very low in order to guaranteed disease free environment.

Malaria is an infectious disease caused by the Plasmodium parasite and transmitted between humans through bites of female Anopheles mosquitoes. A mathematical model describes the dynamics of malaria and human population compartments in terms of mathematical equations and these equations represent the relations between relevant properties of the compartments. The aim of the study is to understand the important parameters in the transmission and spread of endemic malaria disease, and try to find appropriate solutions and strategies for its prevention and control by applying mathematical modelling. The malaria model is developed based on basic mathematical modelling techniques leading to a system of ordinary differential equations (ODEs). Qualitative analysis of the model applies dimensional analysis, scaling, and perturbation techniques in addition to stability theory for ODE systems. We also derive the equilibrium points of the model and investigate their stability. Our results show that if the reproduction number, R0, is less than 1, the disease-free equilibrium point is stable, so that the disease dies out. If R0 is larger than 1, then the disease-free equilibrium is unstable. In that case, the endemic state has a unique equilibrium, re-invasion is always possible, and the disease persists within the human population. Numerical simulations have been carried out applying the numerical software Matlab. These simulations show the behavior of the populations in time and the stability of disease-free and endemic equilibrium points.

Diphtheria is an acute disease that affects the upper respiratory tract caused by Corynebacterium diphtheriae, which can also affect the skin, eyes, and other organs. This article analyzes the stability of the SIQR model of diphtheria disease spread in Mandau District by considering the migration factor. The SIQR model is a development of the SIR model by incorporating the quarantine process as an alternative to reduce morbidity. The purpose of this study is to see the effect of migration on the spread of diphtheria disease in Mandau District through mathematical model simulation. We calculated the disease-free and endemic equilibrium points and the basic reproduction number ( ) of the model. Model parameters were obtained using data from BPS Bengkalis Regency and UPTD Puskesmas Mandau. The calculation resulted in one disease-free equilibrium point and one endemic equilibrium point. If &lt; 1, then the disease-free equilibrium point is asymptotically stable, and if &gt; 1, then the endemic equilibrium point is also asymptotically stable. Based on the results of the data analysis, the value of. This value is less than 1, so the equilibrium point obtained is a disease-free and asymptotically stable equilibrium point. This means that the population will be free from diphtheria and the level of migration affects the presence of diphtheria disease in Mandau District.

Abstrak Kanker paru-paru adalah pertumbuhan sel kanker yang tidak terkendali dalam jaringan paru-paru yang disebabkan oleh sejumlah karsinogen lingkungan, terutama asap rokok. Kematian akibat kanker paru-paru sebagian besar disebabkan oleh rokok dan risiko kanker paru-paru meningkat secara signifikan sesuai dengan durasi dan jumlah rokok yang dikonsumsi. Penelitian ini membahas model matematika SEITR pada penyebaran penyakit kanker paru-paru akibat asap rokok. Tujuan penelitian ini yaitu membentuk model matematika, mencari titik kesetimbangan dan angka reproduksi dasar, menganalisis kestabilan titik kesetimbangan, serta simulasi numerik model dengan Maple 18. Dari hasil tersebut diperoleh Teorema 1 yaitu jika maka hanya ada titik kesetimbangan bebas penyakit yang bernilai positif dan jika maka ada titik kesetimbangan bebas penyakit dan endemik yang bernilai positif serta diperoleh Teorema 2 yaitu titik kesetimbangan bebas penyakit stabil asimtotik lokal jika dan titik kesetimbangan endemik stabil asimtotik lokal jika . Selanjutnya dari simulasi numerik pada model dengan menggunakan Maple 18 diperoleh beberapa fakta, yaitu semakin kecil nilai laju individu rentan menjadi perokok ringan dan peluang individu terinfeksi kanker paru-paru serta semakin besar nilai laju individu perokok ringan mengalami kesembuhan alami dan laju individu terinfeksi kanker paru-paru menjalani pengobatan kemoterapi akan mempercepat laju pertumbuhan individu pada setiap subpopulasi stabil pada titik kesetimbangan bebas penyakit, artinya penyakit kanker paru-paru akan semakin cepat menghilang dari populasi. Kata Kunci: Pemodelan Matematika, Rokok, Kanker Paru-Paru. Abstract Lung cancer is the uncontrolled growth of cancer cells in lung tissue caused by a number of environmental carcinogens, particularly cigarette smoke. Lung cancer deaths are mostly caused by smoking, and the risk of lung cancer increases significantly with the duration and number of cigarettes smoked. This study discusses the SEITR mathematical model of the spread of lung cancer by cigarette smoke. The purpose of this study is to formulate a mathematical model, find the equilibrium point and the basic reproduction number, analyze the stability of the equilibrium point, and numerically simulate the model using Maple 18. From these results, Theorem 1 is obtained, namely, if , then there is only a positive disease-free equilibrium point and if , then there is a positive disease-free and endemic equilibrium point, and Theorem 2 is obtained, namely, the disease-free equilibrium point is locally asymptotically stable if and the endemic equilibrium point is locally asymptotically stable if . Furthermore, several facts are obtained from numerical simulations of the model using Maple 18, Namely, the smaller the values of the rate of susceptible individuals becoming light smokers and the probability of individuals being infected with lung cancer , and the larger the values of the rate of individual light smokers experiencing natural recovery and the rate of individuals infected with lung cancer undergoing chemotherapy treatment , the faster the rate of individual growth in each stable subpopulation at the disease-free equilibrium point, i.e., the faster lung cancer will disappear from the population. Keywords: Mathematical Modeling, Cigarette, Lung Cancer.

This article discusses the mathematical model of the spread of COVID-19 by considering vaccination and population migration. The former model is analyzed by determining the equilibrium point, basic reproduction number, analyzing the stability of the equilibrium point, sensitivity analysis, and accompanied by numerical simulation. Analysis of the stability of disease-free and endemic equilibrium points using the Routh-Hurwitz Criteria and the Castillo-Chaves and Song theorems. The results of the analysis show that there are two equilibrium points, namely a disease-free equilibrium point (T1), which is locally asymptotically stable when R0 1, and an endemic equilibrium point (T2), which is locally asymptotically stable when R0 1. Furthermore, the sensitivity analysis showed that the most sensitive parameters to changes in the basic reproduction number were the emigration rate parameter (m2) and the infection probability parameter after contact between infected and susceptible individuals without vaccination (h). In addition, the numerical simulation results show that the sensitive parameter values, namely m2, h, zse, g, and # have a significant effect on the basic reproduction numbers. Suppressing the chance of infection in susceptible individuals and the rate of contact between susceptible and exposed individuals, as well as increasing the number of individuals who emigrate and who are vaccinated, can reduce the transmission of COVID-19.

This article discusses modifications to the SEIL model that involve logistical growth. This model is used to describe the dynamics of the spread of tuberculosis disease in the population. The existence of the model's equilibrium points and its local stability depends on the basic reproduction number. If the basic reproduction number is less than unity, then there is one equilibrium point that is locally asymptotically stable. The equilibrium point is a disease-free equilibrium point. If the basic reproduction number ranges from one to three, then there are two equilibrium points. The two equilibrium points are disease-free equilibrium and endemic equilibrium points. Furthermore, for this case, the endemic equilibrium point is locally asymptotically stable.

In this work a modified SEIR-SVEVIV model that described the dynamics transmission of malaria disease was proposed and analyzed. The standard method is used to analyze the behaviors of the proposed model. The results shown that there were two equilibrium points; disease free and endemic equilibrium point. The qualitative results are depended on a basic reproductive number(R0). We obtained the basic reproductive number by using the next generation method technique and finding the spectral radius. Routh-Hurwitz criteria is used for determining the stabilities of the model. If R0 <1, then the diseases free equilibrium point is local asymptotically stable: that is the disease will died out, but if R0 >1 then the endemic equilibrium is local asymptotically stable. After that the SEIR-SVEVIV model is modified from the first model by adding the optimal control functions that includes two times – dependent control functions with one minimizing the contract between the susceptible human and the infected vector and the other, minimizing the population of the infected human. The result from the numerical solutions of the models are shown and compared for supporting the analytic results.

This paper discusses the stability analysis of the model for the spread of tuberculosis and the effects of treatment. The authors analyze the dynamic behavior of the model to investigate the local stability properties of the model equilibrium point. The Routh-Hurwitz criterion is used to analyze local stability at the disease-free equilibrium point, while the Transcritical Bifurcation theorem is used to investigate the local stability properties of the endemic equilibrium point. The results of the discussion show that the stability properties of the equilibrium point depend on the value of the basic reproduction number which is calculated based on the Next Generation Matrix (NGM). When the basic reproduction number value is less than one, then the disease-free equilibrium point is locally asymptotically stable, whereas if it is more than one, then the endemic equilibrium point is locally asymptotically stable. Numerical simulations are included to explain the dynamic behavior of disease spread and to understand the effectiveness of tuberculosis treatment in a given population. The simulation results show that treatment in the infected individual phase is known to be more effective than treatment in latent individuals.

Human Immunodeficiency Virus (HIV) co-existing with Tuberculosis (TB) in individuals remains a major global health challenges, with an estimated 1.4 million patients worldwide. These two diseases are enormous public health burden, and unfortunately, not much has been done in terms of modeling the dynamics of HIV-TB co-infection at a population level. We formulated new fifteen (15) compartmental models to gain more insight into the effect of treatment and detection of infected undetected individuals on the dynamical spread of HIV- TB co-infection. Sub models of HIV and TB only were considered first, followed by the full HIV-TB co-infection model. Existence and uniqueness of HIV and TB only model were analyzed quantitatively, and we shown that HIV model only and TB only model have solutions, moreover, the solutions are unique. Stability of HIV model only, TB model only and full model of HIV-TB co-infection were analyzed for the existence of the disease free and endemic equilibrium points. Basic reproduction number () was analyzed, using next generation matrix method (NGM), and it has been shown that the disease free equilibrium point is locally asymptotically stable whenever and unstable whenever this threshold exceeds unity. i.e., Numerical simulation was carried out by maple software using differential transformation method, to show the effect of treatment and detection of infected undetected individuals on the dynamical spread of HIV-TB co-infection. Significantly, all the results obtained from this research show the importance of treatment and detection of infected undetected individuals on the dynamical spread of HIV-TB co-infection. Detection rate of infected undetected individuals reduce the spread of HIV-TB co-infections.

Wiener and Lévy processes to prevent disease outbreaks: Predictable vs stochastic analysis