Dynamic modelling and analysis of human brucellosis on age heterogeneity in Jinzhou, China.

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.
 Keywords: Mathematical Model, Biomathematics, Reproduction Number, Disease Free Equilibrium, Endemic Equilibrium Point

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.

We develop a mathematical model for the dynamics of Cassava Mosaic Disease (CMD), which is driven by both planting of infected cuttings and whitefly transmission. We use the model to analyze the dynamics of a CMD outbreak and to identify the most cost-effective policy for controlling it. The model uses the reproduction number $ \mathscr{R}_0 $ as a threshold, calculated using the Next-Generation Method. A locally-asymptotically-stable disease-free equilibrium is established when $ \mathscr{R}_0 < 1 $, proved by the Routh-Hurwitz criterion. The globally-asymptotically-stable disease-free and endemic-equilibrium points are obtained using Lyapunov's method and LaSalle's invariance principle. Our results indicate that the disease-free equilibrium point is globally-asymptotically-stable when $ \mathscr{R}_0 \leq 1 $, while the endemic-equilibrium point is globally-asymptotically-stable when $ \mathscr{R}_0 > 1 $. Our sensitivity analysis shows that $ \mathscr{R}_0 $ is most sensitive to the density of whitefly. Numerical simulations confirmed the effectiveness of whitefly control for limiting an outbreak while minimizing costs.

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.

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 is aimed to develop a new COVID-19 mathematical model involving viruses in the environment. In this mathematical model, the human population is divided into five subpopulations: susceptible, exposed, infected, hospitalized, and cured individuals. In addition, the model also contains the virus population in the environment. Infection in the model occurs due to interactions between susceptible individual subpopulations and infected individuals and hospitalizations, as well as the spread of the virus in the environment. Based on the results of dynamic analysis, this model has two equilibrium points, the disease-free and endemic equilibrium points. The disease-free equilibrium point always exists, and both equilibrium points are locally asymptotically stable if they meet the Routh-Hurwitz criteria. Model sensitivity analysis was carried out on model parameters that affect the basic reproduction number with the most sensitive parameters are the natural death rate, the recruitment rate, the transmission rate of the virus in the environment, the virus clearance rate, and the rate of wearing PPE (Personal Protective Equipment), as well as the parameter that does not affect the basic reproduction number that is the rate of leaving the recovered population. Numerical simulations performed show results in accordance with the analysis, also from the simulations can be concluded that the increase (or decrease) of the transmission rate of the virus in an environment that has a higher sensitivity index has more significant influences on the basic reproduction number and the number of infected population than the transmission rate of hospitalized individuals. Keywords: Basic Reproduction Number, Dynamics Analysis, Epidemic Models of COVID-19, Local Stability Analysis, Sensitivity Analysis.

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 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.

A foodborne disease called listeriosis is brought on by the bacteria Listeria monocytogenes which typically infects people after consuming contaminated food. Listeriosis mostly affects people with weakened immune systems, pregnant women and newborns. In this paper, we developed and analyzed a risk-structured mathematical model describing the dynamics of Listeriosis using ordinary differential equations. Three equilibrium points were obtained, viz; disease free equilibrium point, , bacteria free equilibrium point, , and endemic equilibrium point, . Contaminated food threshold was established as . The disease-free equilibrium and Bacteria-free equilibrium points are found to be locally asymptotically stable whenever the contaminated food threshold is less than unity (). Also, the endemic equilibrium point is found to be locally asymptotically stable using the Routh-Hurwitz criterion whenever the food safety index is less than unity (). Global stability analysis of the disease-free equilibrium point using Castillo-Chavez method revealed that the disease-free equilibrium point, is globally asymptotically stable.

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.

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.

Successful treatment of COVID-19 that outbroke worldwide since the beginning of 2020 has demonstrated the importance of effective isolation, which is aimed at asymptomatic and symptomatic infected persons in the incubation period. In this paper, to further analyze the transmission dynamics behavior of epidemics with the latent state, we construct a class of health state - latent state - infection - recovery state (SEIR) infectious disease model with heterogeneity and time delay characteristic based on considering the nonlinear incidence rate formed by psychological inhibition factors. Also, the dynamics of the epidemic, the threshold condition, and stability are studied by creating Lyapunov functions reasonably, applying LaSalle's Invariance Principle and mean-field equation theory. The research shows that, the basic reproduction number R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> of the system depends on birth rate, death rate, recovery rate, disease transmission rate, and network topology. If R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> <; 1, the system is stable at the disease-free equilibrium point E <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup> , and if R <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> > 1, the system is sound at the endemic equilibrium point E*. Moreover, it is also proved that latent delay and psychological inhibitory factors can influence the peak and rate of the infected nodes in the system before their convergence to the equilibrium point, but not the system's global stability. Meanwhile, the theoretical results are verified by numerical simulation finally.

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

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.

The outbreak of COVID-19 was first experienced in Wuhan City, China, during December 2019 before it rapidly spread over globally. This paper has proposed a mathematical model for studying its transmission dynamics in the presence of face mask wearing and hospitalization services of human population in Tanzania. Disease-free and endemic equilibria were determined and subsequently their local and global stabilities were carried out. The trace-determinant approach was used in the local stability of disease-free equilibrium point while Lyapunov function technique was used to determine the global stability of both disease-free and endemic equilibrium points. Basic reproduction number, , was determined in which its numerical results revealed that, in the presence of face masks wearing and medication services or hospitalization as preventive measure for its transmission, while in their absence . This supports its analytical solution that the disease-free equilibrium point is asymptotically stable whenever , while endemic equilibrium point is globally asymptotically stable for . Therefore, this paper proves the necessity of face masks wearing and hospitalization services to COVID-19 patients to contain the disease spread to the population.