Dynamic Analysis of a Mathematical Model of the Anti-Tumor Immune Response

This study discusses the dynamic analysis, the Hopf bifurcation, and numerical simulations. The mathematical model of the anti-tumor immune response consists of three compartments namely Immature T Lymphocytes (L1), Mature T Lymphocytes (L2) and Tumor Cells (T). This research was conducted to represent the behavior between immune cells and tumor cells in the body with five γ conditions. Where γ is the intrinsic growth rate of mature T lymphocytes. This study produces R0 > 1 in conditions 1 to 4 while in condition 5 produces R0 < 1. The disease-free equilibrium point is stable only in condition 5, while the endemic equilibrium point is stable only in conditions 2 and 4. Hopf bifurcation occurs at the endemic equilibrium point. Numerical simulation graph in condition 1 shows that tumor cells will increase uncontrollably. In condition 2 the graph show that the endemic equilibrium point for large tumors is stable. In condition 3 the graph show that there will be a bifurcation from the endemic equilibrium point by the disturbance of the parameter value γ. In condition 4 the graph show the small tumor endemic equilibrium point is stable. Finally, in condition 5, the graph show a stable disease-free equilibrium point

A mathematical model for interaction macrophages, T Lymphocytes and Cytokines at infection of mycobacterium tuberculosis with age influence

Tuberculosis (TB) is still a health problem in the world, because of the increasing prevalence and treatment outcomes are less satisfactory. This is presumably due to a complete lack of understanding of the role of the immune system to infecti on Mycobacterium tuberculosis (Mtb). Currently it is known that the immune response plays a role in controlling the development of the germ cells are macrophages, T lymphocytes and cytokines. This study has made a mathematical model of the interaction between macrophages, T lymphocytes and cytokines with Mtb infection in the lungs. Effect of age was observed through disparate data, young (3 months) and old (18 months). Runge Kutta method order-4 is used to solve the system of non-linear differential equations of the first order. Growth of bacteria (extracellular and intracellular) tends to increase up to 3 months, either in old mice and young mice. Behavior of T cells dropped drastically. Concentration of IL-2 and IL-4 is likely to increase at the beginning of infection, TNF-a , IL-10, IFN-g and IL-12 increased. Resting macrophages tend to fall, infected macrophages tended to rise and activated macrophages fluctuate in the first 3 months

Model penyebaran penyakit demam berdarah di Malang

A mathematic model plays an important role in the understanding of epidemiology and the spreader of dengue. This model describes briefly about the non-linear process and the infection process of dengue to a person and the emergence of this disease in certain population. An important definition is taken from –this a dynamic analysis, which is focused on the incidence or prevalence of the spreader of a disease in a certain population. It also includes epidemic process in a steady state population. This research explains mathematic models of the spreader of dengue in Malang that covers the number of outbreaks to cope with DBD, the stability of the disease, the endemic area and the prediction of the number of people that are infected by this disease in the future. The dynamic analysis of the host and vector were derived from the formula of differential equation system, and then defined as mathematic model. From this, we can get basic reproduction number (Ra). It is a number that indicate whether a dynamic system from the model is stable

Analysis Von Bertalanffy equation with variation coefficient

Growth is the increase of size, both its length and weight at a specific time period. In studying the behavior of the growth of fish used a growth model that is Von Bertalanffy models. However Von Bertalanffy models showed that the growth rate is a constant function. Meanwhile these assumptions can only describe the dynamics of growth of marine life in an environment of constant and fail to describe the dynamics of the growth of marine organisms whose growth varies seasonally or by time. Von Bertalanffy models thus developed with a coefficient of variation which gives additional biological realism of Von Bertalanffy models into a population that enables the growth rate by changing the size of the body with seasonal variations. This study aims to find solutions of to the equations Von Bertalanffy Model with a coefficient of variation and actualize the model on Wader pari fish and skipjack, with the incorporation coefficient time varied significantly will increase the the ability of Von Bertalanffy’s model to describe the changes of fish’s body size up to the long asymtot by considering the growth factors such as temperature, water temperature and food availability that exist within the environment

Mathematical model of interactions immune system with Micobacterium tuberculosis

Tuberculosis (TB) remains a public health problem in the world, because of the increasing prevalence and treatment outcomes are less satisfactory. About 3 million people die each year and an estimated one third of the world's population infected with Mycobacterium Tuberculosis (M.tb) is latent. This is apparently related to incomplete understanding of the immune system in infection M.tb. When this has been known that immune responses that play a role in controlling the development of M.tb is Macrophages, T Lymphocytes and Cytokines as mediators. However, how the interaction between the two populations and a variety of cytokines in suppressing the growth of Mycobacterium tuberculosis germ is still unclear. To be able to better understand the dynamics of infection with M tuberculosis host immune response is required of a model.One interesting study on the interaction of the immune system with M.tb mulalui mathematical model approach. Mathematical model is a good tool in understanding the dynamic behavior of a system. With the mediation of mathematical models are expected to know what variables are most responsible for suppressing the growth of Mycobacterium tuberculosis germ that can be a more appropriate approach to treatment and prevention target is to develop a vaccine. This research aims to create dynamic models of interaction between macrophages (Macrophages resting, macrophages activated and macrophages infected), T lymphocytes (CD4 + T cells and T cells CD8 +) and cytokine (IL-2, IL-4, IL-10,IL-12,IFN-dan TNF-) on TB infection in the lung. To see the changes in each variable used parameter values derived from experimental literature. With the understanding that the variable most responsible for defense against Mycobacterium tuberculosis germs, it can be used as the basis for the development of a vaccine or drug delivery targeted so hopefully will improve the management of patients with tuberculosis. Mathematical models used in building Ordinary Differential Equations (ODE) in the form of differential equation systems Non-linear first order, the equation contains the functions used in biological systems such as the Hill function, Monod function, Menten- Kinetic Function. To validate the system used 4th order Runge Kutta method with the help of software in making the program Matlab or Maple to view the behavior and the quantity of cells of each population

Numerical solution for immunology tuberculosis model using Runge Kutta Fehlberg and Adams Bashforth Moulton method

The Immunology tuberculosis model that has been formulated by (Ibarguen, E., Esteva, L., & Chavez, L, 2011) in the form of a system of nonlinear differential equations first order. In this study, we used to Runge Kutta Fehlberg method and Adams Bashforth Moulton method. This study has been obtained numerical solution of the model. The results showed that the relative error obtained from the Adams Bashforth Moulton method is smaller when compared with the Runge Kutta Fehlber method. There are two methods has a high accuracy in solving systems of nonlinear differential equations

Analysis Von Bertalanffy Equation With Variation Coefficient

Growth is the increase of size, both its length and weight at a specific time period. In studying the behavior of the growth of fish used a growth model that is Von Bertalanffy models. However Von Bertalanffy models showed that the growth rate is a constant function. Meanwhile these assumptions can only describe the dynamics of growth of marine life in an environment of constant and fail to describe the dynamics of the growth of marine organisms whose growth varies seasonally or by time. Von Bertalanffy models thus developed with a coefficient of variation which gives additional biological realism of Von Bertalanffy models into a population that enables the growth rate by changing the size of the body with seasonal variations. This study aims to find solutions of to the equations Von Bertalanffy Model with a coefficient of variation and actualize the model on Wader pari fish and skipjack, with the incorporation coefficient time varied significantly will increase the the ability of Von Bertalanffy’s model to describe the changes of fish’s body size up to the long asymtot by considering the growth factors such as temperature, water temperature and food availability that exist within the environment. Keywords: Growth, Von Bertalanffy, Varying Coefficient, actualizatio

Diskritisasi pada sistem persamaan diferensial parsial pola pembentukan sel

Persamaan Meinhardt merupakan sebuah model matematika yang menggambarkan pola pembentukan sel pada hydra. Hans Meinhardt menggunakan jenis persamaan difusi untuk menggambarkan bagaimana variabel-variabel berkembang biak, mati, bergerak dan berinteraksi. Bentuk model yang dirumuskan oleh Meinhardt tersebut merupakan model kontinu, sehingga salah satu studi yang dapat diterapkan pada model Meinhardt adalah dilakukannya diskritrisasi. Diskritisasi merupakan proses kuantisasi sifat-sifat kontinu. Salah satu metode yang dapat memperkirakan bentuk diferensial kontinu menjadi bentuk diskrit ialah metode beda hingga. Sehingga dalam penelitian ini akan dilakukan proses diskritisasi pada model pembentukan sel. Metode yang digunakan adalah beda hingga skema Crank-Nicolson yang merupakan pengembangan dari skema eksplisit dan implisit. Kelebihan dari skema Crank-Nicolson adalah nilai error yang lebih kecil dari pada skema eksplisit dan implisit. Dalam penelitian ini digunakan beda hingga maju untuk turuna

Model matematika pada proses hematopoiesis dengan perlambatan proses proliferasi

Proses produksi sel darah (hematopoiesis) pada kondisi normal diformulasikan dalam bentuk sistem persamaan diferensial nonlinier dengan waktu perlambatan. Waktu perlambatan menunjukkan durasi atau waktu yang diperlukan sel punca berada pada fase proliferasi. Penelitian ini bertujuan untuk menganalisis model matematika pada proses produksi sel darah meliputi analisis titik tetap dan perilaku populasi sel punca hematopoietik. Untuk mempelajari perilaku dinamik model, dilakukan dengan mempelajari persamaan karakteristik dari model tersebut. Hasil simulasi numerik menunjukkan bahwa untuk titik tetap nontrivial model mengalami osilasi. Osilasi pada model matematika proses hematopoiesis mengindikasikan bahwa hematopoiesis yang terjadi tidak stabil sehingga nantinya dapat diimplementasikan pada analisa adanya penyakit-penyakit yang mempengaruhi sel darah

Analisis dinamik model Fitzhugh-Nagumo pada penjalaran impuls sel saraf menggunakan Transformasi Lienard

Model Fitzhugh-Nagumo adalah persamaan diferensial biasa yang menggambarkan rangsangan dan pemulihan beda potensial pada jalannya impuls sel saraf. Berdasarkan suatu aproksimasi, Fitzhugh membuat persamaan yang berasal dari persamaan van der Pol. Dalam penelitian ini digunakan metode transformasi Lienard untuk mengubah persamaan van der Pol menjadi sistem dua dimensi persamaan diferensial biasa nonlinier ordo satu, sehingga didapatkan model BVP (Bonhoeffer van der Pol). Fitzhugh menambahkan ekstra kuantitas pada model BVP, sehingga didapatkan model BVP-FN (Bonhoeffer van der Pol Fitzhugh-Nagumo). Dalam penelitian ini penulis melakukan linierisasi pada model BVP-FN dengan menggunakan ekspansi Taylor dan didapatkan fixed point dan model linier dari BVP-FN. Kemudian penulis melakukan simulasi numerik dan analisis bidang fase dari model BVP-FN. Penulis menggunakan parameter tetap sesuai yang ada di dalam jurnal, yaitu a=0.7,b=0.8, dan c=3, sementara yang divariasikan adalah besarnya arus eksternal (I) yang diberikan. Analisis bidang fase telah berhasil digunakan untuk melihat visualisasi model BVP-FN ketika terjadi perubahan rangsangan atau saat nilai I berubah-ubah. Berdasarkan simulasi numerik dan juga analisis bidang fase model BVP-FN disimpulkan bahwa perilaku dinamika neuron menjadi tidak stabil ketika arus eksternal (I) berada pada interval -1.4<I<-0.4. Sementara pada interval I≥-0.3 dan I≤-1.5 grafik stabil dan menuju ke titik ekuilibrium