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1

Analysis of global dynamics for HIV-infection of CD4+T cells and its treatment

Bodzioch M., Choiński M., Foryś U.

Mathematica Applicanda

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2018

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Vol. 46, no. 1

35--48

EN

Antiviral therapy for HIV-infected patients has greatly improved in recent years. Administration of drug combinations consisting of two or more different drugs can reduce and maintain virus load below detection level in many patients. Cyclic administration of the immune activator interleukin-2 (IL-2) in combination with highly active antiretroviral therapy (HAART) has been suggested as an effective strategy to realize long-term control of HIV replication in vivo. In this article, we formulate a mathematical model of the immune response for HIV-infected individual in the presence of HAART and IL-2. We look for the conditions under which the immune system recovers by applying IL-2 as an immune activator along with HAART. From the analytical point of view this means global stability of the disease-free equilibrium. Comprehensive numerical simulations are presented to illustrate the analytical results.

PL

Przeciwwirusowe terapie dla pacjentów z wirusem HIV są stale ulepszane. Podawanie kombinacji dwóch lub więcej leków powoduje spadek liczby wirusów poniżej poziomu detekcji u wielu pacjentów. W szczególności połączenie wysoce efektywnej terapii antyretrowirusowej (HAART) z aktywatorem immunologicznym (interleukina 2, IL-2) wydaje się stanowić skuteczną metodę długookresowej kontroli replikacji wirusa HIV in vivo. W naszym artykule zaproponowaliśmy matematyczny model odpowiedzi odpornościowej dla pacjentów z wirusem HIV przy zastosowaniu połączonej terapii HAART i IL-2. Zbadaliśmy dynamikę modelu w poszukiwaniu warunków, przy których układ odpornościowy odnawia się dzięki takiej terapii. Z analitycznego punktu widzenia oznacza to globalną stabilność stanu stacjonarnego odpowiadającego zdrowemu organizmowi. Analiza matematyczna została uzupełniona symulacjami komputerowymi.

2

Analysis of a predator-prey model with disease in the predator species

Radziński P., Foryś U.

Mathematica Applicanda

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2018

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Vol. 46, no. 1

137--147

EN

In the paper we analyse a diffusive predator-prey model with disease In predator species proposed by Qiao et al.(2014). In the original article there appears a mistake in the procedure of the model undimensionalisation. We make a correction in this procedure and show that some changes in the model analysis are necessary to obtain results similar to those presented by Qiao et al.(2014) We propose corrected conditions for global stability of one of the existing equilibria – disease free steady state and endemic state in the case without diffusion as well as in the model with diffusion. On the basis of the corrected analysis we present New stability results.

PL

Przeanalizowaliśmy dyfuzyjny model drapieżnik-ofiara z chorobą w populacji drapieżników zaproponowany przez Qiao et al.(2014). W oryginalnym artykule popełniono błąd podczas ubezwymiarowienia modelu. W niniejszej publikacji poprawiliśmy błędy w ubezwymiarowieniu i pokazaliśmy, że małe zmiany są potrzebne aby uzyskać wyniki podobne do wyników uzyskanych przez Qiao et al.(2014). Zaproponowaliśmy poprawione warunki na globalną stabilność jednego z istniejących stanów stacjonarnych – stanu wolnego od choroby lub stanu endemicznego, zarówno w przypadku z dyfuzją jak i bez niej. Na podstawie poprawionej analizy zaproponowaliśmy nowe warunki na stabilność.

3

A modified van der Pol equation with delay in a description of the heart action

Zduniak B., Bodnar M., Foryś U.

International Journal of Applied Mathematics and Computer Science

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2014

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Vol. 24, no. 4

853--863

EN

In this paper, a modified van der Pol equation is considered as a description of the heart action. This model has a number of interesting properties allowing reconstruction of phenomena observed in physiological experiments as well as in Holter electrocardiographic recordings. Our aim is to study periodic solutions of the modified van der Pol equation and take into consideration the influence of feedback and delay which occur in the normal heart action mode as well as in pathological modes. Usage of certain values for feedback and delay parameters allows simulating the heart action when an accessory conducting pathway is present (Wolff–Parkinson–White syndrome).

4

Stability analysis and comparison of the models for carcinogenesis mutations in the case of two stages of mutations

Foryś U.

Journal of Applied Analysis

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2005

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Vol. 11, nr 2

283-302

EN

The present paper is focused on the analysis of three very simple models of carcinogenesis mutations that are based on reaction-diffusion systems and Lotka-Volterra food chains. We consider the case with two stages of mutations and study the systems of three reaction-diffusion equations with zero-flux boundary conditions. We focus on the Turing instability and show that this type of instability is not possible for these models. We also propose some modifications of the considered equations. Results are illustrated by computer simulations.

5

Logistic Equations in Tumour Growth Modelling

Foryś U., Marciniak-Czochra A.

International Journal of Applied Mathematics and Computer Science

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2003

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Vol. 13, no 3

317-325

EN

The aim of this paper is to present some approaches to tumour growth modelling using the logistic equation. As the first approach the well-known ordinary differential equation is used to model the EAT in mice. For the same kind of tumour, a logistic equation with time delay is also used. As the second approach, a logistic equation with diffusion is proposed. In this case a delay argument in the reaction term is also considered. Some mathematical properties of the presented models are studied in the paper. The results are illustrated using computer simulations.

6

Modele matematyczne w epidemiologii i immunologii

Foryś U.

Matematyka Stosowana : matematyka dla społeczeństwa

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2000

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Nr 1

35-67

7

Behaviour of Solutions to Marchuk's Model Depending on a Time Delay

Bodnar M., Foryś U.

International Journal of Applied Mathematics and Computer Science

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2000

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Vol. 10, no 1

97-112

EN

Marchuk's model of an immune reaction is a system of differential equations with a time delay. The aim of this paper is to study the behaviour of solutions to Marchuk's model depending upon the delay of immune reaction and the history of an illness. We study Marchuk's model without delays, with aconstant delay and with an infinite delay. A continuous dependence on thedelay is considered. Bifurcation points are found using computer simulations.