Research

15 pages, 1173 KiB

Open AccessArticle

Antiangiogenic Therapy Efficacy Can Be Tumor-Size Dependent, as Mathematical Modeling Suggests

by Maxim Kuznetsov and Andrey Kolobov

Mathematics 2024, 12(2), 353; https://doi.org/10.3390/math12020353 - 22 Jan 2024

Viewed by 679

Antiangiogenic therapy (AAT) is an indirect oncological modality that is aimed at the disruption of cancer cell nutrient supply. Invasive tumors have been shown to possess inherent resistance to this treatment, while compactly growing benign tumors react to it by shrinking. It is [...] Read more.

Antiangiogenic therapy (AAT) is an indirect oncological modality that is aimed at the disruption of cancer cell nutrient supply. Invasive tumors have been shown to possess inherent resistance to this treatment, while compactly growing benign tumors react to it by shrinking. It is generally accepted that AAT by itself is not curative. This study presents a mathematical model of non-invasive tumor growth with a physiologically justified account of microvasculature alteration and the biomechanical aspects of importance during tumor growth and AAT. In the untreated setting, the model reproduces tumor growth with saturation, where the maximum tumor volume depends on the level of angiogenesis. The outcomes of the AAT simulations depend on the tumor size at the moment of treatment initiation. If it is close to the stable size of an avascular tumor grown in the absence of angiogenesis, then the tumor is rapidly stabilized by AAT. The treatment of large tumors is accompanied by the displacement of normal tissue due to tumor shrinkage. During this, microvasculature undergoes distortion, the degree of which depends on the displacement distance. As it affects tumor nutrient supply, the stable size of a tumor that undergoes AAT negatively correlates with its size at the beginning of treatment. For sufficiently large initial tumors, the long-term survival of tumor cells is compromised by competition with normal cells for the severely limited inflow of nutrients, which makes AAT effectively curative. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis of Problems in Ecology, Epidemiology and Oncology)

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22 pages, 4923 KiB

Open AccessArticle

Modelling Infectious Disease Dynamics: A Robust Computational Approach for Stochastic SIRS with Partial Immunity and an Incidence Rate

by Amani S. Baazeem, Yasir Nawaz, Muhammad Shoaib Arif, Kamaleldin Abodayeh and Mae Ahmed AlHamrani

Mathematics 2023, 11(23), 4794; https://doi.org/10.3390/math11234794 - 27 Nov 2023

Cited by 2 | Viewed by 839

Abstract

For decades, understanding the dynamics of infectious diseases and halting their spread has been a major focus of mathematical modelling and epidemiology. The stochastic SIRS (susceptible–infectious–recovered–susceptible) reaction–diffusion model is a complicated but crucial computational scheme due to the combination of partial immunity and [...] Read more.

For decades, understanding the dynamics of infectious diseases and halting their spread has been a major focus of mathematical modelling and epidemiology. The stochastic SIRS (susceptible–infectious–recovered–susceptible) reaction–diffusion model is a complicated but crucial computational scheme due to the combination of partial immunity and an incidence rate. Considering the randomness of individual interactions and the spread of illnesses via space, this model is a powerful instrument for studying the spread and evolution of infectious diseases in populations with different immunity levels. A stochastic explicit finite difference scheme is proposed for solving stochastic partial differential equations. The scheme is comprised of predictor–corrector stages. The stability and consistency in the mean square sense are also provided. The scheme is applied to diffusive epidemic models with incidence rates and partial immunity. The proposed scheme with space’s second-order central difference formula solves deterministic and stochastic models. The effect of transmission rate and coefficient of partial immunity on susceptible, infected, and recovered people are also deliberated. The deterministic model is also solved by the existing Euler and non-standard finite difference methods, and it is found that the proposed scheme forms better than the existing non-standard finite difference method. Providing insights into disease dynamics, control tactics, and the influence of immunity, the computational framework for the stochastic SIRS reaction–diffusion model with partial immunity and an incidence rate has broad applications in epidemiology. Public health and disease control ultimately benefit from its application to the study and management of infectious illnesses in various settings. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis of Problems in Ecology, Epidemiology and Oncology)

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26 pages, 2843 KiB

Open AccessArticle

Mathematical Model Predicting the Kinetics of Intracellular LCMV Replication

by Julia Sergeeva, Dmitry Grebennikov, Valentina Casella, Paula Cebollada Rica, Andreas Meyerhans and Gennady Bocharov

Mathematics 2023, 11(21), 4454; https://doi.org/10.3390/math11214454 - 27 Oct 2023

Viewed by 982

Abstract

The lymphocytic choriomeningitis virus (LCMV) is a non-cytopathic virus broadly used in fundamental immunology as a mouse model for acute and chronic virus infections. LCMV remains a cause of meningitis in humans, in particular the fatal LCMV infection in organ transplant recipients, which [...] Read more.

The lymphocytic choriomeningitis virus (LCMV) is a non-cytopathic virus broadly used in fundamental immunology as a mouse model for acute and chronic virus infections. LCMV remains a cause of meningitis in humans, in particular the fatal LCMV infection in organ transplant recipients, which highlights the pathogenic potential and clinical significance of this neglected human pathogen. Paradoxically, the kinetics of the LCMV intracellular life cycle has not been investigated in detail. In this study, we formulate and calibrate a mathematical model predicting the kinetics of biochemical processes, including the transcription, translation, and degradation of molecular components of LCMV underlying its replication in infected cells. The model is used to study the sensitivity of the virus growth, providing a clear ranking of intracellular virus replication processes with respect to their contribution to net viral production. The stochastic formulation of the model enables the quantification of the variability characteristics in viral production, probability of productive infection and secretion of protein-deficient viral particles. As it is recognized that antiviral therapeutic options in human LCMV infection are currently limited, our results suggest potential targets for antiviral therapies. The model provides a currently missing building module for developing multi-scale mathematical models of LCMV infection in mice. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis of Problems in Ecology, Epidemiology and Oncology)

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17 pages, 339 KiB

Open AccessArticle

Optimal Treatment of Prostate Cancer Based on State Constraint

by Wenhui Luo, Xuewen Tan, Xiufen Zou and Qing Tan

Mathematics 2023, 11(19), 4025; https://doi.org/10.3390/math11194025 - 22 Sep 2023

Viewed by 765

Abstract

As a new tumor therapeutic strategy, adaptive therapy involves utilizing the competition between cancer cells to suppress the growth of drug-resistant cells, maintaining a certain tumor burden. However, it is difficult to determine the appropriate time and drug dose. In this paper, we [...] Read more.

As a new tumor therapeutic strategy, adaptive therapy involves utilizing the competition between cancer cells to suppress the growth of drug-resistant cells, maintaining a certain tumor burden. However, it is difficult to determine the appropriate time and drug dose. In this paper, we consider the competition model between drug-sensitive cells and drug-resistant cells, propose the problem of drug concentration, and provide two state constraints: the upper limit of the maximum allowable drug concentration and the tumor burden. Using relevant theories, we propose the best treatment strategy. Through a numerical simulation and quantitative analysis, the effects of drug concentrations and different tumor burdens on treatments are studied, and the effects of cell-to-cell competitive advantage on cell changes are taken into account. The clinical dose titration method is further simulated; the results show that our therapeutic regimen can better suppress the growth of drug-resistant cells, control the tumor burden, limit drug toxicity, and extend the effective treatment time. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis of Problems in Ecology, Epidemiology and Oncology)

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13 pages, 1094 KiB

Open AccessFeature PaperArticle

Predictability of Population Fluctuations

by Rodrigo Crespo-Miguel and Francisco J. Cao-García

Mathematics 2022, 10(17), 3176; https://doi.org/10.3390/math10173176 - 03 Sep 2022

Cited by 1 | Viewed by 1303

Abstract

Population dynamics is affected by environmental fluctuations (such as climate variations), which have a characteristic correlation time. Strikingly, the time scale of predictability can be larger for the population dynamics than for the underlying environmental fluctuations. Here, we present a general mechanism leading [...] Read more.

Population dynamics is affected by environmental fluctuations (such as climate variations), which have a characteristic correlation time. Strikingly, the time scale of predictability can be larger for the population dynamics than for the underlying environmental fluctuations. Here, we present a general mechanism leading to this increase in predictability. We considered colored environmental fluctuation acting on a population close to equilibrium. In this framework, we derived the temporal auto and cross-correlation functions for the environmental and population fluctuations. We found a general correlation time hierarchy led by the environmental-population correlation time, closely followed by the population autocorrelation time. The increased predictability of the population fluctuations arises as an increase in its autocorrelation and cross-correlation times. These increases are enhanced by the slow damping of the population fluctuations, which has an integrative effect on the impact of correlated environmental fluctuations. Therefore, population fluctuation predictability is enhanced when the damping time of the population fluctuation is larger than the environmental fluctuations. This general mechanism can be quite frequent in nature, and it largely increases the perspectives of making reliable predictions of population fluctuations. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis of Problems in Ecology, Epidemiology and Oncology)

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29 pages, 1165 KiB

Open AccessArticle

Mathematical Modelling of Harmful Algal Blooms on West Coast of Sabah

by Fatin Nadiah Yussof, Normah Maan, Mohd Nadzri Md Reba and Faisal Ahmed Khan

Mathematics 2022, 10(16), 2836; https://doi.org/10.3390/math10162836 - 09 Aug 2022

Viewed by 1298

Abstract

Algal bloom is a condition in which there is a massive growth of algae in a certain region and it is said to be harmful when the bloom causes damage effects. Due to the tremendous impact of harmful algal bloom (HAB) on some [...] Read more.

Algal bloom is a condition in which there is a massive growth of algae in a certain region and it is said to be harmful when the bloom causes damage effects. Due to the tremendous impact of harmful algal bloom (HAB) on some aspects, this research proposes the mathematical modelling of an HAB model to describe the process of HAB together with population dynamics. This research considers the delay terms in the modelling since the liberation of toxic chemicals by toxin-producing phytoplankton (TPP) is not an instantaneous process in which the species need to achieve their maturity. A model of fish interaction is also being studied to show the effect of HAB on fish species. Time delay is incorporated for the mortality of fish due to the consumption of toxic zooplankton. Stability analysis is conducted and numerical simulations are applied to obtain the analytical results which highlight the critical values for the delay parameters. The existence of Hopf bifurcation is established when the delay passes the threshold value. The results of both models show that the inclusion of the delay term affects the model by stabilizing and destabilizing the model. Therefore, this research shows the effect of an inclusion delay term on the model and also gives knowledge and an understanding of the process of HAB occurrence as well as the effect of HAB on fish populations. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis of Problems in Ecology, Epidemiology and Oncology)

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22 pages, 961 KiB

Open AccessArticle

Quiescence Generates Moving Average in a Stochastic Epidemiological Model with One Host and Two Parasites

by Usman Sanusi, Sona John, Johannes Mueller and Aurélien Tellier

Mathematics 2022, 10(13), 2289; https://doi.org/10.3390/math10132289 - 30 Jun 2022

Cited by 1 | Viewed by 1168

Abstract

Mathematical modelling of epidemiological and coevolutionary dynamics is widely being used to improve disease management strategies of infectious diseases. Many diseases present some form of intra-host quiescent stage, also known as covert infection, while others exhibit dormant stages in the environment. As quiescent/dormant [...] Read more.

Mathematical modelling of epidemiological and coevolutionary dynamics is widely being used to improve disease management strategies of infectious diseases. Many diseases present some form of intra-host quiescent stage, also known as covert infection, while others exhibit dormant stages in the environment. As quiescent/dormant stages can be resistant to drug, antibiotics, fungicide treatments, it is of practical relevance to study the influence of these two life-history traits on the coevolutionary dynamics. We develop first a deterministic coevolutionary model with two parasite types infecting one host type and study analytically the stability of the dynamical system. We specifically derive a stability condition for a five-by-five system of equations with quiescence. Second, we develop a stochastic version of the model to study the influence of quiescence on stochasticity of the system dynamics. We compute the steady state distribution of the parasite types which follows a multivariate normal distribution. Furthermore, we obtain numerical solutions for the covariance matrix of the system under symmetric and asymmetric quiescence rates between parasite types. When parasite strains are identical, quiescence increases the variance of the number of infected individuals at high transmission rate and vice versa when the transmission rate is low. However, when there is competition between parasite strains with different quiescent rates, quiescence generates a moving average behaviour which dampen off stochasticity and decreases the variance of the number of infected hosts. The strain with the highest rate of entering quiescence determines the strength of the moving average and the magnitude of reduction of stochasticity. Thus, it is worth investigating simple models of multi-strain parasite under quiescence/dormancy to improve disease management strategies. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis of Problems in Ecology, Epidemiology and Oncology)

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14 pages, 307 KiB

Open AccessArticle

Some Contributions to the Class of Branching Processes with Several Mating and Reproduction Strategies

by Manuel Molina-Fernández and Manuel Mota-Medina

Mathematics 2022, 10(12), 2061; https://doi.org/10.3390/math10122061 - 14 Jun 2022

Viewed by 1061

Abstract

This work deals with mathematical modeling of dynamical systems. We consider a class of two-sex branching processes with several mating and reproduction strategies. We provide some probabilistic and statistical contributions. We deduce general expressions for the probability generating functions underlying the probability model, [...] Read more.

This work deals with mathematical modeling of dynamical systems. We consider a class of two-sex branching processes with several mating and reproduction strategies. We provide some probabilistic and statistical contributions. We deduce general expressions for the probability generating functions underlying the probability model, we derive some properties concerning the behavior of the states of the process and we determine estimates for the offspring mean vectors governing the reproduction phase. Furthermore, we extend the two-sex model considering immigration of female and male individuals from external populations. The results are illustrated through simulated examples. The investigated two-sex models are of particular interest to mathematically describe the population dynamics of biological species with a single reproductive episode before dying (semalparous species). Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis of Problems in Ecology, Epidemiology and Oncology)

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28 pages, 1548 KiB

Open AccessArticle

Global Stability of a Humoral Immunity COVID-19 Model with Logistic Growth and Delays

by Ahmed M. Elaiw, Abdullah J. Alsaedi, Afnan Diyab Al Agha and Aatef D. Hobiny

Mathematics 2022, 10(11), 1857; https://doi.org/10.3390/math10111857 - 28 May 2022

Cited by 16 | Viewed by 1756

Abstract

The mathematical modeling and analysis of within-host or between-host coronavirus disease 2019 (COVID-19) dynamics are considered robust tools to support scientific research. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is the cause of COVID-19. This paper proposes and investigates a within-host COVID-19 dynamics [...] Read more.

The mathematical modeling and analysis of within-host or between-host coronavirus disease 2019 (COVID-19) dynamics are considered robust tools to support scientific research. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is the cause of COVID-19. This paper proposes and investigates a within-host COVID-19 dynamics model with latent infection, the logistic growth of healthy epithelial cells and the humoral (antibody) immune response. Time delays can affect the dynamics of SARS-CoV-2 infection predicted by mathematical models. Therefore, we incorporate four time delays into the model: (i) delay in the formation of latent infected epithelial cells, (ii) delay in the formation of active infected epithelial cells, (iii) delay in the activation of latent infected epithelial cells, and (iv) maturation delay of new SARS-CoV-2 particles. We establish that the model’s solutions are non-negative and ultimately bounded. This confirms that the concentrations of the virus and cells should not become negative or unbounded. We deduce that the model has three steady states and their existence and stability are perfectly determined by two threshold parameters. We use Lyapunov functionals to confirm the global stability of the model’s steady states. The analytical results are enhanced by numerical simulations. The effect of time delays on the SARS-CoV-2 dynamics is investigated. We observe that increasing time delay values can have the same impact as drug therapies in suppressing viral progression. This offers some insight useful to develop a new class of treatment that causes an increase in the delay periods and then may control SARS-CoV-2 replication. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis of Problems in Ecology, Epidemiology and Oncology)

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24 pages, 4570 KiB

Open AccessArticle

Dynamics of Oxygen-Plankton Model with Variable Zooplankton Search Rate in Deterministic and Fluctuating Environments

by Sudeshna Mondal, Guruprasad Samanta and Manuel De la Sen

Mathematics 2022, 10(10), 1641; https://doi.org/10.3390/math10101641 - 11 May 2022

Cited by 2 | Viewed by 2228

Abstract

It is estimated by scientists that 50–80% of the oxygen production on the planet comes from the oceans due to the photosynthetic activity of phytoplankton. Some of this production is consumed by both phytoplankton and zooplankton for cellular respiration. In this article, we [...] Read more.

It is estimated by scientists that 50–80% of the oxygen production on the planet comes from the oceans due to the photosynthetic activity of phytoplankton. Some of this production is consumed by both phytoplankton and zooplankton for cellular respiration. In this article, we have analyzed the dynamics of the oxygen-plankton model with a modified Holling type II functional response, based on the premise that zooplankton has a variable search rate, rather than constant, which is ecologically meaningful. The positivity and uniform boundedness of the studied system prove that the model is well-behaved. The feasibility conditions and stability criteria of each equilibrium point are discussed. Next, the occurrence of local bifurcations are exhibited taking each of the vital system parameters as a bifurcation parameter. Numerical simulations are illustrated to verify the analytical outcomes. Our findings show that (i) the system dynamics change abruptly for a low oxygen production rate, resulting in depletion of oxygen and plankton extinction; (ii) the proposed system has oscillatory behavior in an intermediate range of oxygen production rates; (iii) it always has a stable coexistence steady state for a high oxygen production rate, which is dissimilar to the outcome of the model of a coupled oxygen-plankton dynamics where zooplankton consumes phytoplankton with classical Holling type II functional response. Lastly, the effect of environmental stochasticity is studied numerically, corresponding to our proposed system. Full article

(This article belongs to the Special Issue Mathematical Modeling and Analysis of Problems in Ecology, Epidemiology and Oncology)

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