International Journal of Biomathematics

In this paper, we have studied about the sensitivity analysis of the human immunodeficiency virus (HIV) protease inhibitor (PI) model and estimated the length of the delay. We have fabricated an HIV PI model accompanied with humoral immunity. Stability analysis of the constructed model about its steady states has been deliberated. We have evaluated some numerical simulations for PI model with humoral immunity by using the existing patient data.

Gene regulatory networks (GRNs) control the production of proteins in cells. It is well-known that this process is not deterministic. Numerous studies employed a non-deterministic transition structure to model these networks. However, it is not realistic to expect state-to-state transition probabilities to remain constant throughout an organism’s lifetime. In this work, we focus on modeling GRN state transition (edge) variability using an ever-changing set of propensities. We suspect that the source of this variation is due to internal noise at the molecular level and can be modeled by introducing additional stochasticity into GRN models. We employ a beta distribution, whose parameters are estimated to capture the pattern inherent in edge behavior with minimum error. Additionally, we develop a method for obtaining propensities from a pre-determined network.

In the United States, prostitution is considered illegal in all but one state; Nevada allows some legal activities in exchange for substantial guidelines. In 2010, approximately 43,600 females were arrested for prostitution. Numerous intervention programs were established in order to obstruct the lifestyle of a prostitute (PRP, Project ROSE, etc.). There are many documentations and programs that share their forethought on prostitution; however, few target prostitution directly. To determine the dynamics of prostitution, this paper constructs a four-class compartmental model that focuses on the effectiveness of government intervention and rehabilitation of prostitutes mathematically. The basic reproductive number, ${\mathcal{ℛ}}_0$, helps to discover the threshold values for the dynamics of prostitution to become both prevalent or absent in society. This paper predominately observes government intervention to curtail a prostitution prevalent society. Various parameters and variables help to define and indicate the dynamics of prostitution to construct viable simulations. Successful prostitution interaction prevention deemed essential in prostitution prevention; however, government intervention corresponding with successful rehabilitation competitively challenges prostitution interaction prevention in reducing basic reproductive values.

We propose a multi-scale modeling framework to investigate the transmission dynamics of cholera. At the population level, we employ a SIR model for the between-host transmission of the disease. At the individual host level, we describe the evolution of the pathogen within the human body. The between-host and within-host dynamics are connected through an environmental equation that characterizes the growth of the pathogen and its interaction with the hosts outside the human body. We put a special emphasis on the within-host dynamics by making a distinction for each individual host. We conduct both mathematical analysis and numerical simulation for our model in order to explore various scenarios associated with cholera transmission and to better understand the complex, multi-scale disease dynamics.

The present paper investigates the dynamics of pine wilt disease with saturated incidence rate. The proposed model is stable both locally and globally. The local stability of the disease-free equilibrium is determined by the basic reproduction ${{ℛ}}_0$. The disease-free equilibrium is stable locally and globally whenever ${{ℛ}}_0<1$. If ${{ℛ}}_0>1$, then the endemic state is stable both locally and globally. Further, a brief discussion with conclusion on the numerical results of the proposed model is presented.

In this paper, we investigate the global existence of nonnegative solutions of a two-species Keller–Segel model with Lotka–Volterra competitive source terms. By raising the regularity of a solution from $L^1$ to $L^p(p>1)$, the existence and uniqueness of the classical global in time solution to this chemotaxis model is proved for any chemotactic coefficients ${\chi }_1,{\chi }_2>0$ when the space dimension is one. Furthermore, it is shown that the model has a unique classical global solution in two and three space dimensions if the chemotactic coefficients ${\chi }_1$ and ${\chi }_2$ are small as compared to the diffusion coefficient $d_3$ of the chemoattractant.

In this paper, we explore the long time behavior of a multigroup Susceptible–Infected–Susceptible (SIS) model with stochastic perturbations. The conditions for the disease to die out are obtained. Besides, we also show that the disease is fluctuating around the endemic equilibrium under some conditions. Moreover, there is a stationary distribution under stronger conditions. At last, some numerical simulations are applied to support our theoretical results.

Pathogenic bacteria in human system mature through the bio-synthesis of protective layer known as cell wall. This bacterial cell wall growth occurs in the presence of enzyme released by it. After maturation by the cell wall formation, pathogenic bacteria become harmful for human body as they are responsible for different diseases. Antibiotics or drugs are employed for curing bacterial diseases through the inhibition of this maturation process and it occurs by the binding progression of antibiotics with the released enzyme. But nowadays, drugs or antibiotics like $\beta$-lactum family (Amoxcillin) which are generally used for inhibition of bio-synthesis of cell wall become ineffective due to evolution of antibiotic resistance by the bacteria. Antibiotic resistance occurs when an antibiotic has lost its ability to effectively control or kill bacterial growth. As a result, the bacteria becomes “resistant” and continue to multiply for the generation of robust pathogenic bacteria in spite of drug administration. This is due to the release of another type of enzyme by the resistant bacteria which binds with the active antibiotic or drug making it ineffective. Hence, another type of drug (Clauvanic acid) is combined to resist the activity of drug hydrolyzing enzyme so that the initial drug can act effectively. Hence a combination of drug therapy is applied to cure the bacterial diseases successfully. We developed a mathematical model based on the bacterial enzyme and bacterial cell wall proliferation mechanism and showed how we can reduce the bacterial infection in the resistant cases with application of combination drugs (Amoxcillin and Clauvanic acid) to restore normal health. Based on the enzymatic activity and individual drug dynamics we studied the overall system under the single drug and combinational drug administration through our formulated model analysis. We also demonstrated the different dosing time interval and dosing concentration to evaluate the optimized drug administration for arresting the cell wall formation completely. Sensitivity of the different kinetic rate constant also has been performed with subject to drug hydrolyzing enzyme. Our analytical and numerical studies also confirm the efficiency of the combinational drug treatment compared to single drug treatment being more effective in drug resistant cases providing recovery from bacterial disease.

Mathematical models and computer simulations are useful experimental tools for building and testing theories. Many mathematical models in biology can be formulated by a nonlinear system of ordinary differential equations. This work deals with the numerical solution of the hantavirus infection model, the human immunodeficiency virus (HIV) infection model of CD4${}^{+}$T cells and the susceptible–infected–removed (SIR) epidemic model using a new reliable algorithm based on shifted Boubaker Lagrangian (SBL) method. This method reduces the solution of such system to a system of linear or nonlinear algebraic equations which are solved using the Newton iteration method. The obtained results of the proposed method show highly accurate and valid for an arbitrary finite interval. Also, those are compared with fourth-order Runge–Kutta (RK4) method and with the solutions obtained by some other methods in the literature.

An HIV infection model with saturated infection rate and double delays is investigated. First, the existence of the infection-free equilibrium $E_0$, the immune-exhausted equilibrium $E_1$ and the infected equilibrium $E_2$ with immunity in different conditions is shown. By analyzing the characteristic equation, we study the locally asymptotical stability of the trivial equilibrium, and the existence of Hopf bifurcations when two delays are used as the bifurcation parameter. Furthermore, we apply the Nyquist criterion to estimate the length of delay for which stability continues to hold. Then with suitable Lyapunov function and LaSalle’s invariance principle, the global stability of the three equilibriums is obtained. Finally, numerical simulations are presented to illustrate the main mathematical results.

In this paper, a system of fractional differential equations that model the synchronized bioluminescence behavior of a set of fireflies put on two spatial arrangements is presented; the alternative representation of these equations contains fractional operators of Liouville–Caputo type. The objective of the model is to qualitatively recover synchronization and show that it is persistent. It is shown that the effort made by each firefly glow changes with respect to the number of male competitors and the distance between them. The conditions on biological parameters are interpreted.

In this paper, our main objective is to find out the necessary and sufficient conditions for a cyclic code of arbitrary length over the ring of four elements $R_1={𝔽}_2+u{𝔽}_2$ $(u^2=1)$ to be a reversible cyclic code. We also obtain the structure of cyclic DNA codes of odd length over the ring $R={𝔽}_2[u,v]/〈u^2-1,v^3-v,uv-vu〉$, which plays an important role in Computational Biology. Furthermore, we establish a direct link between the elements of ring $R$ and 64 codons used in the amino acids of living organisms by introducing a Gray map from $R$ to $R_1$. Among others, binary images of cyclic codes over $R$ are also investigated. As applications, some cyclic DNA codes over $R$ using the Gray map are provided.

The afferent arteriole (AA) of rat kidney exhibits the myogenic response, in which the vessel constricts in response to an elevation in blood pressure and dilates in response to a pressure reduction. Additionally, the AA exhibits spontaneous oscillations in vascular tone at physiological luminal pressures. These time-periodic oscillations stem from the dynamic exchange of Ca${}^{2+}$ between the cytosol and the sarcoplasmic reticulum, coupled to the stimulation of Ca${}^{2+}$-activated potassium and chloride channels, and to the modulation of voltage-gated L-type Ca${}^{2+}$ channels. The effects of physiological factors, including blood pressure and vasoactive substances, on AA vasomotion remain to be well characterized. In this paper, we analyze a mathematical model of Ca${}^{2+}$ signaling in an AA smooth muscle cell. The model represents detailed transmembrane ionic transport, intracellular Ca${}^{2+}$ dynamics as well as kinetics of nitric oxide (NO) and superoxide (O${}_2^{-}$) formation, diffusion and reaction. NO is an important factor in the maintenance of blood pressure and O${}_2^{-}$ has been shown to contribute significantly to the functional alternations of blood vessels in hypertension. We perform a bifurcation analysis of the model equations to assess the effect of luminal pressure, NO and O${}_2^{-}$ on the behaviors of limit cycle oscillations.

This paper studies the local and global stability of solutions for a spatially spread SEI epidemic model with immigration of individuals using a Lyapunov functional. It is shown that in the presence of diffusion, the unique steady state remains globally stable. Numerical results obtained through Matlab simulations are presented to confirm the findings of this study.

Network and equation-based (EB) models are two prominent methods used in the study of epidemics. While EB models use a global approach to model aggregate population, network models focus on the behavior of individuals in the population. The two approaches have been used in several areas of research, including finance, computer science, social science and epidemiology. In this study, epidemiology is used to contrast EB models with network models. The methods are based on the assumptions and properties of compartmental models. In EB models we solve a system of ordinary differential equations and in network models we simulate the spread of epidemics on contact networks using bond percolation. We examine the impact of network structures on the spread of infection by considering various networks, including Poisson, Erdős Rényi, Scale-free, and Watts–Strogatz small-world networks, and discuss how control measures can make use of the network structures. In addition, we simulate EB assumptions on Watts–Strogatz networks to determine when the results are similar to that of EB models. As a case study, we use data from the 1918 Spanish flu pandemic and that from measles outbreak to validate our results.