International Journal of Biomathematics

In our daily life problem we face uncertainties in making right decisions. In this study, we propose two different decision-making problems in medical field. The first problem is fever diagnosing and second problem is mouth cancer risk analysis. In the first problem, we use fuzzy soft similarity measures and fuzzy soft matrix operations to diagnose the type of fever. We consider a hypothetical case study and manipulate similarity measures on it. Our work diagnoses different patients having similar symptoms. We also develop a small application using JAVA. In the second problem, we perform risk analysis of mouth cancer. The proposed fuzzy soft expert system takes two biochemical parameters as inputs that is, serum total malondialdehyde (MDA), and serum proton donors capacity (donors_protons) and determines the risk of mouth cancer. Our study facilitates doctors by diagnosing mouth cancer at its earlier stages. There are four main components of our fuzzy soft expert system. The first component is named as fuzzification which converts crisp input into linguistic variables and formulates fuzzy sets. The second component transforms fuzzy sets into their respective fuzzy soft sets. The third component determines indispensable parameters and performs parameter reduction. The fourth component performs risk analysis by using algorithm. We use exemplary dataset and run all the components of fuzzy soft expert system to compute cancer risk.

The aim of this paper is to use topological concepts in the construction of flexible mathematical models in the field of biological mathematics. Also, we will build new topographic types to study recombination of deoxyribonucleic acid (DNA) and ribonucleic acid (RNA). Finally, we study the topographical properties of constructed operators and the associated topological spaces of DNA and RNA.

We consider a special class of Lotka–Volterra systems where the associated interaction matrix is cyclic, but asymmetric, with a perturbation term on each row. After some discussion of the dynamics under a general setting, we focus our attention on 3D systems for a more detailed study. We derive sufficient conditions for the existence and stability of the nontrivial interior equilibrium. We also show that Hopf bifurcation occurs when the size of the perturbation is large. Such analysis can be similarly extended to higher dimensional systems, and we mention some results in 4D case.

In this paper, an efficient numerical method is introduced for solving the fractional (Caputo sense) Fisher equation. We use the spectral collocation method which is based upon Chebyshev approximations. The properties of Chebyshev polynomials of the third kind are used to reduce the proposed problem to a system of ODEs, which is solved by the finite difference method (FDM). Some theorems about the convergence analysis are stated and proved. A numerical simulation is given and the results are compared with the exact solution.

In this paper, we investigate a possible applicability of the newly established fractional differentiation in the field of epidemiology. To do this, we extend the model describing the Lassa hemorrhagic fever by changing the derivative with the time fractional derivative for the inclusion of memory. Detailed analysis of existence and uniqueness of exact solution is presented using the Banach fixed point theorem. Finally, some numerical simulations are shown to underpin the effectiveness of the used derivative.

This paper is devoted to studying the dynamics of a susceptible-infective-latent-infective (SILI) epidemic model that is subject to the combined effects of environmental noise and intervention strategy. We extend the classical SILI epidemic model from a deterministic framework to a stochastic one. For the deterministic case, the global stability analysis of the solution is carried out in terms of the basic reproduction number. For the stochastic case, sufficient conditions for the extinction of diseases are obtained. Then, the existence of stationary distribution and asymptotic behavior of the solution are further studied to illustrate the cycling phenomena of recurrent diseases. Numerical simulations are conducted to verify these analytical results. It is shown that both stochastic noise and intervention strategy contribute to the control of diseases.

A susceptible–infected–vaccinated epidemic model with proportional vaccination and generalized nonlinear rate is formulated and investigated in the paper. We show that the stochastic epidemic model admits a unique and global positive solution with probability one when constructing a proper $C^2$-function therewith. Then a sufficient condition that guarantees the disappearances of diseases is derived when the indicator $R_0<1$. Further, if $R_0>1$, then we obtain that the solution is weakly permanent with probability one. We also derived the sufficient conditions of the persistence in the mean for the susceptible and infected when another indicator ${\tilde{R}}_0>1$.

Neuron is a fundamental unit of the brain, which is specialized to transmit information throughout the body through electrical and chemical signals. Calcium (${\mathrm{\text{Ca}}}^{2+}$) ions are known as second messengers which play important roles in the movement of the neurotransmitter. Calbindin-$D_{28k}$ is a ${\mathrm{\text{Ca}}}^{2+}$ binding protein which is involved in regulation of intracellular ${\mathrm{\text{Ca}}}^{2+}$ ions and maintains ${\mathrm{\text{Ca}}}^{2+}$ homeostasis level, it also alters the cytosolic calcium concentration ([${\mathrm{\text{Ca}}}^{2+}$]) in nerve cells to keep the cell alive. Parkinson’s disease (PD) is a chronic progressive neurodegenerative brain disorder of the nervous system. Several regions of the brain indicate the hallmark of the PD. The symptoms of PD are plainly linked with the degeneration and death of dopamine neurons in the substantia nigra pars compacta located in midbrain which is accompanied by depletion in calbindin-$D_{28k}$. In the present paper, the neuroprotective role of calbindin-$D_{28k}$ in the cytoplasmic [${\mathrm{\text{Ca}}}^{2+}$] distribution is studied. The elicitation in [${\mathrm{\text{Ca}}}^{2+}$] is due to the presence of low amount of calbindin-$D_{28k}$ which can be portrayed and is a hallmark of PD. A one-dimensional space time fractional reaction diffusion equation is designed by keeping in mind the physiological condition taking place inside Parkinson’s brain. Computational results are performed in MATLAB.

This paper deals with a qualitative and computational study of a class of integro-differential model, with a nonlinear growth rate. The model is applied to immunology problems, in particular to the competition between tumors and immune system. Our purpose is to take into account the activity transfer processes between cells. The first part of this paper deals with the modeling, while the second part develops the existence and uniqueness of global solutions to the initial boundary value problem related to the application of the model. Finally, numerical simulations are used to illustrate the results obtained for a dynamic of tumor cells contrasted by the immune system.

Epidemic transmission has a substantial effect on the dynamics and stability of the predator–prey system, in which the transmission rate plays an important role. The probabilistic cellular automaton (PCA) approach is used to investigate the spatiotemporal dynamics of a predator–prey system with the infected predator. Remarkably, it is impossible to achieve a coexistence state of prey, susceptible predators, and infected predators in a spatial population. This is different from the analysis from a non-spatial population with the mean-field approximation, where Hopf bifurcation arises and the interior equilibrium becomes unstable, and a periodic solution appears with the increasing infection rate. The results show that the introduction of the infected predator with a high transmission rate is beneficial for the persistence of the prey population in space. However, a low transmission rate will promote the coexistence state of the prey and the susceptible predator populations. In summary, it is possible to develop management strategies to manipulate the transmission rate of the infected predator for the benefit of biological control.

In this paper, a multi-compartment 0D model of the blood flow is considered to study the vessel abnormalities (stenoses and aneurysms) in the human systemic circulation (SC). In the complete SC, different levels of stenosis and aneurysms are artificially created by decreasing and increasing the vessel diameters respectively and their effects on pressure and flow are studied using sensitivity analysis (SA). Normalized local sensitivity analysis (LSA) is used to study the impact of stenosis and aneurysms on pressure and flow wave pattern. Furthermore, global sensitivity analysis (GSA), Sobol’s method is used to quantify the overall influence of stenoses and aneurysms in the complete SC.

The results of global sensitivity analysis revealed that the impact of both stenoses and aneurysms is strong within the individual structures (arm, legs, carotid bifurcation, aorta), while, aortic stenoses and aneurysms have effect on almost all downstream nodes. Moreover, the study could be useful for medical doctors, teachers and students to observe the hemodynamical changes in the SC with respect to vessel abnormalities, which could further help in making any clinical decision for patients having different levels of vessel abnormalities in any part or structure of the SC.

We study the reaction–diffusion Ebola PDE model that consists of equations that govern the evolution of susceptible, infected, recovered and deceased human individuals, as well as Ebola virus pathogens in the environment, with diffusive terms in all except the equation of the deceased human individuals. Under the setting of a spatial domain that is bounded, we prove the global well-posedness of the system; in contrast to the previous work on similar models such as cholera, avian influenza, malaria and dengue fever, diffusion coefficients may be different. Moreover, we derive its basic reproduction number, and under the condition that the diffusion coefficients of the susceptible and infected hosts are same, we prove the global stability of the disease-free-equilibrium, and uniform persistence in cases when the basic reproduction number lies beneath and above one, respectively. Again, we do not require that the diffusion coefficients of the recovered hosts be the same as the diffusion coefficients of the susceptible and infected hosts, in contrast to previous work on other models of infectious diseases. Another technical difficulty in our model is that the solution semiflow is not compact due to the lack of diffusion in the equation of the deceased human individuals; we overcome this difficulty using functional analysis techniques concerning Kuratowski measure of non-compactness.

In this paper, an efficient and accurate method is presented to solve continuous population models for single and interacting species using spectral collocation method with exponential Chebyshev (EC) functions. The first problem is a logistic growth model in a population, while the second problem is a prey–predator model: Lotka–Volterra system, the third is a simple 2-species Lotka–Volterra competition model, and the final one is a prey–predator model with limit cycle periodic behavior. The high accuracy of this method is verified through some numerical examples. The obtained numerical results are compared with other methods, showing that the proposed method gives higher accuracy.

This paper introduces a non-conventional approach with multi-dimensional random sampling to solve a cocaine abuse model with statistical probability. The mean Latin hypercube finite difference (MLHFD) method is proposed for the first time via hybrid integration of the classical numerical finite difference (FD) formula with Latin hypercube sampling (LHS) technique to create a random distribution for the model parameters which are dependent on time $t$. The LHS technique gives advantage to MLHFD method to produce fast variation of the parameters’ values via number of multidimensional simulations (100, 1000 and 5000). The generated Latin hypercube sample which is random or non-deterministic in nature is further integrated with the FD method to complete one cycle of LHS-FD simulation iteration. This process is repeated until $n$ final iterations of LHS-FD are obtained. The means of these $n$ final solutions (MLHFD solutions) are tabulated, graphed and analyzed. The numerical simulation results of MLHFD for the SEIR model are presented side-by-side with deterministic solutions obtained from the classical FD scheme and homotopy analysis method with Pade approximation (HAM-Pade). The present MLHFD results are also compared with the previous non-deterministic statistical estimations from 1995 to 2015. Good agreement between the two is perceived with small errors. MLHFD method can be used to predict future behavior, range and prediction interval for the epidemic model solutions. The expected profiles of the cocaine abuse subpopulations are projected until the year 2045. Both the statistical estimations and the deterministic results of FD and HAM-Pade are found to be within the MLHFD prediction intervals for all the years and for all the subpopulations considered.

In this paper, I consider two species feeding on limiting substrate in a chemostat taking into account some possible effects of each species on the other one. System of differential equations is proposed as model of these effects with general inter-specific density-dependent growth rates. Three cases were considered. The first one for a mutual inhibitory relationship where it is proved that at most one species can survive which confirms the competitive exclusion principle. Initial concentrations of species have great importance in determination of which species is the winner. The second one for a food web relationship where it is proved that under general assumptions on the dilution rate, both species persist for any initial conditions. Finally, a third case dealing with an obligate mutualistic relationship was discussed. It is proved that initial condition has a great importance in determination of persistence or extinction of both species.

In this paper, a delayed mathematical model was developed based on experimental data to understand how the time delays required for transcription and translation in Mdm2 gene expression affect the kinetic behavior of the p53-Mdm2 network. Taking the time delays as the main research parameters, the stability of the system at the positive equilibrium was studied by using the theoretical method of delay differential equation. We found that such delays can induce oscillations by undergoing a supercritical Hopf bifurcation. Then, we used the normal form theory and the center manifold reduction to study the direction and stability of the bifurcation in detail. Furthermore, we also studied the effects of the length of time delays and the model parameters by numerical simulations. We found that time delays in Mdm2 synthesis are required for p53 oscillations and the length of such delays can determine the amplitude and period of the oscillations. In addition, the model parameters can also change the stability of the system. These results illustrate that the repair process after DNA damage can be regulated by varying time delays and the model parameters.

In this paper, a mathematical model of conversion from a drug-sensitivity strain to a drug-resistant strain is given to investigate how antibiotic usage may be optimized to preserve or restore antibiotic effectiveness. This novel theoretical framework could result in an optimal criterion on how to reduce the drug resistance to a reasonable range by using the antibiotic dressing strategy. The sufficient conditions of existence of order-1 periodic solution are obtained in view of the geometrical theory of the semi-continuous dynamical system and the qualitative properties of the corresponding continuous system. The stability of the order-1 periodic solution is proved by means of [H. J. Guo, L. S. Chen and X. Y. Song, Qualitative analysis of impulsive state feedback control to an algae-fish system with bistable property, Appl. Math. Comput. 271 (2015) 905–922]. Finally, our results are confirmed by means of numerical simulations.

Salton Sea (a destination resort) is to be saved from being converted into a skeleton-filled wasteland. The critical amount of water flowing into the sea to maintain its level and salinity has been diverted since January 2018. This will lead to shrinking volumes and increasing salinities. Ecological consequences and public health impacts of altered conditions will be phenomenal. We design and analyze a minimal eco-epidemiological model to figure out future journey of this sea; a way station for fish-eating migratory birds. The mathematical model has been assembled in terms of prey-predator interaction. The salient feature of the proposed model is its seasonally varying contact rate which represents rate of conversion of susceptible fishes into infectives. We have analytically investigated the global stability, disease persistence and periodic solutions of the proposed model system. Susceptible prey-induced periodic solution is globally asymptotically stable when ${\bar{R}}_{01}<1$, otherwise unstable and hence disease persists for ${\bar{R}}_{01}>1$.

Global stability and Hopf bifurcation (HB) analysis help us extract parameter values to explore the dynamical behavior of the model system. Two-dimensional parameter scans and bifurcation diagrams reveal that the model displays propensity towards chaotic dynamics, which is associated with extinction-sized population densities. In the presence of stochastic external forces, this implies extinction of most of the fish species. This, in turn, suggests that resident birds will have to migrate to other destinations. The fish-eating migratory birds will be forced to switch over to invertebrates. Ecological consequences and public health impacts of this transition would be severe. Conservation groups are solicited to draw attention of the government to avert this impounding danger. It is important to plan for an ecosystem-wide transition such that impacts on birds and on human inhabitants living adjacent to the shrinking and salinizing sea are minimized.

In this paper, the general nonlinear multi-strain Tuberculosis model is controlled using the merits of Jacobi spectral collocation method. In order to have a variety of accurate results to simulate the reality, a fractional order model of multi-strain Tuberculosis with its control is introduced, where the derivatives are adopted from Caputo’s definition. The shifted Jacobi polynomials are used to approximate the optimality system. Subsequently, Newton’s iterative method will be used to solve the resultant nonlinear algebraic equations. A comparative study of the values of the objective functional, between both the generalized Euler method and the proposed technique is presented. We can claim that the proposed technique reveals better results when compared to the generalized Euler method.

A delayed intraguild predation (IGP) model with intraspecific competition is considered. It is shown that the delay has a destabilizing effect and induces oscillations. The global existence results of periodic solutions bifurcating from the positive equilibrium are established. It is shown that there exists at least one nontrival periodic solution when the delay passes through a certain critical value. Numerical simulations are performed to illustrate our theoretical results and show that intraspecific competition can also affect the stability of the positive equilibrium of the system.