International Journal of Biomathematics

We are constrained by widespread cancerous diseases to improve treatment methods which save patients and provide better living conditions during and after the treatment period. Because of the complexity of the treatment process, mathematical models need to be used in order to have a better understanding of the process. However, deriving an adequate complex model that can capture the disease pattern which could be confirmed by simulations and experiments has its own barriers. In this paper, a new mathematical model is developed concerning immune system effect on cancer. The model is introduced using nonlinear ordinary differential equations. Also, the qualitative behavior of the proposed system is studied in order to examine the extent of the model with respect to the nature of tumor evolution. Thus, number and status of equilibria points in line with the existence of limit cycles are obtained for sub-systems and the whole system. Meanwhile, possible bifurcations are mentioned, and the consequent evolutions are described. It is shown that the model conforms well to natural possibilities, cancer growth or remission. Thus, the model would be fit for further studies for prediction and contemplating treatment method, especially for immune stimulating drugs and immunotherapy.

This paper deals with dengue virus transmitted by the mosquito Aedes aegypti. A reaction–diffusion model is proposed incorporating both human and the A. aegypti population, where we divided mosquitoes into two sub-populations: the winged population and an aquatic form, with the winged mosquito population divided into the susceptible and infected classes. The threshold $Q_0$ depending on spatial heterogeneity is introduced for the mosquito population. Moreover, we introduce the basic reproduction number $R_0$ for the infected classes, as well as its analytical properties. $Q_0$ and $R_0$ are utilized to investigate the stability of the mosquito-established equilibriums. Our results show that the mosquitoes persist if $Q_0>1$, while the transmission of dengue virus depends on the condition whether $R_0<1$ or $R_0>1$.

Coronavirus Disease 2019 (COVID-19), caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infections, was declared a pandemic by the WHO on March 12, 2020. By April 10, 2020, it has spread to almost 215 UN Territories, affected more than 1,600,000 people, and become fatal to more than 100,000 people. We propose a compartmental model to investigate the dynamics of the COVID-19 pandemic and analyze the effects of governments and health officials’ intervention strategies, using data in the USA, Italy, Spain, Germany, and France. These are the countries with the most confirmed cases of the COVID-19 as of April 11, 2020. We perform parameter estimations, sensitivity, and predictive analysis and compare the COVID-19 trend in these understudy countries. Based on the model, we compute the basic reproduction number of the pandemic in these countries. The results indicate that the most sensitive parameters are the contact rate and degree of intervention; these parameters have high value in containing the pandemic. The basic reproduction number in each of the countries under study is more than unity. Based on our findings, the COVID-19 pandemic will continue to spread; but higher compliance with the intervention strategies will help minimize the disease’s spread.

In this paper, the dynamic behaviors are studied for a stochastic delayed avian influenza model with mutation and temporary immunity. First, we prove the existence and uniqueness of the global positive solution for the stochastic model. Second, we give two different thresholds ${\mathcal{ℛ}}_{01}^s$ and ${\mathcal{ℛ}}_{02}^s$, and further establish the sufficient conditions of extinction and persistence in the mean for the avian-only subsystem and avian-human system, respectively. Compared with the corresponding deterministic model, the thresholds affected by the white noises are smaller than the ones of the deterministic system. Finally, numerical simulations are carried out to support our theoretical results. It is concluded that the vaccination immunity period can suppress the spread of avian influenza during poultry and human populations, while prompt the spread of mutant avian influenza in human population.

Human immunodeficiency virus (HIV) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses that infect the susceptible CD$4^{+}$T cells. It is known that HIV and HTLV-I have in common a way of transmission through direct contact with certain body fluids related to infected individuals. Therefore, it is not surprising that a mono-infected person with one of these viruses can be co-infected with the other virus. In the literature, a great number of mathematical models has been presented to describe the within-host dynamics of HIV or HTLV-I mono-infection. However, the within-host dynamics of HIV/HTLV-I co-infection has not been modeled. In this paper, we develop a new within-host HIV/HTLV-I co-infection model. The model includes the impact of Cytotoxic T lymphocytes (CTLs) immune response, which is important to control the progression of viral co-infection. The model describes the interaction between susceptible CD$4^{+}$T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. We first show the nonnegativity and boundedness of the model’s solutions and then we calculate all possible equilibria. We derive the threshold parameters which govern the existence and stability of all equilibria of the model. We prove the global asymptotic stability of all equilibria by utilizing Lyapunov function and LaSalle’s invariance principle. We have presented numerical simulations to illustrate the effectiveness of our main results. In addition, we discuss the effect of HTLV-I infection on the HIV-infected patients and vice versa.

The price fluctuation of commodities affects their demand, and it strongly influences the development of industry and the stability of the international economy. A price-output model with two price-dependent impulses is presented in this paper to describe their mutual restrictions and the price adjustment strategies. The equilibria and dynamic properties of the impulsive-free price-output model are qualitatively analyzed, and the results show the hazard of free price fluctuation. For the price-output model with two price-dependent impulses, conditions for the existence of the order-2 periodic solution are obtained, and the orbital asymptotic stability is proved. Those results reveal the positive role of price adjustment strategies. Finally, some numerical simulations are carried out to verify the theoretical results.

In this paper, we analyze a stochastic rabies epidemic model which is perturbed by both white noise and telegraph noise. First, we prove the existence of the unique global positive solution. Second, by constructing an appropriate Lyapunov function, we establish a sufficient condition for the existence of a unique ergodic stationary distribution of the positive solutions to the model. Then we establish sufficient conditions for the extinction of diseases. Finally, numerical simulations are introduced to illustrate our theoretical results.

In this paper, a neuro-optimized numerical method is presented for approximation of HIV virus progression model in the human body. The model is composed of coupled nonlinear system of differential equations (DEs) containing healthy and infected T-Cells and HIV free virus particles. The coupled system is transformed into feedforward artificial neural network (ANN) with Mexican hat wavelet function in the hidden layers. Two meta-heuristic algorithms based on chaotic particle swarm optimization (CPSO) and its hybrid version with local search technique are exploited to tune the parameters of ANN in an unsupervised manner of error function. A comprehensive testbed is established to observe the virus growth per day with performance metric containing fitness value, computational time complexity and convergence. The proposed solutions are compared with state of art Runge–Kutta method and Legendre Wavelet Collocation Method (LWCM). The core advantages of the proposed scheme are getting the solution on continuous grid, consistent convergence, simplicity in implementation and handling strong nonlinearity effectively.

We consider a SEIR epidemic model with infectious force in latent period and infected period under discontinuous treatment. The treatment rate has at most a finite number of jump discontinuities in every compact interval. By using Lyapunov theory for discontinuous differential equations and other techniques on non-smooth analysis, the basic reproductive number $R_0$ is proved to be a sharp threshold value which completely determines the dynamics of the model. If $R_0\le 1$, then there exists a disease-free equilibrium which is globally stable. If $R_0>1$, the disease-free equilibrium becomes unstable and there exists an endemic equilibrium which is globally stable. We discuss that the disease will die out in a finite time which is impossible for the corresponding SEIR model with continuous treatment. Furthermore, the numerical simulations indicate that strengthening treatment measure after infective individuals reach some level is beneficial to disease control.

Towards the end of 2019, the world witnessed the outbreak of Severe Acute Respiratory Syndrome Coronavirus-2 (COVID-19), a new strain of coronavirus that was unidentified in humans previously. In this paper, a new fractional-order Susceptible–Exposed–Infected–Hospitalized–Recovered (SEIHR) model is formulated for COVID-19, where the population is infected due to human transmission. The fractional-order discrete version of the model is obtained by the process of discretization and the basic reproductive number is calculated with the next-generation matrix approach. All equilibrium points related to the disease transmission model are then computed. Further, sufficient conditions to investigate all possible equilibria of the model are established in terms of the basic reproduction number (local stability) and are supported with time series, phase portraits and bifurcation diagrams. Finally, numerical simulations are provided to demonstrate the theoretical findings.

A birth–death process is considered as an epidemic model with recovery and transmittance from outside. The fraction of infected individuals is for huge population sizes approximated by a solution of an ordinary differential equation taking values in $[0,1]$. For intermediate size or semilarge populations, the fraction of infected individuals is approximated by a diffusion formulated as a stochastic differential equation. That diffusion approximation however needs to be killed at the boundary $\{0\}\cup \{1\}$. An alternative stochastic differential equation model is investigated which instead allows a more natural reflection at the boundary.

The integer-valued generalized autoregressive conditional heteroskedastic (INGARCH) model is often utilized to describe data in biostatistics, such as the number of people infected with dengue fever, daily epileptic seizure counts of an epileptic patient and the number of cases of campylobacterosis infections, etc. Since the structure of such data is generally high-order and sparse, studies about order shrinkage and selection for the model attract many attentions. In this paper, we propose a penalized conditional maximum likelihood (PCML) method to solve this problem. The PCML method can effectively select significant orders and estimate the parameters, simultaneously. Some simulations and a real data analysis are carried out to illustrate the usefulness of our method.