Narrow Results

We propose predator-prey-parasite models to study the effects of parasites upon the predator-prey interaction. There are two parameters that are used to model the effectiveness of the infected prey and infected predator. For the spatial homogeneous system, the asymptotic dynamics depend on the reproductive number of the parasite. The parasite can persist in the population if this reproductive number is larger than one. Numerical simulations suggest that less competitiveness of the infected predator can make the predator-prey interaction less stable. The dynamics may move from coexisting steady state to oscillations. For the spatial heterogeneous system, diffusion may destabilize the homogeneous interior steady state for a particular set of diffusion coefficients. However, both systems do not exhibit complicated dynamical behavior.

An age-structured mathematical model for malaria is presented. The model explicitly includes the human and mosquito populations, structured by chronological age of humans. The infected human population is divided into symptomatic infectious, asymptomatic infectious and asymptomatic chronic infected individuals. The original partial differential equation (PDE) model is reduced to an ordinary differential equation (ODE) model with multiple age groups coupled by aging. The basic reproduction number is derived for the PDE model and the age group model in the case of general n age groups. We assume that infectiousness of chronic infected individuals gets triggered by bites of even susceptible mosquitoes. Our analysis points out that this assumption contributes greatly to the expression and therefore needs to be further studied and understood. Numerical simulations for n = 2 age groups and a sensitivity/uncertainty analysis are presented. Results suggest that it is important not only to consider asymptomatic infectious individuals as a hidden cause for malaria transmission, but also asymptomatic chronic infections (>60%), which often get neglected due to undetectable parasite loads. These individuals represent an important reservoir for future human infectiousness. By considering age-dependent immunity types, the model helps generate insight into effective control measures, by targeting age groups in an optimal way.

The prevalence of infection with hepatitis C virus (HCV) is higher among prison inmates compared with the general population because of the high proportion of intravenous drug misusers. In this paper, a mathematical model for the spread of HCV in correctional institutions is developed and analyzed. The impact of educational campaigns and screening of HCV infected individuals is examined. In the presence of screening, the study suggests that targeted education campaigns of more than 50% will be highly effective in controlling the sharing of injection equipment among intravenous drug users, thereby reducing HCV prevalence within the correctional institutions. Further, we applied optimal control theory to the proposed model. The controls represent educational campaigns and screening. The objective is based on minimizing cumulative latent and infectious cases within the correctional institution, while maximizing the number of susceptible nonintravenous drug misusers. The optimal control is characterized and numerically solved. Overall, the application of optimal control theory in this study suggests that, more effort should be devoted to educational campaigns compared to screening.

For a mathematical model for the spread of HIV by sexual transmission in a heterosexual population we analyse the existence and stability of equilibrium solutions. The model is designed to investigate the effects of a fundamental constraint in any social/sexual mixing process for heterogeneous populations. The group contact constraint conserves the number of new sexual partnerships formed per unit time between the sexes, and will have at least a quantitative influence on the dynamics of the model. The analysis has been carried out for general and specific forms of the sexual activity rates (the mean number of sexual partners per unit time for a typical individual). In general we define a threshold parameter R₀, the Reproductive Number, which is a key determinant of the behaviour of the model. We show that in general if R₀>1 there is at least one endemic equilibrium. The specific cases of a dominance sexual activity rate and proportional sexual activity rates are discussed in more detail. Multiple endemic equilibria can occur for the former but not the latter.

Huanglongbing infection (HLB) or citrus greening and more rarely yellow shoot disease is a fatal bacterial disease of citrus. Our objective in this study is to propose a mathematical model to predict the outbreak of Huanglongbing (HLB) infection with Logistic growth in psyllid insect vectors to predict the transmission of this disease among citrus. Indeed, we considered the periodic environment to model the seasonality of the spread of the disease. In this regard, an explicit expression of the reproductive number $R_0$ is obtained, which has been used for analyzing the behavior of the model in the cases when $R_0>1$ and $R_0<1$. The mathematical findings are supported using numerical illustrations.

A sex-structured model for heterosexual transmission of HIV/AIDS for addressing the epidemic as a gender-based issue in African heterosexual settings is presented. The epidemic threshold and equilibria for the model are determined and stabilities are investigated. Comprehensive qualitative analysis of the model including positivity and boundedness of solutions, as well as persistence are carried out. The epidemic threshold for the model is computed and used to assess the effects of gender-inequality affecting women in heterosexual settings. The obtained gender-inequality-induced reproductive number is greater than the reproductive number in the absence of gender-inequality suggesting that gender-inequality affecting women in heterosexual settings enlarges the HIV/AIDS epidemic. Numerical simulations are carried out using demographic and epidemiological parameters for Zimbabwe and the obtained results confirm that gender-inequality increases HIV/AIDS prevalence in heterosexual settings. We conclude from the study that gender-inequality affecting women among heterosexuals should be properly addressed for the effective control of the HIV/AIDS epidemic.

A deterministic model for evaluating the impact of voluntary testing and treatment on the transmission dynamics of tuberculosis is formulated and analyzed. The epidemiological threshold, known as the reproduction number is derived and qualitatively used to investigate the existence and stability of the associated equilibrium of the model system. The disease-free equilibrium is shown to be locally-asymptotically stable when the reproductive number is less than unity, and unstable if this threshold parameter exceeds unity. It is shown, using the Centre Manifold theory, that the model undergoes the phenomenon of backward bifurcation where the stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction number is less than unity. The analysis of the reproduction number suggests that voluntary tuberculosis testing and treatment may lead to effective control of tuberculosis. Furthermore, numerical simulations support the fact that an increase voluntary tuberculosis testing and treatment have a positive impact in controlling the spread of tuberculosis in the community.

Hepatitis C virus (HCV) is a blood-borne infection that can lead to progressive liver failure, cirrhosis, hepatocellular carcinoma and death. A deterministic mathematical model for assessing the impact of daily intravenous drug misuse on the transmission dynamics of HCV is presented and analyzed. A threshold quantity known as the reproductive number has been computed. Stability of the steady states has been investigated. The dynamical analysis reveals that the model has globally asymptotically stable steady states. The impact of daily intravenous drug misuse on the transmission dynamics of HCV has been discussed through the basic reproductive number and numerical simulations.

In this paper, we develop a fractional-order smoking model by considering relapse class. First, we formulate the model and find the unique positive solution for the proposed model. Then we apply the Grünwald–Letnikov approximation in the place of maintaining a general quadrature formula approach to the Riemann–Liouville integral definition of the fractional derivative. Building on this foundation avoids the need for domain transformations, contour integration or involved theory to compute accurate approximate solutions of fractional-order giving up smoking model. A comparative study between Grünwald–Letnikov method and Runge–Kutta method is presented in the case of integer-order derivative. Finally, we present the obtained results graphically.

Internet worms have received a lot of attention because of their massive hazards to the internet. As the internet worms propagate quickly, therefore, the automatic control on the internet is required. In this paper, we present a new epidemic model called susceptible-breaking out-quarantine-treatment-recovered (SBQTR) model which integrates dynamic quarantine and treatment approaches to control the spread of internet worms. By using the SBQTR model, we can calculate the basic reproduction number $R_0$, which determines the existence of worms. The model has two equilibrium states: virus-free equilibrium (VFE) and viral equilibrium (VE). The stability analysis of both the equilibrium points is carried out and impact of treatment is examined under various parametric values. The sensitivity analysis of model is carried out to determine the impact of model parameters on virus transmission. Furthermore, we established nonstandard finite difference (NSFD) schemes and Runge–Kutta algorithm of order 4 (RK-4) for SBQTR model. The stated NSFD technique retains all of the significant properties of system (1), whereas RK-4 mechanism gives negative solutions and doesn’t converge to the equilibrium points of system (1). Numerical results together with comparison for various values of step size ($h$) are available. Bifurcation value of infection coefficient is also investigated. It is shown that an appropriate treatment is essential to control the spread of worms in computer network.

Corona virus (CoV) is a group of viruses with non-bifurcated, single-stranded, and positive-sense RNA genomes. Apart from infecting several economically significant vertebrates (such as pigs and chickens), it is reported in the recent literature that six main types of CoVs infect the human hosts and cause lung infections. In animals, CoVs cause several diseases, including pneumonia, gastrointestinal tract, and central nervous system diseases. In humans, the CoVs work as respiratory tract diseases, and the new CoVs can penetrate the barrier between other species and humans and can cause a high mortality rate. In the course of this study, a novel approach to networking, based on the density-dependent differential equations, is adopted for the precise explanation of the propagation of the virus and the effect of quarantine on it. An infectious disease model with a time delay is suggested based on the conventional infectious disease model. To describe the viral infection period and treatment time, the time differential is used. Using the epidemic data released in real-time, the minimum error is obtained firstly through the inversion of the numerical simulation parameter; then we simulate the development pattern of the epidemic according to the dynamics system; finally, the effectiveness of quarantine steps is compared and analyzed. With the help of a discrete model, the transformations are documented in detail that is difficult to evaluate numerically. The provided numerical results are in close agreement with the experimental findings. The modeling of Petri nets (PNs) used has proven to be a successful method. The current research strategy can help the public to gain awareness of the disease spread, which is highly desired.