International Journal of Biomathematics

A co-infection model for human papillomavirus (HPV) and syphilis with cost-effectiveness optimal control analysis is developed and presented. The full co-infection model is shown to undergo the phenomenon of backward bifurcation when a certain condition is satisfied. The global asymptotic stability of the disease-free equilibrium of the full model is shown not to exist when the associated reproduction number is less than unity. The existence of endemic equilibrium of the syphilis-only sub-model is shown to exist and the global asymptotic stability of the disease-free and endemic equilibria of the syphilis-only sub-model was established, for a special case. Sensitivity analysis is also carried out on the parameters of the model. Using the syphilis associated reproduction number, ${{ℛ}}_{0s}$, as the response function, it is observed that the five-ranked parameters that drive the dynamics of the co-infection model are the demographic parameter $\mu$, the effective contact rate for syphilis transmission, ${\beta }_s$, the progression rate to late stage of syphilis ${\sigma }_2$, and syphilis treatment rates: ${\tau }_1$ and ${\tau }_2$ for co-infected individuals in compartments $H_i$ and $H_l$, respectively. Moreover, when the HPV associated reproduction number, ${{ℛ}}_{0h}$, is used as the response function, the five most dominant parameters that drive the dynamics of the model are the demographic parameter $\mu$, the effective contact rate for HPV transmission, ${\beta }_h$, the fraction of HPV infected who develop persistent HPV ${\rho }_1$, the fraction of individuals vaccinated against incident HPV infection $\varphi$ and the HPV vaccine efficacy ${\pi }_h$. Numerical simulations of the optimal control model showed that the optimal control strategy which implements syphilis treatment controls for singly infected individuals is the most cost-effective of all the control strategies in reducing the burden of HPV and syphilis co-infections.

In this paper, we analyze a higher-order stochastically perturbed multigroup staged-progression model for the transmission of HIV with saturated incidence rate. We obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a suitable stochastic Lyapunov function. In addition, we make up adequate conditions for complete eradication and wiping out the infectious disease. In a biological interpretation, the existence of a stationary distribution implies that the disease will prevail and persist in the long term. Finally, examples and numerical simulations are introduced to validate our theoretical results.

In this paper, we study a drug epidemic model based on epidemiology by dividing the human population into four classes at time $t$: susceptibles (S), drug users (I), drug users who are treated in isolation and temporarily quit drugs$(Q_T)$ and drug users who are treated in isolation and permanently quit drugs $(Q_p)$. We obtain the basic reproduction number $R_0$ of the model and perform its sensitivity analysis. We show that if $R_0<\frac{{\beta }_1\mathrm{\Lambda }}{\mu (\mu +{\delta }_1)}<1$, then the drug-free equilibrium is globally asymptotically stable, and if $R_0>1$, there exists an drug-abuse equilibrium and it is locally asymptotically stable. The proposed model may possess forward and backward bifurcations. Moreover, three different control strategies and numerical results are presented. Through different adjustments to obtain graphical results, we obtain the best strategy to control the drug epidemic.

We investigate an optimal control problem for a hierarchical size-structured population model, which incorporates the inoculation delay into the globally feedbacked boundary condition. After the establishment of the continuity of the states with respect to the harvesting efforts, the existence of the optimal solutions is proved by a result in convex analysis, and the maximum principle is obtained via conjugate system and tangent-normal cones.

Evolutionary game theory has extensively investigated situations in which several games are competing against each other at the same time, but the model only assumes symmetric interactions in homogeneous environments. Now, the population is considered in heterogeneous environments, individuals in the population occupy a different quality of patches, and individual fitness depends not only on the interaction between individuals, but also on the quality of the environment. Here, by establishing a mathematical framework, we analyze the natural selection between two strategies and two games in heterogeneous environments. Furthermore, we analyze the natural selection of Prisoner’s Dilemma and Hawk–Dove games theoretically to demonstrate the dynamics of cooperators and defectors in their choice of environment and their respective games. As expected, the distribution of games and strategies changes with time. Based on different initial population compositions, we also discuss the invasion problem of games from different perspectives. To one’s surprise, we can find that good quality patches attract all individuals; the long-term dynamics in invariant rich environments is the same as the dynamics of symmetric interactions in homogeneous environments.

In this paper, we propose and investigate an SIRS model with age structure and two delays. Both the infected and the recovered individuals have age structure, the infection rate (from the infective to the susceptible) and the immune loss rate (from the recovered to the susceptible) are related to two independent time delays, respectively. We prove that the proposed age structured SIRS model is well-posed by using the $C_0$-semigroup theory. The basic reproduction number ${\Re }_0$ is given, and the unique endemic equilibrium exists when ${\Re }_0>1$, while the disease-free equilibrium always exists. A rigorous mathematical analysis for the stability of two equilibria is provided. The disease-free equilibrium is local asymptotically stable if ${\Re }_0<1$, and the endemic equilibrium is local asymptotically stable if ${\Re }_0>1$ and ${\tau }_l=0$. Finally, we give numerical simulations to verify our results.

In this paper, we propose detailed and reasonable viral dynamics by using a multi-compartment model that incorporating the age since the infection of multiple infected cells, multiple target cells (Langerhans-cells and CD4${}^{+}$ T-cells), multiple viral strains (CCR5 and CXR4 HIV) and multiple infection routes (cell-to-cell and cell-to-virus). The basic reproduction number, $R_0$, of the whole model is derived from two transmission mechanisms: one is the potential trigger from the infection routes for a single target cell and other is the joint effect of multiple viral infections for multi-target cells. Accordingly, we study the global stability of the steady states for the single target model. For the whole model, we prove that the infection-free steady state is globally asymptotically stable if $R_0<1$, whereas viruses persist uniformly if $R_0>1$. Numerical simulations are carried out to illustrate the theoretical results. Sensitive analyses expound the effect of model parameters on the comprehensive reproduction number. It is remarkable to find that simultaneous control of HIV infection for two target cells can effectively reduce the viral loads within-host. Finally, our work suggests that the synergetic mechanism of multi-target cells and multi-strain cannot be ignored during treatment.

In this paper, we study the linear determinacy of the minimal speed of traveling wavefronts for a three-component competition system with nonlocal dispersal. We first transform this system into a cooperative system. Then, by constructing suitable upper solutions, we give some general conditions to ensure the linear determinacy of the minimal speed. Finally, we provide some more precise conditions that only rely on the parameters of the system such that the linear determinacy of the minimal speed is assured.

The world’s oceans have played an important role in sequestering atmospheric carbon dioxide through solubility and the action of algae. Fixation of atmospheric carbon dioxide by photoautotrophic algal cultures has the potential to diminish the release of carbon dioxide into the atmosphere, thereby helping to alleviate the trend toward global warming. This work investigates the role of algae in controlling the level of atmospheric carbon dioxide. Partial Rank Correlation Coefficients (PRCCs) technique is used to address how the concentration of atmospheric carbon dioxide is affected by changes in a specific parameter disregarding the uncertainty over the rest of the model parameters. Parameters related to algal growth are shown to significantly reduce the level of atmospheric CO₂. Further, we explore the dynamics of nonautonomous system by incorporating the seasonal variations of some ecologically important model parameters. Our nonautonomous system exhibits globally attractive positive periodic solution, and also the appearance of double periodic solution is observed. Moreover, by letting the seasonally forced parameters as almost periodic functions of time, we show almost periodic behavior of the system. Our findings suggest that the policy makers should focus on continuous addition of nutrients in the ocean to accelerate the algal growth thereby reducing the level of carbon dioxide in the atmosphere.

An Impulsive Multidirectional Associative Memory Neural Network (IMAMNN) with time-varying and leakage delays is proposed. Through the use of a continuation theorem of coincidence degree theory and differential inequality techniques we establish new conditions for the existence and exponential stability of anti-periodic solutions for the model considered in this work. Moreover, two examples and its numerical simulations are presented to show the validity and the effectiveness of the results.

This work provides analytical results towards applications in the field of invasive-invaded systems modeled with nonlinear diffusion and with advection. The results focus on showing regularity, existence and uniqueness of weak solutions using the condition of a nonlinear slightly positive parabolic operator and the reaction-absorption monotone properties. The coupling in the reaction-absorption terms, that characterizes the species interaction, impedes the formulation of a global comparison principle that is shown to exist locally. Additionally, this work provides analytical solutions obtained as selfsimilar minimal and maximal profiles. A propagating diffusive front is shown to exist until the invaded specie notes the existence of the invasive. When the desertion of the invaded starts, the diffusive front vanishes globally and the nonlinear diffusion concentrates only on the propagating tail which exhibits finite speed. Finally, the invaded specie is shown to exhibit an exponential decay along a characteristic curve. Such exponential decay is not trivial in the nonlinear diffusion case and confirms that the invasive continues to feed on the invaded during the desertion.

Considering the impact of environmental white noise on the quantity and behavior of vector of disease, a stochastic differential model describing the transmission of Dengue fever between mosquitoes and humans, in this paper, is proposed. By using Lyapunov methods and Itô’s formula, we first prove the existence and uniqueness of a global positive solution for this model. Further, some sufficient conditions for the extinction and persistence in the mean of this stochastic model are obtained by using the techniques of a series of stochastic inequalities. In addition, we also discuss the existence of a unique stationary distribution which leads to the stochastic persistence of this disease. Finally, several numerical simulations are carried to illustrate the main results of this contribution.