Narrow Results

In this paper, a new mathematical model of the interactions between a growing tumor and an immune system is presented by incorporating the danger model. The populations involved are tumor cells, CD8⁺ T-cells, natural killer cells (NK-cells), dendritic cells (DCs) and cytokine interleukin-12 (IL-12). A key feature of this work is the inclusion of the danger model into the dynamics of the immune system, which is rarely considered by previous works. Regarding the constructed mathematical model, both the location of equilibria and their stability properties are discussed, which are useful not only to gain a broad understanding of the specific system dynamics, but also to help guide the development of therapies. Moreover, numerical simulations of the system with chemotherapy and immunotherapy by using specific parameters are presented to illustrate that proper therapy is able to eliminate the entire tumor. In addition, we illustrate cases for which neither chemotherapy nor immunotherapy alone are able to control tumor growth, but a combination treatment is sufficient to eliminate the tumor cells.

In this paper, we develop a two-strain SIS model on heterogeneous networks with demographics for disease transmission. We calculate the basic reproduction number $R_0$ of infection for the model. We prove that if $R_0<1$, the disease-free equilibrium is globally asymptotically stable. If $R_0>1$, the conditions of the existence and global asymptotical stability of two boundary equilibria and the existence of endemic equilibria are established, respectively. Numerical simulations illustrate that the degree distribution of population varies with time before it reaches the stationary state. What is more, the basic reproduction number $R_0$ does not depend on the degree distribution like in the static network but depend on the demographic factors.

A mathematical model considering two strains of hepatitis B virus (HBV) chronic carriers, to assess the impact of dose-structured imperfect vaccine, in a population, is designed and analyzed. The model is shown to have a locally and globally asymptotically stable disease-free equilibrium (DFE) whenever its associated reproduction number is numerically less than unity. Numerical analysis of the model shows that with the expected 50% minimum efficacy of the first vaccine dose, vaccinating 55% of the susceptible population with the first vaccine dose will be sufficient to effectively control the spread of hepatitis B infection. Such effective control can also be achieved if 50% of the first vaccine dose recipients take the second dose. Threshold analysis reveals that an imperfect HBV vaccine should have positive or negative population-level effect. Latin hypercube sampling–PRCC analysis illustrates that disease transmission rate, birth rate, natural death rate and proportion of children born with maternal immunity are most influential parameters in the disease dynamics. In this paper, the sensitivity analysis based on mathematical and in addition statistical techniques have been performed to determine the significance of the model parameters. It is observed that a number of the parameters play an important role to determine the magnitude of the basic reproduction number. Sensitivity analysis is achieved to determine model parameters’ importance in disease dynamics. It is observed that the reproduction number is the most responsive quantity to the potent transmission rate of HBV and in addition also vital to control the spread of the disease.

Tuberculosis (TB) is a transmittable bacterial infection, and it is one of the main health problems worldwide. This infection is preventable and can be cured with an appropriate treatment. The mathematical modeling technique could be applied successfully to investigate the transmission dynamics and provide the proper control measures of transferable diseases inclusive TB. In this paper, we developed a mathematical model of TB with drug resistance TB (DR-TB) in the presence of optimal control. We considered the two diseases — drug sensitive TB (DS-TB) and drug resistance TB. They affect the country Ethiopia. The DS-TB can be cured by first-line anti-TB drugs. However, once the tubercle bacilli begins it resistance to one or more anti-TB drugs, the DR-TB appears. This type of TB is difficult for the physicians to detect the strains. It is also expensive to treat. We analyzed the model and discussed the basic elements such as equilibrium points, basic reproduction number, stabilities of equilibrium points and possibility of bifurcation. The analytical result showed that if the threshold quantity $(R_0)<1$, the disease-free equilibrium is stable, whereas if $R_0>1$, the endemic equilibrium (EE) is stable. When $R_0=1$, a backward bifurcation appears. We extended the model by proposing strategies such as preventive effort, case finding control and case holding control. In this study, four different strategies are introduced based on different combination measures. Moreover, the optimal control problem is examined both analytically and numerically. The finding suggested that optimal combination strategies are used to reduce both the disease burden and the cost of intervention.

The dynamic behavior of tri-trophic food chains consisting of resources, prey, predator and top-predator is dealt with. We compare a formulation whereby the prey growth is logistic, with a mass balance formulation. In the case of the mass balance formulation both the linear and the hyperbolic functional response are discussed. The consequences of the different formulations on the dynamics of a microbial food chain in chemostat situation are described. Bifurcation diagrams for the nonlinear dynamic systems are given. When the prey grows logistically there is no coexistence of the three species for biologically realistic parameter values for a microbial food chain. The same holds for the mass balance equations with a linear functional response for the prey. For a hyperbolic functional response, however, there is a stable equilibrium for the whole food chain in a rather large region of the parameter space. Furthermore, this model shows more complex dynamic behaviors; besides point attractors, limit cycles and chaotic attractors.