International Journal of Biomathematics

Environmental perturbations are unavoidable in the propagation of infectious diseases. In this paper, we introduce the stochasticity into the susceptible–infected–recovered (SIR) model via the parameter perturbation method. The stochastic disturbances associated with the disease transmission coefficient and the mortality rate are presented with two perturbations: Gaussian white noise and Lévy jumps, respectively. This idea provides an overview of disease dynamics under different random perturbation scenarios. By using new techniques and methods, we study certain interesting asymptotic properties of our perturbed model, namely: persistence in the mean, ergodicity and extinction of the disease. For illustrative purposes, numerical examples are presented for checking the theoretical study.

The contact rate is defined as the average number of contacts adequate for disease transmission by an individual per unit time and it is usually assumed to be constant in time. However, in reality, the contact rate is not always constant throughout the year due to different factors such as population behavior, environmental factors and many others. In the case of serious diseases with a high level of infection, the population tends to reduce their contacts in the hope of reducing the risk of infection. Therefore, it is more realistic to consider it to be a function of time. In particular, the study of models with contact rates decreasing in time is well worth exploring. In this paper, an SIR model with a time-varying contact rate is considered. A new form of a contact rate that decreases in time from its initial value till it reaches a certain level and then remains constant is proposed. The proposed form includes two important parameters, which represent how far and how fast the contact rate is reduced. These two parameters are found to play important roles in disease dynamics. The existence and local stability of the equilibria of the model are analyzed. Results on the global stability of disease-free equilibrium and transcritical bifurcation are proved. Numerical simulations are presented to illustrate the theoretical results and to demonstrate the effect of the model parameters related to the behavior of the contact rate on the model dynamics. Finally, comparisons between the constant, variable contact rate and variable contact rate with delay in response cases are presented.

In this paper, Health-related quality of life has not been adequately measured in bladder cancer. A recently developed reliable and disease-specific quality of life instrument (Bladder Cancer Index, BCI) was used to measure. Progressive type II censoring schemes have potential usefulness in practice where budget constraints in place or there is a necessity for the speedy test. To test the process capability, the lifetime performance index $C_L$ is widely recommended for evaluating the performance of the product’s lifetime and evaluating the lifetime performance index $C_L$ for the three-parameter Weighted-Lomax distribution (WLx) under progressive type-II censoring sample for a lower specification limit ($L$). The statistical inference concerning $C_L$ is conducted via obtaining the maximum likelihood of $C_L$ on the base of progressive type-II censoring. The asymptotic normal distribution of the MLE of $C_L$ and the confidence interval is proposed. Moreover, the hypothesis testing of $C_L$ for evaluating the lifetime performance of WLx data is conducted. Also, assuming the conjugate prior distribution and squared error loss function, this study constructs a Bayes estimator of $C_L$. The Bayes estimator of $C_L$ is then utilized to develop a credible interval in the condition of known $L$. Moreover, we propose a Bayesian test to assess the lifetime performance of products. We also propose a Bayesian test to assess the lifetime performance of products. Finally, two examples are given, one of them is considering a real life data of the remission times of bladder cancer patients in endurance lifetime test and the other is a simulated example to illustrate the usage of the proposed procedure.

In this paper, a stochastic delayed differential algebraic predator–prey model with Michaelis–Menten-type prey harvesting is proposed. Due to the influence of gestation delay and stochastic fluctuations, the proposed model displays a complex dynamics. Criteria on the local stability of the interior equilibrium are established, and the effect of gestation delay on the model dynamics is discussed. Taking the gestation delay and economic profit as bifurcation parameters, Hopf bifurcation and singularity induced bifurcation can occur as they cross through some critical values, respectively. Moreover, the solution of the model will blow up in a limited time when delay $\tau >{\tau }_0$. Then, we calculate the fluctuation intensity of the stochastic fluctuations by Fourier transform method, which is the key to illustrate the effect of stochastic fluctuations. Finally, we demonstrate our theoretical results by numerical simulations.

In this paper, we analyze two stochastic predator–prey models with distributed delay and stage structure for prey. For the nonautonomous periodic case of the model, by using Khasminskii’s theory of periodic solution, we show that the system has at least one positive $T$-periodic solution. For the model which is disturbed by both white and telegraph noises, we obtain sufficient criteria for positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching. The positive recurrence implies that both prey and predator populations will be persistent in the long term.

We study a conflict of two individuals over a valuable resource. We construct a sequential game where the first individual, the Owner, decides whether to defend the resource and the other individual, the Intruder, decides whether to attack and try to steal the resource. Individuals know the value of the resource to themselves. This provides the individuals an estimate of the value of the resource to the opponent. We build a mathematical model that allows us to quantify and vary the accuracy of this estimate. We study how the outcomes of the game depend on the accuracy. We show that, in our setting, the accuracy does not matter to the Intruder but it does to the Owner. For resources of a large value, the Owner benefits from a smaller accuracy. However, for resources of a small value, the Owner benefits from a bigger accuracy.

This paper presents a mathematically formalized description of the process of crossing individuals of different populations as a break of the first kind of functional dependencies of the population state vector components. In this case, the validity of the quasispecies model as a model of population development was assumed. The practical interpretation of a mathematically formalized description is considered on the example of an artificial population of Lymantria dispar. The results of the work can be used to assess the development parameters of both natural and artificial populations.

Migration can be divided into temporary and permanent migration, which is related to the residence time of people in the patch, thus we consider an SIS epidemic model with migration and residence time in a patchy environment. If $R_0\le 1$, the disease-free equilibrium is globally asymptotically stable and the disease dies out. With the same migration rate of susceptible and infectious individuals and without disease-induced death, when $R_0>1$, the endemic equilibrium is unique and globally asymptotically stable. Numerical simulations are carried out to show the effects of residence time and the migration rate on disease prevalence.

In this paper, we propose a delayed prey-predator-scavenger system with fear effect and linear harvesting. First, we discuss the existence and stability of all possible equilibria. Next, we investigate the existence of Hopf bifurcation of the delayed system by regarding the gestation period of the scavenger as a bifurcation parameter. Furthermore, we obtain the direction of Hopf bifurcation and the stability of bifurcating periodic solutions by using the normal form theory and the central manifold theorem. In addition, we give the optimal harvesting strategy of the delayed system based on Pontryagin’s maximum principle with delay. Finally, some numerical simulations are carried out to verify our theoretical results.

In the actual microbial fermentation process, excessive or insufficient substrate can produce inhibitory effects on cells growth. The artificial substrate feeding rules by past experiences have great blindness to keep substrate concentration in a given appropriate range. This paper considers that alkali feed depends on pH value of the solution and glycerol feed depends on glycerol concentration of the solution in the uncoupled microbial fed-batch fermentation process, and establishes a state-dependent switched system in which the flow rates of glycerol and alkali, the number of mode switches, the mode sequence and the switching times are prior unknown. To maximize the yield of target product 1,3-Propanediol (1,3-PD), we formulate a switching optimal control problem with the flow rates of glycerol and alkali, the number of mode switches, the mode sequence and the switching times as decision variables, which is a mixed-integer dynamic programming problem. For solving the mixed-integer dynamic programming problem, the control parametrization technique, the time scaling transformation and the embedded system technology are used to obtain an approximate parameter optimization problem. By using a parallel optimization algorithm, we obtain the optimal control strategies. Under the obtained optimal control strategies, the 1,3-PD yield at the terminal time is increased significantly compared with the previous results.