International Journal of Biomathematics

In this paper, a discrete-time SIR epidemic model with nonlinear incidence and recovery rates is obtained by using the forward Euler’s method. The existence and stability of fixed points in this model are well studied. The center manifold theorem and bifurcation theory are applied to analyze the bifurcation properties by using the discrete time step and the intervention level as control parameters. We discuss in detail some codimension-one bifurcations such as transcritical, period-doubling and Neimark–Sacker bifurcations, and a codimension-two bifurcation with 1:2 resonance. In addition, the phase portraits, bifurcation diagrams and maximum Lyapunov exponent diagrams are drawn to verify the correctness of our theoretical analysis. It is found that the numerical results are consistent with the theoretical analysis. More interestingly, we also found other bifurcations in the model during the numerical simulation, such as codimension-two bifurcations with 1:1 resonance, 1:3 resonance and 1:4 resonance, generalized period-doubling and fold-flip bifurcations. The results show that the dynamics of the discrete-time model are richer than that of the continuous-time SIR epidemic model. Such a discrete-time model may not only be widely used to detect the pathogenesis of infectious diseases, but also make a great contribution to the prevention and control of infectious diseases.

Recent evidences show that individuals who recovered from COVID-19 can be reinfected. However, this phenomenon has rarely been studied using mathematical models. In this paper, we propose an SEIRE epidemic model to describe the spread of the epidemic with reinfection. We obtain the important thresholds $R_0$ (the basic reproduction number) and $R_c$ (a threshold less than one). Our investigations show that when $R_0>1$, the system has an endemic equilibrium, which is globally asymptotically stable. When $R_c<R_0<1$, the epidemic system exhibits bistable dynamics. That is, the system has backward bifurcation and the disease cannot be eradicated. In order to eradicate the disease, we must ensure that the basic reproduction number $R_0$ is less than $R_c$. The basic reinfection number is obtained to measure the reinfection force, which turns out to be a new tipping point for disease dynamics. We also give definition of robustness, a new concept to measure the difficulty of completely eliminating the disease for a bistable epidemic system. Numerical simulations are carried out to verify the conclusions.

Mitigation of the enhanced greenhouse gas (GHG) concentrations in the Earth’s atmosphere is imperative to meet the climate change mitigation objective. Governments of many countries are developing and implementing various mitigation strategies to reduce their GHG emissions. However, a time delay between the formulation and implementation of these mitigation policies can affect their effectiveness in controlling greenhouse gas levels in the atmosphere. This work presents black a nonlinear mathematical model to investigate the effect of application of mitigation strategies and the delay involved in their implementation over the reduction of atmospheric greenhouse gases. In model formulation, it is assumed that the mitigation strategies work two-fold; first they black reduce the GHG emission rate from the anthropogenic source and second they increase the black removal rate of greenhouse gas from the atmosphere. black A comprehensive stability analysis of the proposed model system is made to examine its long-term behavior. The model analysis shows that an increase in the implementation rate of mitigation strategies and their efficiencies to cut down the GHG emission rate from point sources and increase the GHG uptake rate lead to reduction in equilibrium GHG concentration. It is found that a long delay in the execution of mitigation policies can destabilize the system dynamics and leads to the generation of periodic oscillations. The expression for the threshold value of the delay parameter at which periodic oscillations arise via Hopf-bifurcation is determined. The stability and direction of bifurcating periodic solutions are discussed. A sensitivity analysis is performed to investigate the effect of changes in key parameters over system dynamics.

Quantum process may conduce to brain function. Particularly, it was proposed that the Posner molecules, Ca_g(PO₄)₆, can serve as neural qubits, which can maintain quantum entanglement between phosphorus nuclei for quite a long time. We study the process from entangled Posner molecules to the synchronized activities of brain. A diffusion model of Posner molecules with appropriate boundary conditions and the mean first arriving time of Posner molecules from one area of brain to another are obtained. We establish linkages between the quantum entangled connectivity of brain and the remaining time of entangled Posner molecules before decoherence. Based on these results, we find direct correlations between entangled Posner molecules and the functional connectivity of human brain. Furthermore, we discover that the quantum entangled connectivity depends on age.

The predation process plays a significant role in advancing life evolution and the maintenance of ecological balance and biodiversity. Hunting cooperation in predators is one of the most remarkable features of the predation process, which benefits the predators by developing fear upon their prey. This study investigates the dynamical behavior of a modified LV-type predator–prey system with Michaelis–Menten-type harvesting of predators where predators adopt cooperation strategy during hunting. The ecologically feasible steady states of the system and their asymptotic stabilities are explored. The local codimension one bifurcations, viz. transcritical, saddle-node and Hopf bifurcations, that emerge in the system are investigated. Sotomayors approach is utilized to show the appearance of transcritical bifurcation and saddle-node bifurcation. A backward Hopf-bifurcation is detected when the harvesting effort is increased, which destabilizes the system by generating periodic solutions. The stability nature of the Hopf-bifurcating periodic orbits is determined by computing the first Lyapunov coefficient. Our analyses revealed that above a threshold value of the harvesting effort promotes the coexistence of both populations. Similar periodic solutions of the system are also observed when the conversion efficiency rate or the hunting cooperation rate is increased. We have also explored codimension two bifurcations viz. the generalized Hopf and the Bogdanov–Takens bifurcation exhibit by the system. To visualize the dynamical behavior of the system, numerical simulations are conducted using an ecologically plausible parameter set. The existence of the bionomic equilibrium of the model is analyzed. Moreover, an optimal harvesting policy for the proposed model is derived by considering harvesting effort as a control parameter with the help of Pontryagins maximum principle.

This paper investigates multiple bifurcations analyses and strong resonances of the Bazykin–Berezovskaya predator–prey model in depth using analytical and numerical bifurcation analysis. The stability conditions of fixed points, codim-1 and codim-2 bifurcations to include multiple and generic bifurcations are studied. This model exhibits transcritical, flip, Neimark–Sacker, and $1:2$, $1:3$, $1:4$ strong resonances. The normal form coefficients and their scenarios for each bifurcation are examined by using the normal form theorem and bifurcation theory. For each bifurcation, various types of critical states are calculated, such as potential transformations between the one-parameter bifurcation point and different bifurcation points obtained from the two-parameter bifurcation point. To validate our analytical findings, the bifurcation curves of fixed points are determined by using MatcontM.

Waterborne disease threatens public health globally. Previous studies mainly consider that the birth of pathogens in water sources arises solely by the shedding of infected individuals. However, for free-living pathogens, intrinsic growth without the presence of hosts in environment could be possible. In this paper, a stochastic waterborne disease model with a logistic growth of pathogens is investigated. We obtain the sufficient conditions for the extinction of disease and also the existence and uniqueness of an ergodic stationary distribution if the threshold ${{ℛ}}_0^s>1$. By solving the Fokker–Planck equation, an exact expression of probability density function near the quasi-endemic equilibrium is obtained. Results suggest that the intrinsic growth in bacteria population induces a large reproduction number to determine the disease dynamics. Finally, theoretical results are validated by numerical examples.

This paper considered a multi-stage sheep brucellosis model with incomplete immunity. First, we established a deterministic model, calculated the basic reproduction number ${{ℛ}}_0$, set out the conditions for the global stability of the disease-free equilibrium and endemic equilibrium. Second, considering the influence of environmental white noise on brucellosis infection, we further established the stochastic version of the model. By constructing a suitable Lyapunov function, we proved the existence and uniqueness of the global positive solution. Further, we got the sufficient conditions for disease extinction and the existence of ergodic stationary distribution. Finally, we carried out some numerical simulations to verify the theoretical results.

In this paper, a stage structure predator–prey model consisting of three nonlinear ordinary differential equations is proposed and analyzed. The prey populations are divided into two parts: juvenile prey and adult prey. From extensive experimental data, it has been found that prey fear of predators can alter the physiological behavior of individual prey, and the fear effect reduces their reproductive rate and increases their mortality. In addition, we also consider the presence of constant ratio refuge in adult prey populations. Moreover, we consider the existence of intraspecific competition between adult prey species and predator species separately in our model and also introduce the gestation delay of predators to obtain a more realistic and natural eco-dynamic behaviors. We study the positivity and boundedness of the solution of the non-delayed system and analyze the existence of various equilibria and the stability of the system at these equilibria. Next by choosing the intra-specific competition coefficient of adult prey as bifurcation parameter, we demonstrate that Hopf bifurcation may occur near the positive equilibrium point. Then by taking the gestation delay as bifurcation parameter, the sufficient conditions for the existence of Hopf bifurcation of the delayed system at the positive equilibrium point are given. And the direction of Hopf bifurcation and the stability of the periodic solution are analyzed by using the center manifold theorem and normal form theory. What’s more, numerical experiments are performed to test the theoretical results obtained in this paper.

Optimal control technique is crucial to improve the yield of microbial fermentation production. In this paper, we propose a nonlinear control system with state-dependent impulses, where the impulsive volume of feeding glycerol and the critical concentration of glycerol for occurring impulse are the control variables, to formulate 1,3-propanediol (1,3-PD) fed-batch production process. We also discuss a quantity of important properties for this control system. Then, we analyze the sensitivity of system state with respect to the kinetic parameters. We further propose a constrained optimal control model governed by the control system with state-dependent impulses. The existence of the optimal impulsive controls is established. For solving this problem, we utilize an exact penalty method to transform the problem into an optimization problem with only box constraints. Moreover, an improved differential evolution method is developed to seek the optimal impulsive strategy. Finally, numerical simulation results demonstrate that, by using the optimal impulsive strategies, final 1,3-PD concentration is considerably increased under the nominal parameter values and disturbances of kinetic parameters have significant effects on the optimal final 1,3-PD yield.