Cognitive Familiarity‐Driven Rumor Diffusion: A Dynamic Compartmental Model With Stratified Susceptibility

This article discusses the mathematical model of the spread of COVID-19 by considering vaccination and population migration. The former model is analyzed by determining the equilibrium point, basic reproduction number, analyzing the stability of the equilibrium point, sensitivity analysis, and accompanied by numerical simulation. Analysis of the stability of disease-free and endemic equilibrium points using the Routh-Hurwitz Criteria and the Castillo-Chaves and Song theorems. The results of the analysis show that there are two equilibrium points, namely a disease-free equilibrium point (T1), which is locally asymptotically stable when R0 1, and an endemic equilibrium point (T2), which is locally asymptotically stable when R0 1. Furthermore, the sensitivity analysis showed that the most sensitive parameters to changes in the basic reproduction number were the emigration rate parameter (m2) and the infection probability parameter after contact between infected and susceptible individuals without vaccination (h). In addition, the numerical simulation results show that the sensitive parameter values, namely m2, h, zse, g, and # have a significant effect on the basic reproduction numbers. Suppressing the chance of infection in susceptible individuals and the rate of contact between susceptible and exposed individuals, as well as increasing the number of individuals who emigrate and who are vaccinated, can reduce the transmission of COVID-19.

In this study, we present a unified mathematical model for tuberculosis (TB) that integrates key interventions: Mask use and media campaigns to raise community awareness and promote vaccine booster uptake. The model also incorporates slow-fast disease progression and limited treatment capacity. A mathematical analysis was conducted to determine the existence and stability of equilibrium points. From the mathematical analysis on the stability criteria of the TB-free equilibrium point, we show that TB can be eradicated if the basic reproduction number is below one. However, due to insufficient treatment capacity, a backward bifurcation may occur when the reproduction number equals one, enabling the coexistence of endemic and disease-free equilibria even when the reproduction number is below one. The parameter estimation is based on TB incidence data per 100,000 individuals in Indonesia. Sensitivity analysis reveald that although both interventions are effective, media campaigns combined with vaccine boosters are more impactful in reducing TB transmission than the use of masks. Numerical simulations further suggest the possibility of periodic outbreaks, indicating potential seasonal TB patterns. To explore adaptive intervention strategies, we extended the model using an optimal control framework. Our findings suggested that combined implementation of face masks and media campaigns is more effective than using either alone, particularly when the likelihood of rapid disease progression increases.

This article presents a mathematical model of tuberculosis (TB) that incorporates non-linear incidence rate, relapse, and reinfection to capture the complexity of TB transmission dynamics. The non-linear incidence rate is introduced to capture the significant impact of population ignorance on the dangers of TB, which can lead to its rapid spread. In this study, the existence and stability of equilibrium points are analyzed both analytically and numerically. Our findings indicate that a basic reproduction number less than one is not sufficient to ensure TB elimination within a population. The model exhibits complex dynamics, including forward and backward bifurcation with hysteresis, as well as the potential for multiple stable equilibria (bistability) due to the effects of nonlinear incidence rates and reinfection. Bistability is a common phenomenon in Tuberculosis transmission models, characterized by unique features such as relapse and reinfection processes. Bistability enables both TB-free and TB-endemic equilibria to coexist, even when a stable TB-free equilibrium exists. The occurrence of three endemic equilibria adds complexity to the model, illustrating the challenges in TB control. When bistability occurs, we analyzed the potential shifts in stability trajectories from the endemic equilibrium to the disease-free equilibrium through specific interventions. Our global sensitivity analysis of the infected population emphasizes that primary infection and recovery rates are crucial parameters for reducing TB transmission. These insights highlight the importance of controlling primary infection through the use of preventive measures and optimizing recovery strategies to support the efforts taken toward TB eradication. This analysis offers a nuanced perspective on the challenges of achieving TB eradication, particularly in settings with high relapse and reinfection risks, and underscores the need for the implementation of comprehensive intervention strategies in public health programs. A numerical simulation using an adjustable infection rate step function was conducted to explore the optimal combination of intervention intensity, timing, and duration required for effective TB elimination. We illustrate how optimal timing and intervention intensity can shift the solution trajectory from a TB-endemic to a TB-free equilibrium when bistability occurs.

Bifurcation is an indispensable tool to describe the behavior of the system at steady states. Recently, the forward bifurcation showed the existence of both local and global stability of equilibrium points obtained from epidemiological models. It is known that the computing process to show the global stability of endemic equilibrium is tricky. But, in this chapter, we incorporate the principles that support the simplification of computation and give the exact existence of global stability of endemic equilibrium point. The most important issue is the application of forward bifurcation diagram obtained from endemic equilibrium and basic reproduction number. For illustration purposes, the mathematical modeling of HBV transmission dynamics is built in this study. The generated HBV model’s well-posedness is confirmed, and the equilibrium points are determined. Additionally, a next-generation matrix approach is used to calculate the basic reproduction number from infected compartments, and numerical simulations are used to demonstrate the occurrence of forward bifurcation at R0=1. If R0<1, the disease-free equilibrium point is both locally and globally asymptotically stable, and if R0>1, the endemic equilibrium is both locally and globally asymptotically stable. The MATLAB platform is used to facilitate numerical simulation.

Dengue is one of the major health problems in Bangladesh and many people are died in recent years due to the severity of this disease. Therefore, in this paper, a SIRS model for the human and SI model for vector population with saturated incidence rate and constant treatment function has been presented to describe the transmission of dengue. The equilibrium points and the basic reproduction number have been computed. The conditions which lead the disease free equilibrium and the endemic equilibrium have been determined. The local stability for the equilibrium points has been established based on the eigenvalues of the Jacobian matrix and the global stability has been analyzed by using the Lyapunov function theory. It is found that the stability of equilibrium points can be controlled by the reproduction number. In order to calculate the infection rate, data for infected human populations have been collected from several health institutions of Bangladesh. Numerical simulations of various compartments have been generated using MATLAB to investigate the influence of the key parameters for the transmission of the disease and to support the analytical results. The effect of treatment function over the infected compartment has been illustrated. The sensitivity of the reproduction number concerning the parameters of the model has been analyzed. Finally, the most sensitive parameter that has the highest effect over reproduction number has been identified.

<abstract><p>Infectious diseases have a great impact on the economy and society. Dynamic models of infectious diseases are an effective tool for revealing the laws of disease transmission. Quarantine and nonlinear innate immunity are the crucial factors in the control of infectious diseases. Currently, there no mathematical models that comprehensively study the effect of both innate immunity and quarantine. In this paper, we propose and analyze an SEIQR epidemic model with nonlinear innate immunity. The boundedness and positivity of the solutions are discussed. Employing the next-generation matrix, we compute the expression of the basic reproduction number. Under certain conditions, the phenomenon of backward bifurcation may occur. That is to say, the stable disease-free equilibrium point and the stable endemic equilibrium point coexist when the basic reproduction ratio is less than one. And the basic reproduction number is no longer the threshold value to determine whether the disease breaks out. We investigate the globally asymptotical stability of the disease-free equilibrium point for the system by constructing Lyapunov function. Also, we research the global stability of the endemic equilibrium by using geometric approach. Numerical simulations are carried out to reveal the theoretical results and find some complex dynamics (for example, the existence of Hopf bifurcation) of the system. Both theoretical and numerical results indicate that the nonlinear innate immunity may cause backward bifurcation and Hopf bifurcation, which makes more difficult to eliminate the disease.</p></abstract>

In this paper, an age-structured model with delay applied to Schistosomiase in the Republic of Niger is studied. First, the model is formulated using delay differential equations [25]. Then, the basic reproduction number is determined. Once this task is completed, the stability of the disease-free equilibrium point is analyzed. The analytical and numerical results obtained show the stability of the disease-free equilibrium point. Subsequently, by substitution, the endemic equilibrium point is determined. Then, its stability is investigated by using the method of Routh Hurwitz [5, 23, 25]. To do this, the dynamic system reflecting the dynamics of the disease is linearized. Then, the characteristic polynomial of the linearized system is determined. A polynomial depending on the incubation time of the disease is obtained [22, 25]. Afterwards, the stability of the equilibrium point depending on the value of the incubation time is discussed. The analytical and numerical results obtained show that the endemic equilibrium point is stable when the time delay is neglected and it is unstable from certain values of the incubation time. From the analytical and simulation results, we observed that a Hopf bifurcation occurs for a certain value of latency time.

Tuberculosis is an air-borne infectious disease which is transmitted to one another through the respiratory system and mostly occurs due to close contact with an infected person. Here, a mathematical of SIR type is proposed for the dynamics of tuberculosis with the effect of treatment and time delay. The level of treatment is assumed proportional to the number of infected people reported to the health organization. The equilibria and stability analysis has been carried out using qualitative theory. This paper provides some vital information such as the basic reproduction number \(R_0\) and the stability of equilibrium points. Hopf bifurcation at the endemic steady states has been analyzed taking delay as the main parameter. Numerical simulations fulfill analytical outcomes. We found that large time delay in treatment can cause problems and it should be avoided.

A model for the dynamic transmission of SEITR pandemics is proposed, incorporating awareness and a mass action incidence function. The stability of both endemic and disease-free equilibrium points was examined, revealing that the disease-free equilibrium is locally and globally asymptotically stable when the Basic Reproduction Number is less than 1, while the endemic equilibrium is stable when . Additionally, it's shown that the Basic Reproduction Number (without awareness) is greater than (with awareness), this means that when , then . A Lyapunov function is constructed to demonstrate the global stability of both disease-free equilibrium when and the endemic equilibrium when . Numerical simulations illustrate the impact of awareness on disease spread, highlighting the importance of awareness in informing and educating individuals about infection risks during outbreaks, thereby reducing the health burden of the epidemic by lowering its peak.

This article deals with the dynamics of prey and predator populations in a two patch environment, a protected area and an unprotected area for fishing. The prey disperses between the two patches and migrates easily. There are two predators, one is in the protected area and another is in the unprotected area. The predators cannot migrate. Both prey and predator in unprotected area are harvested with constant efforts. The dynamical behavior of the populations is stated as a system of differential equations. The existence of a positive equilibrium point and its stability are investigated. We discuss the local stability of the positive equilibrium point. The stable equilibrium point is then associated with optimal harvesting problems. Based on the analysis, we found that there exist a stable positive equilibrium point when there is no harvesting. For model with constant efforts for both prey and predator, we found that over fishing will maximize the profit but the predator in the unprotected area will be extinct. With the help of Pontryagin’s maximum principle in maximizing the present value of revenues, we found the extremal of the efforts that maximize the present value of revenues. This means that both prey and predator in the protected area as well as the prey and predator in the unprotected area are possibly coexist although the prey and the predator in the unprotected area are harvested with constant efforts. Some numerical simulations area given to confirm the result of analysis.

<p style='text-indent:20px;'>We study a predator-prey model with Holling type Ⅰ functional response, an alternative food source for the predator, and multiple Allee effects on the prey. We show that the model has at most two equilibrium points in the first quadrant, one is always a saddle point while the other can be a repeller or an attractor. Moreover, there is always a stable equilibrium point that corresponds to the persistence of the predator population and the extinction of the prey population. Additionally, we show that when the parameters are varied the model displays a wide range of different bifurcations, such as saddle-node bifurcations, Hopf bifurcations, Bogadonov-Takens bifurcations and homoclinic bifurcations. We use numerical simulations to illustrate the impact changing the predation rate, or the non-fertile prey population, and the proportion of alternative food source have on the basins of attraction of the stable equilibrium point in the first quadrant (when it exists). In particular, we also show that the basin of attraction of the stable positive equilibrium point in the first quadrant is bigger when we reduce the depensation in the model.

In this paper we have presented a deterministic model for pneumonia transmission and we have used the model to avail the potential impact of therapy. The model is based on the vaccinated-susceptible-carrier-infected-recovered-susceptible compartmental structure and their possible interventions with the possibility of infected individual recovery from natural immunity. Here, we have modeled Pneumonia considering vaccination, screening and treatment with a system of nonlinear ordinary differential equation. The model reproduction number R0 is derived and the stability of the equilibria are derived. The stability of equilibrium points is analyzed. The results shows that there exists a locally stable disease free equilibrium points, E0 when R0&lt;1 and a unique endemic equilibrium E1, when R0&gt;1. Infection free point was found to be locally stable and if reproduction number is greater than unity, then there is unique endemic equilibrium point and if it is less than unity, the endemic equilibrium point is globally asymptotically stable and pneumonia will be eliminated.

A new mathematical model for the transmission dynamics of herpes simplex virus type 2 (HSV-2), which takes into account disease transmission by infected individuals in the quiescent state and an imperfect HSV-2 vaccine, is designed and qualitatively analysed. In the absence of vaccination, it is shown that the model has a globally asymptotically stable (GAS) disease-free equilibrium (DFE) point whenever an epidemiological threshold, known as the ‘basic reproduction number’, is less than unity. Further, this model has a unique endemic equilibrium whenever the reproduction number exceeds unity. Using a non-linear Lyapunov function, it is shown that the unique endemic equilibrium is GAS (for a special case) when the associated reproduction threshold is greater than unity. On the other hand, the model with vaccination undergoes a vaccine-induced backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium when the reproduction threshold is less than unity. Threshold analysis of the vaccination model reveals that the use of an imperfect HSV-2 vaccine could have positive or negative population-level impact (in reducing disease burden). Simulations of the vaccination model show that an HSV-2 vaccine could lead to effective disease control or elimination if the vaccine efficacy and the fraction of susceptible individuals vaccinated at steady state are high enough (at least 80% each).

Atherosclerosis is a non-communicable disease (NCDs) which appears when the blood vessels in the human body become thick and stiff. The symptoms range from chest pain, sudden numbness in the arms or legs, temporary loss of vision in one eye, or even kidney failure, which may lead to death. Treatment in cases with severe symptoms requires surgery, in which the number of doctors or hospitals is limited in some countries, especially countries with low health levels. This article aims to propose a mathematical model to understand the impact of limited hospital resources on the success of the control program of atherosclerosis spreads. The model was constructed based on a deterministic model, where the hospitalization rate is defined as a time-dependent saturated function concerning the number of infected individuals. The existence and stability of all possible equilibrium points were shown analytically and numerically, along with the basic reproduction number. Our analysis indicates that our model may exhibit various types of bifurcation phenomena, such as forward bifurcation, backward bifurcation, or a forward bifurcation with hysteresis depending on the value of hospitalization saturation parameter and the infection rate for treated infected individuals. These phenomenon triggers a complex and tricky control program of atherosclerosis. A forward bifurcation with hysteresis auses a possible condition of having more than one stable endemic equilibrium when the basic reproduction number is larger than one, but close to one. The more significant value of hospitalization saturation rate or the infection rate for treated infected individuals increases the possibility of the stable endemic equilibrium point even though the disease-free equilibrium is stable. Furthermore, the Pontryagin Maximum Principle was used to characterize the optimal control problem for our model. Based on the results of our analysis, we conclude that atherosclerosis control interventions should prioritize prevention efforts over endemic reduction scenarios to avoid high intervention costs. In addition, the government also needs to pay great attention to the availability of hospital services for this disease to avoid the dynamic complexity of the spread of atherosclerosis in the field.

Dynamical analysis and effects of law enforcement in a social interaction model