Transmission dynamic and backward bifurcation of Middle Eastern respiratory syndrome coronavirus

Middle East respiratory syndrome coronavirus (MERS-CoV) remains an emerging disease threat with regular human cases on the Arabian Peninsula driven by recurring camels to human transmission events. In this paper, we present a new deterministic model for the transmission dynamics of (MERS-CoV). In order to do this, we develop a model formulation and analyze the stability of the proposed model. The stability conditions are obtained in term of R0, we find those conditions for which the model become stable. We discuss basic reproductive number R0 along with sensitivity analysis to show the impact of every epidemic parameter. We show that the proposed model exhibits the phenomena of backward bifurcation. Finally, we show the numerical simulation of our proposed model for supporting our analytical work. The aim of this work is to show via mathematical model the transmission of MERS-CoV between humans and camels, which are suspected to be the primary source of infection.

Assessing the Potential Impact of Immunity Waning on the Dynamics of COVID-19: An Endemic Model of COVID-19

We developed an endemic model of COVID-19 to assess the impact of vaccination and immunity waning on the dynamics of the disease. Our model exhibits the phenomenon of backward bifurcation and bi-stability, where a stable disease-free equilibrium co-exists with a stable endemic equilibrium. The epidemiological implication of this is that the control reproduction number being less than unity is no longer sufficient to guarantee disease eradication. We showed that this phenomenon could be eliminated by either increasing the vaccine efficacy or by reducing the disease transmission rate (adhering to non-pharmaceutical interventions). Furthermore, we numerically investigated the impacts of vaccination and waning of both vaccine-induced immunity and post-recovery immunity on the disease dynamics. Our simulation results show that the waning of vaccine-induced immunity has more effect on the disease dynamics relative to post-recovery immunity waning, and suggests that more emphasis should be on reducing the waning of vaccine-induced immunity to eradicate COVID-19.

Modelling HIV dynamics with cell-to-cell transmission and CTL response

In most HIV models, the emergence of backward bifurcation means that the control for basic reproduction number less than one is no longer effective for HIV treatment. In this paper, we study an HIV model with CTL response and cell-to-cell transmission by using the dynamical approach. The local and global stability of equilibria is investigated, the relations of subcritical Hopf bifurcation and supercritical bifurcation points are revealed, especially, the so-called new type bifurcation is also found with two Hopf bifurcation curves meeting at the same Bogdanov-Takens bifurcation point. Forward and backward bifurcation, Hopf bifurcation, saddle-node bifurcation, Bogdanov-Takens bifurcation are investigated analytically and numerically. Two limit cycles are also found numerically, which indicates that the complex behavior of HIV dynamics. Interestingly, the role of cell-to-cell interaction is fully uncovered, it may cause the oscillations to disappear and keep the so-called new type bifurcation persist. Finally, some conclusions and discussions are also given.

Complex Dynamics of An Epidemic Model With Saturated Media Coverage and Recovery

During the outbreak of emerging infectious diseases, media coverage and medical resource play important roles in affecting the disease transmission. To investigate the effects of the saturation of media coverage and limited medical resources, we proposed a mathematical model with extra compartment of media coverage and two nonlinear functions. We theoretically obtained that saturated recovery significantly contributes the occurrence of backward bifurcation and rich dynamics. Then it is reasonable to only considering nonlinear recovery, we theoretically showed that backward bifurcation can occur and multiple equilibria may coexist under certain conditions in this case. And numerical simulations reveals the rich dynamic behaviors, including forward-backward bifurcation, Hopf bifurcation, Saddle-Node bifurcation, Homoclinic bifurcation and unstable limit cycle. Comparing the system with linear recovery, where the threshold dynamic are almost completely characterized by a threshold condition called the basic reproduction number, we concluded that only saturated media impact hardly induces the complicated dynamics, while the nonlinear recovery function, associated with limitation of medical resources, may induce the coexistence of the disease-free equilibrium (DFE) and a endemic state or multiple endemic states, which means that the limitation of medical resources causes much difficulties in eliminating the infectious diseases.

A Bistable Phenomena Induced by a Mean-Field SIS Epidemic Model on Complex Networks: A Geometric Approach

In this paper, we propose a degree-based mean-field SIS epidemic model with a saturated function on complex networks. First, we adopt an edge-compartmental approach to lower the dimensions of such a proposed system. Then we give the existence of the feasible equilibria and completely study their stability by a geometric approach. We show that the proposed system exhibits a backward bifurcation, whose stabilities are determined by signs of the tangent slopes of the epidemic curve at the associated equilibria. Our results suggest that increasing the management and the allocation of medical resources effectively mitigate the lag effect of the treatment and then reduce the risk of an outbreak. Moreover, we show that decreasing the average of a network sufficiently eradicates the disease in a region or a country.

Mathematical analysis and optimal control interventions for sex structured syphilis model with three stages of infection and loss of immunity

In this study, we develop a nonlinear ordinary differential equation to study the dynamics of syphilis transmission incorporating controls, namely prevention and treatment of the infected males and females. We obtain syphilis-free equilibrium (SFE) and syphilis-present equilibrium (SPE). We obtain the basic reproduction number, which can be used to control the transmission of the disease, and thus establish the conditions for local and global stability of the syphilis-free equilibrium. The stability results show that the model is locally asymptotically stable if the Routh–Hurwitz criteria are satisfied and globally asymptotically stable. The bifurcation analysis result reveals that the model exhibits backward bifurcation. We adopted Pontryagin’s maximum principle to determine the optimality system for the syphilis model, which was solved numerically to show that syphilis transmission can be optimally best control using a combination of condoms usage and treatment in the primary stage of infection in both infected male and female populations.

Dynamics of an SEIRS COVID-19 Epidemic Model with Saturated Incidence and Saturated Treatment Response: Bifurcation Analysis and Simulations

In this work, we study the dynamics of the Coronavirus Disease 2019 pandemic using an SEIRS model with saturated incidence and treatment rates. We derive the basic reproduction number R_0 and study the local stability of the disease-free and endemic states. Since the condition R_0<1 for our model does not determine if the disease will die out, we study the backward bifurcation and Hopf bifurcation to understand the dynamics of the disease at the occurrence of a second wave and the kind of treatment measures needed to curtail it. We present some numerical simulations considering the symptomatic and asymptomatic infections and make a comparison with the reported COVID-19 data for Nigeria. Our results show that the limited availability of medical resources favours the emergence of complex dynamics that complicates the control of the outbreak

Epidemic Model and Mathematical Study of Impact of Vaccination for the Control of Malware in Computer Network

Malware remains a significant threat to computer network. In this paper, we consideredthe problem which computer malware cause to personal computers with its control by proposing a compartmental model SVEIRS (Susceptible Vaccinated-Exposed-infected-Recovered-Susceptible) for malware transmission in computer network using nonlinear ordinary differential equation. Through the analysis of the model, the basic reproduction number were obtained, and the malware free equilibrium was proved to be locally asymptotical stable if is less than unity and globally asymptotically stable if Ro is less than some threshold using a Lyapunov function. Also, the unique endemic equilibrium exists under certain conditions and the model underwent backward bifurcation phenomenon. To illustrate our theoretical analysis, some numerical simulation of the system was performed with RungeKutta fourth order (KR4) method in Mathlab. This was used in analyzing the behavior of different compartments of the model and the results showed that vaccination and treatment is very essential for malware control.