Backward bifurcation and hysteresis in models of recurrent tuberculosis

How Does Vaccine-Induced Immunity Compare to Infection-Acquired Immunity in the Dynamics of COVID-19?

Five years into the COVID-19 pandemic, the availability of effective vaccines has substantially reduced new cases, hospitalizations, and mortality. However, the waning of immunity has been a topic of particular interest in relation to disease control. The objective of this study is to investigate the impact of the decline in vaccine-induced immunity (ω1) and infection-acquired immunity (ω2) on disease dynamics. For this purpose, we use a compartmental model with seven compartments that accounts for differential morbidity, vaccination, and waning immunity. A compartmental model divides a population into distinct groups depending on their disease status. The temporal changes in the compartments are represented through ordinary differential equations (ODEs). The model is mathematically analyzed to show that a backward bifurcation (i.e., a perverse outcome) may occur when the vaccinated reproduction number (Rv) is equal to unity. Both local and global sensitivity analysis on the reproduction number reveal that the vaccine efficacy, waning of vaccine-induced immunity, vaccine coverage rate, coefficients of transmissibility, and the recovery rate for mild infections are the most sensitive parameters. The global sensitivity analysis on the cumulative number of infections shows that ω1 and ω2 are both pivotal parameters, while ω2 has a higher influence. Simulations on infections and mortality suggest that the changes in ω2 result in dynamics that are more pronounced compared to the dynamics resulting from the changes in ω1, thus indicating the importance of the duration of infection-acquired immunity in disease spread.

Impact of media coverage on the transmission dynamics of TB with vaccines and treatment

Tuberculosis (TB) is one of the deadly infectious diseases affecting millions of individuals throughout the world. The main objective of this study is to investigate the impact of media coverage on the transmission dynamics of TB with vaccine and treatment strategy using mathematical model analysis. In the qualitative analysis of the proposed model we proved the existence, uniqueness, positivity, and boundedness of the model solutions, investigated both the disease-free and endemic equilibrium points, computed the basic and effective reproduction numbers using next generation matrix approach, analyzed the stability analysis of the equilibrium points, the backward bifurcation using the Castillo-Chavez and Song theorem and we re-formulated the corresponding optimal control problem and analyzed by applying the Pontryagin's Minimum Principle. In the model quantitative (numerical) analysis part, we performed the model parameters sensitivity analysis and carried out numerical simulation to verify the qualitative analysis results. The findings of the study indicate that if the reproduction number is less than one, the solution converges to the disease-free state, signifying the asymptotic stability of the TB-free steady state. Moreover, the existence of a backward bifurcation shows that the disease-free equilibrium coexists with one or more endemic equilibria, even when the basic reproduction number is less than 1. Furthermore, it is found that as media efficacy increases, the disease infection rate decreases, which consequently leads to an increase in prevention and treatment control strategies.

An ODEs multiscale model with cell proliferation for hepatitis C virus infection treated with direct acting antiviral agents

In a recent study, a mathematically identical ODE model is derived from a multiscale PDE model of hepatitis C virus infection, which helps to overcome the limitations of the PDE model in the analysis. Here, an extended proposed model is formulated for this transformed ODE model by including the hepatocyte proliferation of both uninfected and infected cells. Unlike the transformed model, the proposed model can predict the triphasic viral decline and the virus level after therapy cessation without oscillations. Numerical simulations are performed to investigate the effect of hepatocyte proliferation and therapy with direct-acting antivirals agents (DAAs). The basic reproduction number is obtained, the equilibrium points are specified, and their stability is analysed. A bifurcation analysis is performed to specify the bifurcation points and to study the effect of varying system parameters. Various viral load profiles generated by the model are confirmed to fit with reported data in the literature.

Exploring Hopf-bifurcations and endemic bubbles in a tuberculosis model with behavioral changes and treatment saturation

To manage risks and minimize the transmission of contagious diseases, individuals may reduce their contact with each other and take other precautions as much as possible in their daily lives and workplaces. As a result, the transmission of the infection reduces due to the behavioral changes. These behavioral changes are incorporated into models by introducing saturation in disease incidence. In this article, we propose and analyze a tuberculosis model that incorporates saturated exogenous reinfection and treatment. The stability analysis of the model's steady states is rigorously examined. We observe that the disease-free equilibrium point and the endemic equilibrium point (EEP) are globally asymptotically stable if the basic reproduction number (R0) is less than 1 and greater than 1, respectively, only when exogenous reinfection is not present (p=0) and when treatment is available for all (ω=0). However, even when R0 is less than 1, tuberculosis may persist at a specific level in the presence of exogenous reinfection and treatment saturation, leading to a backward bifurcation in the system. The existence and direction of Hopf-bifurcations are also discussed. Furthermore, we numerically validate our analytical results using different parameter sets. In the numerical examples, we study Hopf-bifurcations for parameters such as β, p, α, and ω. In one example, we observe that increasing β leads to the loss of stability of the unique EEP through a forward Hopf-bifurcation. If β is further increased, the unique EEP restores its stability, and the bifurcation diagram exhibits an interesting structure known as an endemic bubble. The existence of an endemic bubble for the saturation constant ω is also observed.

An SEQAIHR model to study COVID-19 transmission and optimal control strategies in Hong Kong, 2022

During the COVID-19 pandemic, one of the major concerns was a medical emergency in human society. Therefore it was necessary to control or restrict the disease spreading among populations in any fruitful way at that time. To frame out a proper policy for controlling COVID-19 spreading with limited medical facilities, here we propose an SEQAIHR model having saturated treatment. We check biological feasibility of model solutions and compute the basic reproduction number ( R 0 ). Moreover, the model exhibits transcritical, backward bifurcation and forward bifurcation with hysteresis with respect to different parameters under some restrictions. Further to validate the model, we fit it with real COVID-19 infected data of Hong Kong from 19th December, 2021 to 3rd April, 2022 and estimate model parameters. Applying sensitivity analysis, we find out the most sensitive parameters that have an effect on R 0 . We estimate R 0 using actual initial growth data of COVID-19 and calculate effective reproduction number for same period. Finally, an optimal control problem has been proposed considering effective vaccination and saturated treatment for hospitalized class to decrease density of the infected class and to minimize implemented cost.

Nonlinear dynamics of a SIRI model incorporating the impact of information and saturated treatment with optimal control

In this article, we propose and analyze an infectious disease model with reinfection and investigate disease dynamics by incorporating saturated treatment and information effect. In the model, we consider the case where an individual's immunity deteriorates and they become infected again after recovering. According to our findings, multiple steady states and backward bifurcation may occur as a result of treatment saturation. Further, if treatment is available for all, the disease will be eradicated providedR 0 < 1 ; however, because limited medical resources caused saturation in treatment, the disease may persist even ifR 0 < 1 . The global stability of the unique endemic steady state is established using a geometric approach. We also establish certain conditions on the transmission rate for the occurrence of periodic oscillations in the model system. Among nonlinear dynamics, we show supercritical Hopf bifurcation, bi-stability, backward Hopf bifurcation, and double Hopf bifurcation. To illustrate and validate our theoretical results, we present numerical examples. We found that when disease information coverage is high, infection cases fall considerably, and the disease persists when the reinfection rate is high. We then extend our model by incorporating two time-dependent controls, namely inhibitory interventions and treatment. Using Pontryagin's maximum principle, we prove the existence of optimal control paths and find the optimal pair of controls. According to our numerical simulations, the second control is less effective than the first. Furthermore, while implementing a single intervention at a time may be effective, combining both interventions is most effective in reducing disease burden and cost.

Mathematical modeling and optimal control of SARS-CoV-2 and tuberculosis co-infection: a case study of Indonesia

A new mathematical model incorporating epidemiological features of the co-dynamics of tuberculosis (TB) and SARS-CoV-2 is analyzed. Local asymptotic stability of the disease-free and endemic equilibria are shown for the sub-models when the respective reproduction numbers are below unity. Bifurcation analysis is carried out for the TB only sub-model, where it was shown that the sub-model undergoes forward bifurcation. The model is fitted to the cumulative confirmed daily SARS-CoV-2 cases for Indonesia from February 11, 2021 to August 26, 2021. The fitting was carried out using the fmincon optimization toolbox in MATLAB. Relevant parameters in the model are estimated from the fitting. The necessary conditions for the existence of optimal control and the optimality system for the co-infection model is established through the application of Pontryagin’s Principle. Different control strategies: face-mask usage and SARS-CoV-2 vaccination, TB prevention as well as treatment controls for both diseases are considered. Simulations results show that: (1) the strategy against incident SARS-CoV-2 infection averts about 27,878,840 new TB cases; (2) also, TB prevention and treatment controls could avert 5,397,795 new SARS-CoV-2 cases. (3) In addition, either SARS-CoV-2 or TB only control strategy greatly mitigates a significant number of new co-infection cases.

A BEHAVIORAL CHANGE MODEL TO ASSESS VACCINATION-INDUCED RELAXATION OF SOCIAL DISTANCING DURING AN EPIDEMIC

The success of mass vaccination campaigns may be jeopardized by human risky behaviors. For example, high level of vaccination coverage may induce early relaxation of social distancing. In this paper, we focus on the mutual influence between the decline in prevalence, due to the rise in the overall immunization coverage, and the consequent decrease in the compliance to social distancing measures. We consider an epidemic model where both the vaccination rate and the disease transmission rate are influenced by human behavior, which in turn depends on the current and past information about the spread of the disease. We highlight the impact of the information-related parameters on the transient and asymptotic behavior of the system that is on the early stage of the epidemic and its final outcome. Among the main results, we evidence that sustained oscillations may be triggered by the behavioral memory in the prevalence-dependent vaccination rate. However, the relaxation of social distancing may induce a switch from a cyclic regime to damped oscillations.

Nonlinear dynamical behavior of an SEIR mathematical model: Effect of information and saturated treatment

When a disease spreads in a population, individuals tend to change their behavior due to the presence of information about disease prevalence. Therefore, the infection rate is affected and incidence term in the model should be appropriately modified. In addition, a limitation of medical resources has its impact on the dynamics of the disease. In this work, we propose and analyze an Susceptible-Exposed-Infected-Recovered (SEIR) model, which accounts for the information-induced non-monotonic incidence function and saturated treatment function. The model analysis is carried out, and it is found that when R₀ is below one, the disease may or may not die out due to the saturated treatment (i.e., a backward bifurcation may exist and cause multi-stability). Further, we note that in this case, disease eradication is possible if medical resources are available for all. When R₀ exceeds one, there is a possibility of the existence of multiple endemic equilibria. These multiple equilibria give rise to rich and complex dynamics by showing various bifurcations and oscillations (via Hopf bifurcation). A global asymptotic stability of a unique endemic equilibrium (when it exists) is established under certain conditions. An impact of information is shown and also a sensitivity analysis of model parameters is performed. Various cases are considered numerically to provide the insight of model behavior mathematically and epidemiologically. We found that the model shows hysteresis. Our study underlines that a limitation of medical resources may cause bi(multi)-stability in the model system. Also, information plays a significant role and gives rise to a rich and complex dynamical behavior of the model.

Zoonotic MERS-CoV transmission: modeling, backward bifurcation and optimal control analysis

Middle East Respiratory Syndrome Coronavirus (MERS-CoV) can cause mild to severe acute respiratory illness with a high mortality rate. As of January 2020, more than 2500 cases of MERS-CoV resulting in around 860 deaths were reported globally. In the absence of neither effective treatment nor a ready-to-use vaccine, control measures can be derived from mathematical models of disease epidemiology. In this manuscript, we propose and analyze a compartmental model of zoonotic MERS-CoV transmission with two co-circulating strains. The human population is considered with eight compartments while the zoonotic camel population consist of two compartments. The expression of basic reproduction numbers are obtained for both single strain and two strain version of the proposed model. We show that the disease-free equilibrium of the system with single stain is globally asymptotically stable under some parametric conditions. We also demonstrate that both models undergo backward bifurcation phenomenon, which in turn indicates that only keeping R 0 below unity may not ensure eradication. To the best of the authors knowledge, backward bifurcation was not shown in a MERS-CoV transmission model previously. Further, we perform normalized sensitivity analysis of important model parameters with respect to basic reproduction number of the proposed model. Furthermore, we perform optimal control analysis on different combination interventions with four components namely preventive measures such as use of masks, isolation of strain-1 infected people, strain-2 infected people and infected camels. Optimal control analysis suggests that combination of preventive measures and isolation of infected camels will eventually eradicate the disease from the community.

A fractional order HIV-TB co-infection model in the presence of exogenous reinfection and recurrent TB

In this article, a novel fractional order model has been introduced in Caputo sense for HIV-TB co-infection in the presence of exogenous reinfection and recurrent TB along with the treatment for both HIV and TB. The main aim of considering the fractional order model is to incorporate the memory effect of both diseases. We have analyzed both sub-models separately with fractional order. The basic reproduction number, which measures the contagiousness of the disease, is determined. The HIV sub-model is shown to have a locally asymptotically stable disease-free equilibrium point when the corresponding reproduction number, R H , is less than unity, whereas, for R H > 1 , the endemic equilibrium point comes into existence. For the TB sub-model, the disease-free equilibrium point has been proved to be locally asymptotically stable for R T < 1 . The existence of TB endemic equilibrium points in the presence of reinfection and recurrent TB for R T < 1 justifies the existence of backward bifurcation under certain restrictions on the parameters. Further, we numerically simulate the fractional order model to verify the analytical results and highlight the role of fractional order in co-infection modeling. The fractional order derivative is shown to have a crucial role in determining the transmission dynamics of HIV-TB co-infection. It is concluded that the memory effect plays a significant role in reducing the infection prevalence of HIV-TB co-infection. An increment in the number of recovered individuals can also be observed when the memory effect is taken into consideration by introducing fractional order model.

Backward bifurcation in a fractional-order and two-patch model of tuberculosis epidemic with incomplete treatment

In this paper, we consider a tuberculosis model with incomplete treatment and extend the model to a Caputo fractional-order and two-patch version with exogenous re-infection among the treated individuals, in which only susceptible individuals can travel freely between the patches. The model has multiple equilibria. We determine conditions that lead to the appearance of a backward bifurcation. The results show that the TB model can have exogenous reinfection among the treated individuals and, at the same time, does not exhibit backward bifurcation. Also, conditions that lead to the global asymptotic stability of the disease-free equilibrium are obtained. In case without reinfection, the model has four equilibria. In this case, the global asymptotic stability of the equilibria is established using the Lyapunov function theory together with the LaSalle invariance principle for fractional differential equations (FDEs). Numerical simulations confirm the validity of the theoretical results.

Modelling heterogeneity in host susceptibility to tuberculosis and its effect on public health interventions

A tuberculosis (TB) model that accounts for heterogeneity in host susceptibility to tuberculosis is proposed, with the aim of investigating the implications this may have for the effectiveness of public health interventions. The model examines the possibility that recovered individuals treated from active TB and individuals treated with preventive therapy acquire different levels of immunity. This contrasts with recent studies that assume the two cohorts acquire the same level of immunity, and therefore both groups are reinfected at the same rate. The analysis presented here examines the impact of this assumption when designing intervention strategies. Comparison of reinfection rates between cohorts treated with preventive therapy and recovered individuals who were previously treated for active TB provides important epidemiological insights. It is found that the reinfection rate of the cohort treated with preventive therapy is the one that plays the key role in qualitative changes in TB dynamics. By contrast, the reinfection rate of recovered individuals (previously treated from active TB) plays a minor role. Moreover, the study shows that preventive treatment of individuals during early latency is always beneficial regardless of the level of susceptibility to reinfection. Further, if patients have greater immunity following treatment for late latent infection, then treatment is again beneficial. However, if susceptibility increases following treatment for late latent infection, the effect of treatment depends on the epidemiological setting. That is: (i) in (very) low burden settings, the effect on reactivation predominates and the burden declines with treatment; (ii) in moderate to high burden settings the effect of reinfection predominates and burden increases with treatment. The effect is most dominant between the two reinfection thresholds, RT2 and RT1, respectively associated with individuals being treated with preventive therapy and individuals with untreated late latent TB infection.