Existence of periodic solutions of a periodic SEIRS model with general incidence.

For a family of periodic SEIRS models with general incidence, we prove the existence of at least one endemic periodic orbit when some condition related to R0 holds. Additionally, we prove the existence of a unique disease-free periodic orbit, that is globally asymptotically stable when R0 < 1. In particular, our main result generalizes the one in Zhang et al. (2012). We also discuss some examples where our results apply and show that, in some particular situations, we have a sharp threshold between existence and non existence of an endemic periodic orbit

Imported Infections Versus Herd Immunity Gaps; A Didactic Demonstration of Compartment Models Through the Example of a Minor Measles Outbreak in Hungary

Introduction: In Hungary, where MMR vaccine coverage is 99%, in 2017, a minor measles epidemic started from imported cases due to two major factors – latent susceptible cohorts among the domestic population and the vicinity of measles-endemic countries. Suspended immunization activities due to the COVID-19 surge are an ominous precursor to a measles resurgence. This epidemiological demonstration is aimed at promoting a better public understanding of epidemiological data. Materials and Methods: Our previous MMR sero-epidemiological measurements (N of total measles cases = 3919, N of mumps cases = 2132, and N of rubella cases = 2132) were analyzed using open-source epidemiological data (ANTSZ) of a small-scale measles epidemic outbreak (2017, Hungary). A simplified SEIR model was applied in the analysis. Results: In case of measles, due to a cluster-specific inadequacy of IgG levels, the cumulative seropositivity ratios (measles = 89.97%) failed to reach the herd immunity threshold (HIT Measles = 92–95%). Despite the fact that 90% of overall vaccination coverage is just slightly below the HIT, unprotected individuals may pose an elevated epidemiological risk. According to the SEIR model, ≥74% of susceptible individuals are expected to get infected. Estimations based on the input data of a local epidemic may suggest an even lower effective coverage rate (80%) in certain clusters of the population. Conclusion: Serological survey-based, historical and model-computed results are in agreement. A practical demonstration of epidemiological events of the past and present may promote a higher awareness of infectious diseases. Because of the high R0 value of measles, continuous large-scale monitoring of humoral immunity levels is important

Bifurcation of a Delayed SEIS Epidemic Model with a Changing Delitescence and Nonlinear Incidence Rate

This paper is concerned with a delayed SEIS (Susceptible-Exposed-Infectious-Susceptible) epidemic model with a changing delitescence and nonlinear incidence rate. First of all, local stability of the endemic equilibrium and the existence of a Hopf bifurcation are studied by choosing the time delay as the bifurcation parameter. Directly afterwards, properties of the Hopf bifurcation are determined based on the normal form theory and the center manifold theorem. At last, numerical simulations are carried out to illustrate the obtained theoretical results

Epidemiological model of the transmission and spread of Hepatitis B pandemic in Ghana

The world\u27s attention to the burden and spread of hepatitis B has increased signicantly in the recent past. A number of interventions by way of treatment and immunisation have been initiated to fight the pandemic, especially in high prevalence regions such as Africa and Asia. Lack of good quality information about hepatitis B is a major hindrance to policy direction and comprehensive implementation of interventions in Sub-Saharan Africa designated as an endemic region. Limited studies on hepatitis B, coupled with lack of adequate health care systems and infrastructure, have led to ignorance or misconceptions and stigmatisation about the disease and worsened the disease prevalence in Ghana a Sub-Saharan African country. As a contribution, this study employed a SEIR deterministic compartmental model, which incorporates latent period and vertical transmission, to examine the transmission and spread of hepatitis B in the context of the Sub-Saharan Africa with incidence data obtained from Ghana. The SEIR deterministic compartmental modelling divided the human population into separate classes namely susceptible (S), exposed (E), infectious (I) and removed (R) or recovered, and disease progression among population members in the various classes was described using a system of nonlinear ordinary di_erential equations (ODEs). The model has two equilibrium states namely, the disease-free equilibrium Q0 and the endemic equilibrium Q*. Stability analysis indicated that the model has an epidemiological threshold parameter R0 which is defined as the expected proportion of secondary infections generated as a result of introducing a single infected individual into the population. When R0 ≤1 the disease-free equilibrium state is globally asymptotically stable whilst the endemic equilibrium state is unstable and so the disease is brought under control. When R0 \u3e 1, the disease-free equilibrium is unstable whilst the endemic equilibrium state is stable and so the disease persists in the population. Stability of the model was analysed in terms of proportions instead of the absolute number of cases and so disease eradication or persistence referred to the infected proportions vanishing or persisting respectively. A hybrid nonlinear least squares method, that combines a Genetic Algorithm (GA) and a modifed Levenberg-Marquadt (LM) algorithm, was applied to the hepatitis B incidence data to estimate the parameters of the model for Ghana (global) and also for each of the ten regions of Ghana. By numerical simulations, sensitivity analysis was performed to examine the effects of the model parameters on the threshold parameter R0 using MATLAB. Furthermore, the model was modi_ed to include a vaccination component to examine the impact of an intervention on the transmission and spread of the disease. The vaccination model also has an epidemiological threshold parameter Rv0 such that when the vaccination rate k is greater than a threshold value k*, then Rv0 \u3c 1 and the disease decreases; and when k is less than k*, Rv0 \u3e 1 and the disease increases. This indicated that when the rate of vaccination k was increased beyond the threshold value k*, the disease would be kept under control. The threshold parameter was calculated as R0 = 1:6854 for Ghana. This indicated that the endemic proportion equilibrium is asymptotically stable and so hepatitis B persists in the population of Ghana. The contact rate β, latency rate and vertical transmission rate γ were identifed as driving the disease spread in the population. A critical proportion of H = 0:4067 was calculated as the herd immunity threshold value of the population. This means that about 41% of the population are needed to be immune in order to adequately reduce the rates of transmission to keep the disease under control. Variability in the regional threshold parameters R0 indicated significant disparities in the spread and burden of hepatitis B across the ten regions of Ghana. The highest and the least values of (R0 = 3:7212;H = 0:7312) and (R0 = 1:3669;H = 0:2684) were calculated for Upper West and Volta regions respectively. The regional threshold parameters R0 also indicated that the trend of transmission and spread of the disease increase from south to north across the regions of Ghana. A simple regression analysis performed indicated that the increasing trend from south to north is highly associated with poverty and health sector differentials. Another factor that was considered in this study to have potentially impacted the distribution and pattern of spread and burden of hepatitis B in Ghana is prevalence differentials among regions and between Ghana and its neighbouring countries

Backward bifurcation and reinfection in mathematical models of tuberculosis

Mathematical models are widely used for understanding the transmission mechanisms and control of infectious diseases. Numerous infectious diseases such as those caused by bacterial and viral infections do not confer life long immunity after recovering from the first episode. Consequently, they are characterized by partial or complete loss of immunity and subsequent reinfection. This thesis explores the epidemiological implications of loss of immunity using simple and complex mathematical models. First, a simple basic model mimicking transmission mechanisms of tuberculosis (TB) is proposed with the aim of correcting problems that are often repeated by mathematical modellers when determining underlying bifurcation structures. Specifically, the model makes transparent the problems that may arise if one aggregates all the bifurcation parameters when computing backward bifurcation thresholds and structures. The backward bifurcation phenomenon is an important concept for public health and disease management. This is because backward bifurcation signals that disease will not be eliminated even when the basic reproduction number R0 is decreased below unity; rather, for the disease to be eliminated, R0 has to be reduced below another critical threshold. I provide conditions to find the threshold correctly. Secondly, the simple basic TB model is extended to incorporate epidemiological and biological aspects pertinent to TB transmission such as recurrent TB, which is defined as a second episode of TB following successful recovery from a previous episode. I study the conditions for backward bifurcation in this extended model that features recurrent TB. Mathematical techniques based on the center manifold approach, are used to derive an exact backward bifurcation threshold. Furthermore, both analytical and numerical findings reveal that recurrent TB is capable of inducing a new and rare hysteresis effect where TB will persist when the basic reproduction number is below unity even though there is no backward bifurcation. Moreover, when the reinfection pathway among latently infected individuals is switched off, leaving only recurrent TB, the model analysis indicates that recurrent TB can independently induce a backward bifurcation. However, this will only occur if recurrent TB transmission exceeds a certain threshold. Although this threshold seems to be relatively high when realistic parameters are used, it falls within the recent range estimated in the relevant literature. The second TB model is extended by dividing the latent compartment into two: fast (early latent) and slow (late latent) latent compartments, to enhance realism. Individuals in both early and late compartments are subjected to treatment. The proposed TB model is used to investigate how heterogeneity in host susceptibility influences the effectiveness of treatment. It is found that making the assumption that individuals treated with preventive therapy and recovered individuals (previously treated for active TB) acquire equal levels of protection after initial infection, and are therefore reinfected at the same rate, may obscure dynamics that are imperative when designing intervention strategies. Comparison of reinfection rates between cohorts treated with preventive therapy and recovered individuals who were previously treated from active TB provides important epidemiological insights. That is, the reinfection parameter accounting for the relative rate of reinfection of the cohort treated with preventive therapy is the one that plays the key role in generating qualitative changes in TB dynamics. In contrast, the parameter accounting for the risk of reinfection among recovered individuals (previously treated for active TB) does not play a significant role. The study shows that preventive treatment during early latency is always beneficial regardless of the level of susceptibility to reinfection. And 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: (a) for (very) low burden settings, the effect on reactivation predominates and burden declines; (b) for high burden settings, the effect on reinfection predominates and burden increases. This is mostly observed 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. Finally, a mathematical model that examines how heroin addiction spreads in society is formulated. The model has many commonalities with the TB model. The global stability properties of the proposed model are analysed using both the Lyapunov direct method and the geometric approach by Li and Muldowney. It is shown that even for a four dimensional model, the use of two well known nonlinear stability techniques becomes nontrivial. When all the parameters of the model are accounted for, it is difficult if not impossible, to design a Lyapunov function. Here I apply the geometric approach to establish a global condition that accounts for all model parameters. If the condition is satisfied, then heroin persistence within the community is globally stable. However, if the global condition is not satisfied heroin users can oscillate periodically in number. Numerical simulations are also presented to give a more complete representation of the model dynamics. Sensitivity analysis performed by Latin hypercube sampling (LHS) suggests that the effective contact rate in the population, the relapse rate of heroin users undergoing treatment, and the extent of saturation of heroin users, are the key mechanisms fuelling heroin epidemic proliferation. However, in the long term, relapse of heroin users undergoing treatment back to a heroin using career, has the most significant impact

Differential equation and complex network approaches for epidemic modelling

This study consists of three parts. The first part focuses on bifurcation analysis of epidemic models with sub-optimal immunity and saturated treatment/recovery rate as well as nonlinear incidence rate. The second part of the research focuses on estimating the domain of attraction for sub-optimal immunity epidemic models. In the third part of the research, we develop a bond percolation model for community clustered networks with an arbitrarily specified joint degree distribution

Avian–human influenza epidemic model with diffusion, nonlocal delay and spatial homogeneous environment

In this paper, an avian–human influenza epidemic model with diffusion, nonlocal delay and spatial homogeneous environment is investigated. This model describes the transmission of avian influenza among poultry, humans and environment. The behavior of positive solutions to a reaction–diffusion system with homogeneous Neumann boundary conditions is investigated. By means of linearization method and spectral analysis the local asymptotical stability is established. The global asymptotical stability for the poultry sub-system is studied by spectral analysis and by using a Lyapunov functional. For the full system, the global stability of the disease-free equilibrium is studied using the comparison Theorem for parabolic equations. Our result shows that the disease-free equilibrium is globally asymptotically stable, whenever the contact rate for the susceptible poultry is small. This suggests that the best policy to prevent the occurrence of an epidemic is not only to exterminate the asymptomatic poultry but also to reduce the contact rate between susceptible humans and the poultry environment. Numerical simulations are presented to illustrate the main results.http://www.elsevier.com/locate/nonrwahj2023Mathematics and Applied Mathematic

Mathematical models of infectious diseases in ungulate populations

In this thesis we develop a suite of mathematical models to understand the epidemiological dynamics of infectious diseases in ungulate hosts. Using ordinary differential equation frameworks, we explored the key routes of transmission that promote the persistence of the highly virulent African swine fever (ASF) infection in wild boar and tested control strategies that could limit ASF outbreaks and its persistence. These modelling techniques were extended to investigate the impact of an ASF outbreak on endemic tuberculosis in wild boar. The generality of the model framework meant the results could add new perspective on the coexistence of multiple pathogens. Motivated by the work on the persistence of ASF, we used a suite of stochastic continuous-time Markov chain models to show that latent and chronic infection could have a significant impact on the mean time to pathogen extinction. We also developed a model framework to assess how hosts, including ungulates, contribute to tick-borne infections. This expands on previously studied models such that the regulation of tick density is dependent on the density of the specific hosts on which different tick stages feed. Our results outlined the effect host density and composition could have on tick-borne prevalence and incidence levels. The work in this thesis has highlighted how mathematical models are important tools for understanding epidemiological dynamics in wildlife systems with our work having had an impact on the management of key, current, endemic and emerging diseases in ungulates.The UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/L016508/01