We investigate a two-strain disease model with amplification to simulate the prevalence of drug-susceptible (s) and drug-resistant (m) disease strains. We model the emergence of drug resistance as a consequence of inadequate treatment, i.e. amplification. We perform a dynamical analysis of the resulting system and find that the model contains three equilibrium points: a disease-free equilibrium; a mono-existent disease-endemic equilibrium with respect to the drug-resistant strain; and a co-existent disease-endemic equilibrium where both the drug-susceptible and drug-resistant strains persist. We found two basic reproduction numbers: one associated with the drug-susceptible strain $R_{0s}$; the other with the drug-resistant strain $R_{0m}$,and showed that at least one of the strains can spread in a population if ($R_{0s}$,$R_{0m}$) > 1 (epidemic).Furthermore, we also showed that if $R_{0m}$ > max($R_{0s}$,1), the drug-susceptible strain dies out but the drug-resistant strain persists in the population; however if $R_{0s}$ > max($R_{0m}$,1), then both the drug-susceptible and drug-resistant strains persist in the population. We conducted a local stability analysis of the system equilibrium points using the Routh-Hurwitz conditions and a global stability analysis using appropriate Lyapunov functions. Sensitivity analysis was used to identify the most important model parameters through the partial rank correlation coefficient (PRCC) method. We found that the contact rate of both strains had the largest influence on prevalence. We also investigated the impact of amplification and treatment rates of both strains on the equilibrium prevalence of infection; results suggest that poor quality treatment make coexistence more likely but increase the relative abundance of resistant infections.
Starting from common assumptions, we build a rate equation model for multi-strain disease dynamics in terms of immune repertoire classes. We then move to a strain-level description where a low-order closure reminiscent of a pair approximation can be applied. We characterize the endemic equilibrium of the ensuing model in the absence of mutation and discuss the presence of degeneracy regarding the prevalence of the different strains. Finally we study the behavior of the system under the injection of mutant strains.
We formulate a compartmental model for the propagation of a respiratory disease in a patchy environment. The patches are connected through the mobility of individuals, and we assume that disease transmission and recovery are possible during travel. Moreover, the migration terms are assumed to depend on the distance between patches and the perceived severity of the disease. The positivity and boundedness of the model solutions are discussed. We analytically show the existence and global asymptotic stability of the disease-free equilibrium. We study three different network topologies numerically…
We present a model for obligated mutualistic associations, in which two transmissible diseases are allowed to infect just one population. As the general model proves too hard to be fully analytically investigated, some special cases are analysed. Among our findings, the coexistence of the two strains does not appear possible, under the model assumptions. Furthermore, in particularly unfavorable circumstances the ecosystem may disappear. In this respect, an accurate computation of the basin of attraction of the origin is provided using novel techniques. For this obligated mutualistic system the…
The crisis caused by the COVID-19 outbreak around the globe raised an increasing concern about the ongoing emergence of variants of SARS-CoV-2 that may evade the immune response provided by vaccines. New variants appear due to mutation, and as the cases accumulate, the probability of the emergence of a variant of concern increases. In this article, we propose a modified SIR model with waning immunity that captures the competition of two strain classes of an infectious disease under the effect of vaccination with a highly contagious and deadly strain class emerging from a prior strain due to mu…
In this paper we consider a competition system in which two diseases spread by contact. We characterize the system behavior, establishing that only some configurations are possible. In particular we discover that coexistence of the two strains is not possible, under the assumptions of the model. A number of transcritical bifurcations relate the more relevant system's equilibria. Bistability is shown between a situation in which only the disease-unaffected population thrives and another one containing only the second population with endemic disease. An accurate computation of the separating sur…
The bacterium Group A Streptococcus (Streptococcus pyogenes, GAS) is a human-specific pathogen and a major cause of global morbidity and mortality. Despite decades of research our knowledge of GAS infection and immunity is incomplete, hampering vaccine design and other efforts to reduce disease prevalence. Epidemiological studies indicate positive associations between the prevalence of GAS-related disease, the diversity of circulating strains and the degree of poverty in host populations. However, the infection and immune mechanisms underlying these associations are not clear. In this work, we…
In mathematical studies of the dynamics of multi-strain diseases caused by antigenically diverse pathogens, there is a substantial interest in analytical insights. Using the example of a generic model of multi-strain diseases with cross-immunity between strains, we show that a significant understanding of the stability of steady states and possible dynamical behaviours can be achieved when the symmetry of interactions between strains is taken into account. Techniques of equivariant bifurcation theory allow one to identify the type of possible symmetry-breaking Hopf bifurcation, as well as to c…
We develop a new perturbation method for studying quasi-neutral competition in a broad class of stochastic competition models, and apply it to the analysis of fixation of competing strains in two epidemic models. The first model is a two-strain generalization of the stochastic Susceptible-Infected-Susceptible (SIS) model. Here we extend previous results due to Parsons and Quince (2007), Parsons et al (2008) and Lin, Kim and Doering (2012). The second model, a two-strain generalization of the stochastic Susceptible-Infected-Recovered (SIR) model with population turnover, has not been studied pr…
Dengue and Zika incidence data and the latest research have raised questions about how dengue vaccine strategies might be impacted by the emergence of Zika virus. Existing antibodies to one virus might temporarily protect or promote infection by the other through antibody-dependent enhancement (ADE). With this condition, understanding the dynamics of propagation of these two viruses is of great importance when implementing vaccines. In this work, we analyze the effect of vaccination against one strain, in a two-strain model that accounts for cross-immunity and ADE. Using basic and invasion rep…
We develop a variable population age-structured ODE model to investigate the role of Intermittent Preventive Treatment (IPT) in averting malaria-induced mortalities in children, as well as its related cost in promoting the spread of anti-malarial drug resistance. IPT, a malaria control strategy in which a full curative dose of an antimalarial medication is administered to vulnerable asymptomatic individuals at specified intervals, has been shown to have a positive impact on reducing malaria transmission and deaths in children and pregnant women. However, it can also promote drug resistance spr…
In this paper the concept of Critical Community Size (CCS) for the deterministic SIR model is introduced and its consequences for the disease dynamics are stressed. The disease can fade out after an outburst. Also the principle of competitive exclusion holds no longer true. This is exemplified for the dynamics of two competing virus strains. The virus with higher R0 can be eradicated from the population.
Transmission rates are key in understanding the spread of infectious diseases. Using the framework of compartmental models, we introduce a simple method that enables us to reconstruct time series of transmission rates directly from incidence or disease-related mortality data. The reconstruction exploits differential equations, which model the time evolution of infective stages and strains. Being sensitive to initial values, the method produces asymptotically correct solutions. The computations are fast, with time complexity being quadratic. We apply the reconstruction to data of measles (Engla…
Modeling and analysis of a novel two-strain dengue epidemics model considering secondary infections with increased mortality
Johannes Gutenberg-Universität Mainz（约翰内斯·古腾堡美因茨大学）；Institute for Quantitative and Computational Biosciences (IQCB), Johannes Gutenberg University Mainz（约翰内斯·古腾堡美因茨大学定量与计算生物科学研究所）；Ordu University, Faculty of Science and Letters（奥尔杜大学理学院）；Escuela Superior Politécnica del Litoral（厄瓜多尔高等理工学院）；Center for Applied Dynamical Systems and Computational Methods (CADSCOM), Faculty of Natural Sciences and Mathematics, Escuela Superior Politécnica del Litoral（厄瓜多尔高等理工学院自然科学与数学学院应用动力系统与计算方法中心）；TU Dresden（德累斯顿工业大学）；University of Koblenz（科布伦茨大学）
Influenza A is a serious disease that causes significant morbidity and mortality, and vaccines against the seasonal influenza disease are of variable effectiveness. In this paper, we discuss use of the $p_{\rm epitope}$ method to predict the dominant influenza strain and the expected vaccine effectiveness in the coming flu season. We illustrate how the effectiveness of the 2014/2015 A/Texas/50/2012 [clade 3C.1] vaccine against the A/California/02/2014 [clade 3C.3a] strain that emerged in the population can be estimated via pepitope. In addition, we show by a multidimensional scaling analysis o…
Mathematical modelling of epidemiological and coevolutionary dynamics is widely being used to improve disease management strategies of infectious diseases. Many diseases present some form of intra-host quiescent stage, also known as covert infection, while others exhibit dormant stages in the environment. As quiescent/dormant stages can be resistant to drug, antibiotics, fungicide treatments, it is of practical relevance to study the influence of these two life-history traits on the coevolutionary dynamics. We develop first a deterministic coevolutionary model with two parasite types infecting…
Mathematical disease modelling has long operated under the assumption that any one infectious disease is caused by one transmissible pathogen spreading among a population. This paradigm has been useful in simplifying the biological reality of epidemics and has allowed the modelling community to focus on the complexity of other factors such as population structure and interventions. However, there is an increasing amount of evidence that the strain diversity of pathogens, and their interplay with the host immune system, can play a large role in shaping the dynamics of epidemics. Here, we introd…
The rise of antimicrobial drug resistance is an imminent threat to global health that has warranted, and duly received, considerable attention within the medical, microbiological and modelling communities. Outbreaks of drug-resistant pathogens are ignited by the emergence and transmission of mutant variants descended from wild-type strains circulating in the community. In this work we investigate the stochastic dynamics of the emergence of a novel disease strain, introduced into a population in which it must compete with an existing endemic strain. In analogy with past work on single-strain ep…
For many infectious diseases, a small-world network on an underlying regular lattice is a suitable simplified model for the contact structure of the host population. It is well known that the contact network, described in this setting by a single parameter, the small-world parameter $p$, plays an important role both in the short term and in the long term dynamics of epidemic spread. We have studied the effect of the network structure on models of immune for life diseases and found that in addition to the reduction of the effective transmission rate, through the screening of infectives, spatial…
We develop a multiple compartment Susceptible-Infected-Recovered (SIR) model to analyze the spread of several infectious diseases through different geographic areas. Additionally, we propose a data-quality sensitive optimization framework for fitting this model to observed data. We fit the model to the temporal profile of the number of people infected by one of six influenza strains in Europe over $7$ influenza seasons. In addition to describing the temporal and spatial spread of influenza, the model provides an estimate of the inter-country and intra-country infection and recovery rates of ea…
In this work, we study the critical behavior of an epidemic propagation model that considers individuals that can develop drug resistance. In our lattice model, each site can be found in one of four states: empty, healthy, normally infected (not drug resistant) and strain infected (drug resistant) states. The most relevant parameters in our model are related to the mortality, cure and mutation rates. This model presents two distinct stationary active phases: a phase with co-existing normal and drug resistant infected individuals and an intermediate active phase with only drug resistant individ…
In this paper we propose a malaria within-host model with k classes of age for the parasitized red blood cells and n strains for the parasite. We provide a global analysis for this model. A competitive exclusion principle holds. If R0, the basic reproduction number, satisfies R0 </- 1, then the disease-free equilibrium is globally asymptotically stable. On the contrary if R0 > 1, then generically there is a unique endemic equilibrium which corresponds to the endemic stabilization of the most virulent parasite strain and to the extinction of all the other parasites strains. We prove that this e…
Different extensions of the classical single-strain SIR model for the host population, motivated by modeling dengue fever epidemiology, have reported a rich dynamic structure including deterministic chaos which was able to explain the large fluctuations of disease incidences. A comparison between the basic two-strain dengue model, which already captures differences between primary and secondary infections, with the four-strain dengue model, that introduces the idea of competition of multiple strains in dengue epidemics shows that the difference between first and secondary infections drives the…
We consider the dynamics of a virus spreading through a population that produces a mutant strain with the ability to infect individuals that were infected with the established strain. Temporary cross-immunity is included using a time delay, but is found to be a harmless delay. We provide some sufficient conditions that guarantee local and global asymptotic stability of the disease-free equilibrium and the two boundary equilibria when the two strains outcompete one another. It is shown that, due to the immune evasion of the emerging strain, the reproduction number of the emerging strain must be…
Networks provide a mathematically rich framework to represent social contacts sufficient for the transmission of disease. Social networks are often highly clustered and fail to be locally tree-like. In this paper, we study the effects of clustering on the spread of sequential strains of a pathogen using the generating function formulation under a complete cross-immunity coupling, deriving conditions for the threshold of coexistence of the second strain. We show that clustering reduces the coexistence threshold of the second strain and its outbreak size in Poisson networks, whilst exhibiting th…
Dengue control in India is a challenging task due to complex healthcare settings. In yesteryears, an amplification of dengue infections in India posed the need for introspection of existing dengue control policies. Prior understanding of the impacts of control interventions is necessary for their future implementation. In this paper, we propose and analyze a compartmental model of dengue to assess the impact of active case finding (ACF) on dengue disease transmission. Currently, primary prevention of dengue is possible only with vector control and personal protection from the bites of infected…
