A two-strain model of infectious disease spread with asymmetric temporary immunity periods and partial cross-immunity

We introduce a two-strain model with asymmetric temporary immunity periods and partial cross-immunity. We derive explicit conditions for competitive exclusion and coexistence of the strains depending on the strain-specific basic reproduction numbers, temporary immunity periods, and degree of cross-immunity. The results of our bifurcation analysis suggest that, even when two strains share similar basic reproduction numbers and other epidemiological parameters, a disparity in temporary immunity periods and partial or complete cross-immunity can provide a significant competitive advantage. To analyze the dynamics, we introduce a quasi-steady state reduced model which assumes the original strain remains at its endemic steady state. We completely analyze the resulting reduced planar hybrid switching system using linear stability analysis, planar phase-plane analysis, and the Bendixson-Dulac criterion. We validate both the full and reduced models with COVID-19 incidence data, focusing on the Delta (B.1.617.2), Omicron (B.1.1.529), and Kraken (XBB.1.5) variants. These numerical studies suggest that, while early novel strains of COVID-19 had a tendency toward dramatic takeovers and extinction of ancestral strains, more recent strains have the capacity for co-existence.

Estimating Vaccine-Driven Selection in Seasonal Influenza

Vaccination could be an evolutionary pressure on seasonal influenza if vaccines reduce the transmission rates of some ("targeted") strains more than others. In theory, more vaccinated populations should have a lower prevalence of targeted strains compared to less vaccinated populations. We tested for vaccine-induced selection in influenza by comparing strain frequencies between more and less vaccinated human populations. We defined strains in three ways: first as influenza types and subtypes, next as lineages of type B, and finally as clades of influenza A/H3N2. We detected spatial differences partially consistent with vaccine use in the frequencies of subtypes and types and between the lineages of influenza B, suggesting that vaccines do not select strongly among all these phylogenetic groups at regional scales. We did detect a significantly greater frequency of an H3N2 clade with known vaccine escape mutations in more vaccinated countries during the 2014⁻2015 season, which is consistent with vaccine-driven selection within the H3N2 subtype. Overall, we find more support for vaccine-driven selection when large differences in vaccine effectiveness suggest a strong effect size. Variation in surveillance practices across countries could obscure signals of selection, especially when strain-specific differences in vaccine effectiveness are small. Further examination of the influenza vaccine's evolutionary effects would benefit from improvements in epidemiological surveillance and reporting.

Stochastic dynamics for reinfection by transmitted diseases

The use of stochastic models to study the dynamics of infectious diseases is an important tool to understand the epidemiological process. For several directly transmitted diseases, reinfection is a relevant process, which can be expressed by endogenous reactivation of the pathogen or by exogenous reinfection due to direct contact with an infected individual (with smaller reinfection rate σβ than infection rate β). In this paper, we examine the stochastic susceptible, infected, recovered, infected (SIRI) model simulating the endogenous reactivation by a spontaneous reaction, while exogenous reinfection by a catalytic reaction. Analyzing the mean-field approximations of a site and pairs of sites, and Monte Carlo (MC) simulations for the particular case of exogenous reinfection, we obtained continuous phase transitions involving endemic, epidemic, and no transmission phases for the simple approach; the approach of pairs is better to describe the phase transition from endemic phase (susceptible, infected, susceptible (SIS)-like model) to epidemic phase (susceptible, infected, and removed or recovered (SIR)-like model) considering the comparison with MC results; the reinfection increases the peaks of outbreaks until the system reaches endemic phase. For the particular case of endogenous reactivation, the approach of pairs leads to a continuous phase transition from endemic phase (SIS-like model) to no transmission phase. Finally, there is no phase transition when both effects are taken into account. We hope the results of this study can be generalized for the susceptible, exposed, infected, and removed or recovered (SEIR_{I}^{E}) model, for which the state exposed (infected but not infectious), describing more realistically transmitted diseases such as tuberculosis. In future work, we also intend to investigate the effect of network topology on phase transitions when the SIRI model describes both transmitted diseases (σ<1) and social contagions (σ>1).

Periodic solutions in an SIRWS model with immune boosting and cross-immunity

Incidence of whooping cough, an infection caused by Bordetella pertussis and Bordetella parapertussis, has been on the rise since the 1980s in many countries. Immunological interactions, such as immune boosting and cross-immunity between pathogens, have been hypothesised to be important drivers of epidemiological dynamics. We present a two-pathogen model of transmission which examines how immune boosting and cross-immunity can influence the timing and severity of epidemics. We use a combination of numerical simulations and bifurcation techniques to study the dynamical properties of the system, particularly the conditions under which stable periodic solutions are present. We derive analytic expressions for the steady state of the single-pathogen model, and give a condition for the presence of periodic solutions. A key result from our two-pathogen model is that, while studies have shown that immune boosting at relatively strong levels can independently generate periodic solutions, cross-immunity allows for the presence of periodic solutions even when the level of immune boosting is weak. Asymmetric cross-immunity can produce striking increases in the incidence and period. Our study underscores the importance of developing a better understanding of the immunological interactions between pathogens in order to improve model-based interpretations of epidemiological data.

Demographic buffering: titrating the effects of birth rate and imperfect immunity on epidemic dynamics

Host demography can alter the dynamics of infectious disease. In the case of perfectly immunizing infections, observations of strong sensitivity to demographic variation have been mechanistically explained through analysis of the susceptible–infected–recovered (SIR) model that assumes lifelong immunity following recovery from infection. When imperfect immunity is incorporated into this framework via the susceptible–infected–recovered–susceptible (SIRS) model, with individuals regaining full susceptibility following recovery, we show that rapid loss of immunity is predicted to buffer populations against the effects of demographic change. However, this buffering is contrary to the dependence on demography recently observed for partially immunizing infections such as rotavirus and respiratory syncytial virus. We show that this discrepancy arises from a key simplification embedded in the SIR(S) framework, namely that the potential for differential immune responses to repeat exposures is ignored. We explore the minimum additional immunological information that must be included to reflect the range of observed dependencies on demography. We show that including partial protection and lower transmission following primary infection is sufficient to capture more realistic reduced levels of buffering, in addition to changes in epidemic timing, across a range of partially and fully immunizing infections. Furthermore, our results identify key variables in this relationship, including R₀.

Computational Population Biology: Linking the inner and outer worlds of organisms

Computationally complex systems models are needed to advance research and implement policy in theoretical and applied population biology. Difference and differential equations used to build lumped dynamic models (LDMs) may have the advantage of clarity, but are limited in their inability to include fine-scale spatial information and individual-specific physical, physiological, immunological, neural and behavioral states. Current formulations of agent-based models (ABMs) are too idiosyncratic and freewheeling to provide a general, coherent framework for dynamically linking the inner and outer worlds of organisms. Here I propose principles for a general, modular, hierarchically scalable, framework for building computational population models (CPMs) designed to treat the inner world of individual agents as complex dynamical systems that take information from their spatially detailed outer worlds to drive the dynamic inner worlds of these agents, simulate their ecology and the evolutionary pathways of their progeny. All the modeling elements are in place, although improvements in software technology will be helpful; but most of all we need a cultural shift in the way population biologists communicate and share model components and the models themselves, fit, test, refute, and refine models, to make the progress needed to meet the ecosystems management challenges posed by global change biology.

QUALITATIVE STUDY FOR A MULTI-DRUG RESISTANT TB MODEL WITH EXOGENOUS REINFECTION AND RELAPSE

One tuberculosis transmission model is formulated by incorporating exogenous reinfection, relapse, and two treatment stages of infectious TB cases. The global stability of the unique disease-free equilibrium is obtained by applying the comparison principle if the effective reproduction number for the full model is less than unity. The existence and stability of the boundary equilibria are given by introducing the invasion reproduction numbers. Furthermore, the existence and local stability of the endemic equilibrium are addressed under some conditions.

Modeling HIV/AIDS and tuberculosis coinfection

An HIV/AIDS and TB coinfection model which considers antiretroviral therapy for the AIDS cases and treatment of all forms of TB, i.e., latent and active forms of TB, is presented. We begin by presenting an HIV/AIDS-TB coinfection model and analyze the TB and HIV/AIDS submodels separately without any intervention strategy. The TB-only model is shown to exhibit backward bifurcation when its corresponding reproduction number is less than unity. On the other hand, the HIV/AIDS-only model has a globally asymptotically stable disease-free equilibrium when its corresponding reproduction number is less than unity. We proceed to analyze the full HIV-TB coinfection model and extend the model to incorporate antiretroviral therapy for the AIDS cases and treatment of active and latent forms of TB. The thresholds and equilibria quantities for the models are determined and stabilities analyzed. From the study we conclude that treatment of AIDS cases results in a significant reductions of numbers of individuals progressing to active TB. Further, treatment of latent and active forms of TB results in delayed onset of the AIDS stage of HIV infection.

Of mice and men: asymmetric interactions between Bordetella pathogen species

In a recent experiment, we found that mice previously infected with Bordetella pertussis were not protected against a later infection with Bordetella parapertussis, while primary infection with B. parapertussis conferred cross-protection. This challenges the common assumption made in most mathematical models for pathogenic strain dynamics that cross-immunity between strains is symmetric. Here we investigate the potential consequences of this pattern on the circulation of the two pathogens in human populations. To match the empirical dominance of B. pertussis, we made the additional assumption that B. parapertussis pays a cost in terms of reduced fitness. We begin by exploring the range of parameter values that allow the coexistence of the two pathogens, with or without vaccination. We then track the dynamics of the system following the introduction of anti-pertussis vaccination. Our results suggest that (1) in order for B. pertussis to be more prevalent than B. parapertussis, the former must have a strong competitive advantage, possibly in the form of higher infectivity, and (2) because of asymmetric cross-immunity, the introduction of anti-pertussis vaccination should have little effect on the absolute prevalence of B. parapertussis. We discuss the evidence supporting these predictions, and the potential relevance of this model for other pathogens.

Understanding the transmission dynamics of respiratory syncytial virus using multiple time series and nested models

The nature and role of re-infection and partial immunity are likely to be important determinants of the transmission dynamics of human respiratory syncytial virus (hRSV). We propose a single model structure that captures four possible host responses to infection and subsequent reinfection: partial susceptibility, altered infection duration, reduced infectiousness and temporary immunity (which might be partial). The magnitude of these responses is determined by four homotopy parameters, and by setting some of these parameters to extreme values we generate a set of eight nested, deterministic transmission models. In order to investigate hRSV transmission dynamics, we applied these models to incidence data from eight international locations. Seasonality is included as cyclic variation in transmission. Parameters associated with the natural history of the infection were assumed to be independent of geographic location, while others, such as those associated with seasonality, were assumed location specific. Models incorporating either of the two extreme assumptions for immunity (none or solid and lifelong) were unable to reproduce the observed dynamics. Model fits with either waning or partial immunity to disease or both were visually comparable. The best fitting structure was a lifelong partial immunity to both disease and infection. Observed patterns were reproduced by stochastic simulations using the parameter values estimated from the deterministic models.

Pneumococcal carriage in United Kingdom families: estimating serotype-specific transmission parameters from longitudinal data

Repeated observations of pneumococcal infection in 121 United Kingdom families (October 2001-July 2002) were used to explore the transmission properties of five highly prevalent pneumococcal serotypes (6A, 6B, 14, 19F, 23F). A family-based Markov model was developed, and maximum likelihood estimates were produced for model parameters. The authors found higher community acquisition rates among preschool children for all serotypes and higher within-household transmission for 6A and 14. Significant differences in the spontaneous clearance rate were estimated between age categories and serotypes, with 6B being carried for almost 4 months in children. Different mechanisms of competition between serotypes were investigated, and a complete exclusion model (i.e., the resident strain cannot be outcompeted by challengers) was discarded in favor of a competing mechanism that leaves a resident serotype partially or fully susceptible to challengers. Large variation was found in the challenging strength, which was low for 19F and 23F and high for 6A and 6B. Serotype 6B was the only one characterized by high resistance capacity. Only small differences in the transmission characteristics were found when vaccine and nonvaccine serotypes were grouped, suggesting that a serotype-specific analysis is needed to detect distinctive serotype behavior.

Vaccination and the dynamics of immune evasion

Vaccines exert strong selective pressures on pathogens, favouring the spread of antigenic variants. We propose a simple mathematical model to investigate the dynamics of a novel pathogenic strain that emerges in a population where a previous strain is maintained at low endemic level by a vaccine. We compare three methods to assess the ability of the novel strain to invade and persist: algebraic rate of invasion; deterministic dynamics; and stochastic dynamics. These three techniques provide complementary predictions on the fate of the system. In particular, we emphasize the importance of stochastic simulations, which account for the possibility of extinctions of either strain. More specifically, our model suggests that the probability of persistence of an invasive strain (i) can be minimized for intermediate levels of vaccine cross-protection (i.e. immune protection against the novel strain) and (ii) is lower if cross-immunity acts through a reduced infectious period rather than through reduced susceptibility.

Tracking the dynamics of pathogen interactions: Modeling ecological and immune-mediated processes in a two-pathogen single-host system

Traditionally, epidemiological studies have focused on understanding the dynamics of a single pathogen, assuming no interactions with other pathogens. Recently, a large body of work has begun to explore the effects of immune-mediated interactions, arising from cross-immunity and antibody-dependent enhancement, between related pathogen strains. In addition, ecological processes such as a temporary period of convalescence and pathogen-induced mortality have led to the concept of ecological interference between unrelated diseases. There remains, however, the need for a systematic study of both immunological and ecological processes within a single framework. In this paper, we develop a general two-pathogen single-host model of pathogen interactions that simultaneously incorporates these mechanisms. We are then able to mechanistically explore how immunoecological processes mediate interactions between diseases for a pool of susceptible individuals. We show that the precise nature of the interaction can induce either competitive or cooperative associations between pathogens. Understanding the dynamic implications of multi-pathogen associations has potentially important public health consequences. Such a framework may be especially helpful in disentangling the effects of partially cross-immunizing infections that affect populations with a pre-disposition towards immunosuppression such as children and the elderly.

The transmission and control of mastitis in dairy cows: A theoretical approach

A multi-species model that incorporates the transmission of both major and minor mastitis pathogens as well as the interaction between them via coinfection of a quarter is fitted to data from seven dairy herds. The results suggest that major and minor pathogens can interact, on occasion, in a counter-intuitive way with implications for the control of clinical mastitis. The key finding is that delaying culling of cows with major pathogen infections for more than 100 days post infection could result in a higher prevalence of major pathogen infections, whereas early culling would reduce the levels. A theoretical exploration of current and proposed control strategies is carried out, informed by parameters estimated from the model and data. The results at each stage suggest of areas of further research such as: field-testing of the hypotheses presented; the exploration of a stochastic formulation of the model; analysis of the raw repeated measures data; application of control theory to determine the most effective combination of control strategies; inclusion of economic factors into the modelling framework.

Vaccination against multiple HPV types

Vaccines against the most common human papillomavirus (HPV) types are currently under development. Epidemiologic data suggest that the transmission dynamics of different HPV types are not independent. Some studies indicate that interactions among HPV types are synergistic, where infection with one type facilitates concurrent or subsequent infection with another HPV type. Other studies point to antagonistic interference among HPV types. Here we develop a mathematical model to explore how these interactions may either enhance or diminish the effectiveness of vaccination programs designed to reduce the prevalence of the HPV types associated with cervical cancer. We analyze the local stability of the infection-free and boundary equilibria and characterize the conditions leading to a coexistence equilibrium. We also illustrate the results with numerical simulations using realistic model parameters. We show that if interactions among HPV types are synergistic, mass vaccination may reduce the prevalence of types that are not even included in the vaccine.

Strain replacement in an epidemic model with super-infection and perfect vaccination

Several articles in the recent literature discuss the complexities of the impact of vaccination on competing subtypes of one micro-organism. Both with competing virus strains and competing serotypes of bacteria, it has been established that vaccination has the potential to switch the competitive advantage from one of the pathogen subtypes to the other resulting in pathogen replacement. The main mechanism behind this process of substitution is thought to be the differential effectiveness of the vaccine with respect to the two competing micro-organisms. In this article, we show that, if the disease dynamics is regulated by super-infection, strain substitution may indeed occur even with perfect vaccination. In fact we discuss a two-strain epidemic model in which the first strain can infect individuals already infected by the second and, as far as vaccination is concerned, we consider a best-case scenario in which the vaccine provides perfect protection against both strains. We find out that if the reproduction number of the first strain is smaller than the reproduction number of the second strain and the first strain dominates in the absence of vaccination then increasing vaccination levels promotes coexistence which allows the first strain to persist in the population even if its vaccine-dependent reproduction number is below one. Further increase of vaccination levels induces the domination of the second strain in the population. Thus the second strain replaces the first strain. Large enough vaccination levels lead to the eradication of the disease.

AGE-DEPENDENT EPIDEMIOLOGICAL PATTERNS AND STRAIN DIVERSITY IN HELMINTH PARASITES

Field studies of schistosomes and the major intestinal nematodes Trichuris trichiura and Ascaris lumbricoides repeatedly demonstrate that the intensity and prevalence of infection exhibit marked dependency on host age. Peak levels of infection typically occur in hosts aged between 10 and 14 yr in endemically infected communities. It has widely been assumed that the slow acquisition of resistance in adults is caused by repeated exposure to the same antigenic repertoire of a single parasite strain. Consequently, these empirical patterns have previously been taken to suggest that human immunity to helminth parasites confers poor protection against reinfection. Here, an alternative explanation is suggested on the basis of results from a simplified model of helminth transmission. It is proposed that the empirical observations can be attributed to the circulation of multiple helminth strains that each elicit highly protective immunity. If this hypothesis is correct, estimates of epidemiological parameters from field data and the potential for control of helminth diseases might require reevaluation.

The reinfection threshold promotes variability in tuberculosis epidemiology and vaccine efficacy

Population patterns of infection are determined largely by susceptibility to infection. Infection and vaccination induce an immune response that, typically, reduces susceptibility to subsequent infections. With a general epidemic model, we detect a 'reinfection threshold', above which reinfection is the principal type of transmission and, consequently, infection levels are much higher and vaccination fails. The model is further developed to address human tuberculosis (TB) and the impact of vaccination. The bacille Calmette-Guérin (BCG) is the only vaccine in current use against TB, and there is no consensus about its usefulness. Estimates of protection range from 0 to 80%, and this variability is aggravated by an association between low vaccine efficacy and high prevalence of the disease. We propose an explanation based on three postulates: (i) the potential for transmission varies between populations, owing to differences in socio-economic and environmental factors; (ii) exposure to mycobacteria induces an immune response that is partially protective against reinfection; and (iii) this protection is not significantly improved by BCG vaccination. These postulates combine to reproduce the observed trends, and this is attributed to a reinfection threshold intrinsic to the transmission dynamics. Finally, we demonstrate how reinfection thresholds can be manipulated by vaccination programmes, suggesting that they have a potentially powerful role in global control.

Structural identifiability analysis of some highly structured families of statespace models using differential algebra

In this paper we identify biologically relevant families of models whose structural identifiability analysis could not be performed with available techniques directly. The models considered come from both the immunological and epidemiological literature.

On the determinants of population structure in antigenically diverse pathogens

Many pathogens exhibit antigenic diversity and elicit strain-specific immune responses. This potential for cross-immunity structure in the host resource motivates the development of mathematical models, stressing competition for susceptible hosts in driving pathogen population dynamics and genetics. Here we establish that certain model formulations exhibit characteristics of prototype pattern-forming systems, with pathogen population structure emerging as three possible patterns: (i) incidence is steady and homogeneous; (ii) incidence is steady but heterogeneous; and (iii) incidence shows oscillatory dynamics, with travelling waves in strain-space. Results are robust to strain number, but sensitive to the mechanism of cumulative immunity.

Infection with group C rotavirus in a suburban community in Brazil

Group C rotaviruses are associated with sporadic outbreaks of gastroenteritis worldwide. Age-specific seroprevalence of group C rotavirus antibodies was investigated in sera, randomly collected and representative of a suburban community in Brazil which had previously been screened for group A rotavirus antibodies. Antibody prevalence to group C rotavirus was low in children under 5 years and increased slowly with age to 36% seropositivity in adults, reflecting continuous exposure to primary infection in all age groups. This suggests a higher incidence of infection than disease might predict. Adult antibody prevalence was similar to that in other geographical settings. No obvious patterns of infection with group A and group C rotavirus were found within individuals, which suggests independent transmission. However, further epidemiological studies are required to understand group C rotavirus dynamics and possible interactions with group A rotavirus transmission and immunity.

Microparasite population dynamics and continuous immunity

A mathematical model is presented for the transmission of a microparasite where the hosts occupy one of two states, uninfected or infected. In each state, the hosts are distributed over a continuous range of immunity. The immune levels vary within hosts due to the processes of waning of immunity (when uninfected), and increasing immunity (when infected), eventually resulting in recovery. Immunity level also influences the host's ability to infect or be infected. Thus the proposed model incorporates both inter- and intra-host dynamics. It is shown from equilibrium results that this model is a general form of the susceptible-infected-resistant (SIR) and susceptible-infected-susceptible (SIS) family of models, a development that is useful for exploring multistrain pathogen transmission and use of vaccines which confer temporary protection.