Multi-stage Vector-Borne Zoonoses Models: A Global Analysis

A class of models that describes the interactions between multiple host species and an arthropod vector is formulated and its dynamics investigated. A host-vector disease model where the host’s infection is structured into n stages is formulated and a complete global dynamics analysis is provided. The basic reproduction number acts as a sharp threshold, that is, the disease-free equilibrium is globally asymptotically stable (GAS) whenever \({\mathcal {R}}_0^2\le 1\) and that a unique interior endemic equilibrium exists and is GAS if \({\mathcal {R}}_0^2>1\). We proceed to extend this model with m host species, capturing a class of zoonoses where the cross-species bridge is an arthropod vector. The basic reproduction number of the multi-host-vector, \({\mathcal {R}}_0^2(m)\), is derived and shown to be the sum of basic reproduction numbers of the model when each host is isolated with an arthropod vector. It is shown that the disease will persist in all hosts as long as it persists in one host. Moreover, the overall basic reproduction number increases with respect to the host and that bringing the basic reproduction number of each isolated host below unity in each host is not sufficient to eradicate the disease in all hosts. This is a type of “amplification effect,” that is, for the considered vector-borne zoonoses, the increase in host diversity increases the basic reproduction number and therefore the disease burden.     

Edge removal in random contact networks and the basic reproduction number

Understanding the effect of edge removal on the basic reproduction number ${\mathcal{R}_0}$ for disease spread on contact networks is important for disease management. The formula for the basic reproduction number ${\mathcal{R}_0}$ in random network SIR models of configuration type suggests that for degree distributions with large variance, a reduction of the average degree may actually increase ${\mathcal{R}_0}$ . To understand this phenomenon, we develop a dynamical model for the evolution of the degree distribution under random edge removal, and show that truly random removal always reduces ${\mathcal{R}_0}$ . The discrepancy implies that any increase in ${\mathcal{R}_0}$ must result from edge removal changing the network type, invalidating the use of the basic reproduction number formula for a random contact network. We further develop an epidemic model incorporating a contact network consisting of two groups of nodes with random intra- and inter-group connections, and derive its basic reproduction number. We then prove that random edge removal within either group, and between groups, always decreases the appropriately defined ${\mathcal{R}_0}$ . Our models also allow an estimation of the number of edges that need to be removed in order to curtail an epidemic.     

A Bovine Babesiosis Model with Dispersion

Bovine Babesiosis (BB) is a tick borne parasitic disease with worldwide over 1.3 billion bovines at potential risk of being infected. The disease, also called tick fever, causes significant mortality from infection by the protozoa upon exposure to infected ticks. An important factor in the spread of the disease is the dispersion or migration of cattle as well as ticks. In this paper, we study the effect of this factor. We introduce a number, $\mathcal{P}$ , a “proliferation index,” which plays the same role as the basic reproduction number $\mathcal{R}_{0}$ with respect to the stability/instability of the disease-free equilibrium, and observe that $\mathcal{P}$ decreases as the dispersion coefficients increase. We prove, mathematically, that if $\mathcal{P}>1$ then the tick fever will remain endemic. We also consider the case where the birth rate of ticks undergoes seasonal oscillations. Based on data from Colombia, South Africa, and Brazil, we use the model to determine the effectiveness of several intervention schemes to control the progression of BB.     

Modeling optimal treatment strategies in a heterogeneous mixing model

Background

Many mathematical models assume random or homogeneous mixing for various infectious diseases. Homogeneous mixing can be generalized to mathematical models with multi-patches or age structure by incorporating contact matrices to capture the dynamics of the heterogeneously mixing populations. Contact or mixing patterns are difficult to measure in many infectious diseases including influenza. Mixing patterns are considered to be one of the critical factors for infectious disease modeling.

Methods

A two-group influenza model is considered to evaluate the impact of heterogeneous mixing on the influenza transmission dynamics. Heterogeneous mixing between two groups with two different activity levels includes proportionate mixing, preferred mixing and like-with-like mixing. Furthermore, the optimal control problem is formulated in this two-group influenza model to identify the group-specific optimal treatment strategies at a minimal cost. We investigate group-specific optimal treatment strategies under various mixing scenarios.

Results

The characteristics of the two-group influenza dynamics have been investigated in terms of the basic reproduction number and the final epidemic size under various mixing scenarios. As the mixing patterns become proportionate mixing, the basic reproduction number becomes smaller; however, the final epidemic size becomes larger. This is due to the fact that the number of infected people increases only slightly in the higher activity level group, while the number of infected people increases more significantly in the lower activity level group. Our results indicate that more intensive treatment of both groups at the early stage is the most effective treatment regardless of the mixing scenario. However, proportionate mixing requires more treated cases for all combinations of different group activity levels and group population sizes.

Conclusions

Mixing patterns can play a critical role in the effectiveness of optimal treatments. As the mixing becomes more like-with-like mixing, treating the higher activity group in the population is almost as effective as treating the entire populations since it reduces the number of disease cases effectively but only requires similar treatments. The gain becomes more pronounced as the basic reproduction number increases. This can be a critical issue which must be considered for future pandemic influenza interventions, especially when there are limited resources available.

     

Queens stay,workers leave: caste-specific responses to fatal infections in an ant

Background

The intense interactions among closely related individuals in animal societies provide perfect conditions for the spread of pathogens. Social insects have therefore evolved counter-measures on the cellular, individual, and social level to reduce the infection risk. One striking example is altruistic self-removal, i.e., lethally infected workers leave the nest and die in isolation to prevent the spread of a contagious disease to their nestmates. Because reproductive queens and egg-laying workers behave less altruistically than non-laying workers, e.g., when it comes to colony defense, we wondered whether moribund egg-layers would show the same self-removal as non-reproductive workers. Furthermore, we investigated how a lethal infection affects reproduction and studied if queens and egg-laying workers intensify their reproductive efforts when their residual reproductive value decreases (“terminal investment”).

Results

We treated queens, egg-laying workers from queenless colonies, and non-laying workers from queenright colonies of the monogynous (single-queened) ant Temnothorax crassispinus either with a control solution or a solution containing spores of the entomopathogenic fungus Metarhizium brunneum. Lethally infected workers left the nest and died away from it, regardless of their reproductive status. In contrast, infected queens never left the nest and were removed by workers only after they had died. The reproductive investment of queens strongly decreased after the treatment with both, the control solution and the Metarhizium brunneum suspension. The egg laying rate in queenless colonies was initially reduced in infected colonies but not in control colonies. Egg number increased again with decreasing number of infected workers.

Conclusions

Queens and workers of the ant Temnothorax crassispinus differ in their reaction to an infection risk and a reduced life expectancy. Workers isolate themselves to prevent contagion inside the colony, whereas queens stay in the nest. We did not find terminal investment; instead it appeared that egg-layers completely shut down egg production in response to the lethal infection. Workers in queenless colonies resumed reproduction only after all infected individuals had died, probably again to minimize the risk of infecting the offspring.

     

Mathematical Analysis of the Transmission Dynamics of HIV Syphilis Co-infection in the Presence of Treatment for Syphilis

The re-emergence of syphilis has become a global public health issue, and more persons are getting infected, especially in developing countries. This has also led to an increase in the incidence of human immunodeficiency virus (HIV) infections as some studies have shown in the recent decade. This paper investigates the synergistic interaction between HIV and syphilis using a mathematical model that assesses the impact of syphilis treatment on the dynamics of syphilis and HIV co-infection in a human population where HIV treatment is not readily available or accessible to HIV-infected individuals. In the absence of HIV, the syphilis-only model undergoes the phenomenon of backward bifurcation when the associated reproduction number (\({\mathcal {R}}_{T}\)) is less than unity, due to susceptibility to syphilis reinfection after recovery from a previous infection. The complete syphilis–HIV co-infection model also undergoes the phenomenon of backward bifurcation when the associated effective reproduction number (\({\mathcal {R}}_{C}\)) is less than unity for the same reason as the syphilis-only model. When susceptibility to syphilis reinfection after treatment is insignificant, the disease-free equilibrium of the syphilis-only model is shown to be globally asymptotically stable whenever the associated reproduction number (\({\mathcal {R}}_{T}\)) is less than unity. Sensitivity and uncertainty analysis show that the top three parameters that drive the syphilis infection (with respect to the associated response function, \({\mathcal {R}}_{T}\)) are the contact rate (\(\beta _S\)), modification parameter that accounts for the increased infectiousness of syphilis-infected individuals in the secondary stage of the infection (\(\theta _1\)) and treatment rate for syphilis-only infected individuals in the primary stage of the infection (\(r_1\)). The co-infection model was numerically simulated to investigate the impact of various treatment strategies for primary and secondary syphilis, in both singly and dually infected individuals, on the dynamics of the co-infection of syphilis and HIV. It is observed that if concerted effort is exerted in the treatment of primary and secondary syphilis (in both singly and dually infected individuals), especially with high treatment rates for primary syphilis, this will result in a reduction in the incidence of HIV (and its co-infection with syphilis) in the population.     

Threshold dynamics of an infective disease model with a fixed latent period and non-local infections

In this paper, we derive and analyze an infectious disease model containing a fixed latency and non-local infection caused by the mobility of the latent individuals in a continuous bounded domain. The model is given by a spatially non-local reaction–diffusion system carrying a discrete delay associated with the zero-flux condition on the boundary. By applying some existing abstract results in dynamical systems theory, we prove the existence of a global attractor for the model system. By appealing to the theory of monotone dynamical systems and uniform persistence, we show that the model has the global threshold dynamics which can be described either by the principal eigenvalue of a linear non-local scalar reaction diffusion equation or equivalently by the basic reproduction number ${\mathcal{R}_0}$ for the model. Such threshold dynamics predicts whether the disease will die out or persist. We identify the next generation operator, the spectral radius of which defines basic reproduction number. When all model parameters are constants, we are able to find explicitly the principal eigenvalue and ${\mathcal{R}_0}$ . In addition to computing the spectral radius of the next generation operator, we also discuss an alternative way to compute ${\mathcal{R}_0}$ .     

Reproduction Number and Asymptotic Stability for the Dynamics of a Honey Bee Colony with Continuous Age Structure

A system of partial differential equations is derived as a model for the dynamics of a honey bee colony with a continuous age distribution, and the system is then extended to include the effects of a simplified infectious disease. In the disease-free case, we analytically derive the equilibrium age distribution within the colony and propose a novel approach for determining the global asymptotic stability of a reduced model. Furthermore, we present a method for determining the basic reproduction number \(R_0\) of the infection; the method can be applied to other age-structured disease models with interacting susceptible classes. The results of asymptotic stability indicate that a honey bee colony suffering losses will recover naturally so long as the cause of the losses is removed before the colony collapses. Our expression for \(R_0\) has potential uses in the tracking and control of an infectious disease within a bee colony.     

Estimating transmission probability in schools for the 2009 H1N1 influenza pandemic in Italy

Background

Epidemic models are being extensively used to understand the main pathways of spread of infectious diseases, and thus to assess control methods. Schools are well known to represent hot spots for epidemic spread; hence, understanding typical patterns of infection transmission within schools is crucial for designing adequate control strategies. The attention that was given to the 2009 A/H1N1pdm09 flu pandemic has made it possible to collect detailed data on the occurrence of influenza-like illness (ILI) symptoms in two primary schools of Trento, Italy.

Results

The data collected in the two schools were used to calibrate a discrete-time SIR model, which was designed to estimate the probabilities of influenza transmission within the classes, grades and schools using Markov Chain Monte Carlo (MCMC) methods. We found that the virus was mainly transmitted within class, with lower levels of transmission between students in the same grade and even lower, though not significantly so, among different grades within the schools. We estimated median values of R ₀ from the epidemic curves in the two schools of 1.16 and 1.40; on the other hand, we estimated the average number of students infected by the first school case to be 0.85 and 1.09 in the two schools.

Conclusions

The discrepancy between the values of R ₀ estimated from the epidemic curve or from the within-school transmission probabilities suggests that household and community transmission played an important role in sustaining the school epidemics. The high probability of infection between students in the same class confirms that targeting within-class transmission is key to controlling the spread of influenza in school settings and, as a consequence, in the general population.

     

Caspase-1-Mediated Pyroptosis of the Predominance for Driving CD4$$^{}$$ T Cells Death: A Nonlocal Spatial Mathematical Model

Caspase-1-mediated pyroptosis is the predominance for driving CD4\(^{+}\) T cells death. Dying infected CD4\(^{+}\) T cells can release inflammatory signals which attract more uninfected CD4\(^{+}\) T cells to die. This paper is devoted to developing a diffusive mathematical model which can make useful contributions to understanding caspase-1-mediated pyroptosis by inflammatory cytokines IL-1\(\beta \) released from infected cells in the within-host environment. The well-posedness of solutions, basic reproduction number, threshold dynamics are investigated for spatially heterogeneous infection. Travelling wave solutions for spatially homogeneous infection are studied. Numerical computations reveal that the spatially heterogeneous infection can make \(\mathscr {R}_0>1\), that is, it can induce the persistence of virus compared to the spatially homogeneous infection. We also find that the random movements of virus have no effect on basic reproduction number for the spatially homogeneous model, while it may result in less infection risk for the spatially heterogeneous model, under some suitable parameters. Further, the death of infected CD4\(^{+}\) cells which are caused by pyroptosis can make \(\mathscr {R}_0<1\), that is, it can induce the extinction of virus, regardless of whether or not the parameters are spatially dependent.     

Modelling the impact of correlations between condom use and sexual contact pattern on the dynamics of sexually transmitted infections

Background

It is believed that sexually active people, i.e. people having multiple or concurrent sexual partners, are at a high risk of sexually transmitted infections (STI), but they are likely to be more aware of the risk and may exhibit greater fraction of the use of condom. The purpose of the present study is to examine the correlation between condom use and sexual contact pattern and clarify its impact on the transmission dynamics of STIs using a mathematical model.

Methods

The definition of sexual contact pattern can be broad, but we focus on two specific aspects: (i) type of partnership (i.e. steady or casual partnership) and (ii) existence of concurrency (i.e. with single or multiple partners). Systematic review and meta-analysis of published studies are performed, analysing literature that epidemiologically examined the relationship between condom use and sexual contact pattern. Subsequently, we employ an epidemiological model and compute the reproduction number that accounts for with and without concurrency so that the corresponding coverage of condom use and its correlation with existence of concurrency can be explicitly investigated using the mathematical model. Combining the model with parameters estimated from the meta-analysis along with other assumed parameters, the impact of varying the proportion of population with multiple partners on the reproduction number is examined.

Results

Based on systematic review, we show that a greater number of people used condoms during sexual contact with casual partners than with steady partners. Furthermore, people with multiple partners use condoms more frequently than people with a single partner alone. Our mathematical model revealed a positive relationship between the effective reproduction number and the proportion of people with multiple partners. Nevertheless, the association was reversed to be negative by employing a slightly greater value of the relative risk of condom use for people with multiple partners than that empirically estimated.

Conclusions

Depending on the correlation between condom use and the existence of concurrency, association between the proportion of people with multiple partners and the reproduction number can be reversed, suggesting the sexually active population is not necessary a primary target population to encourage condom use (i.e., sexually less active individuals could equivalently be a target in some cases).

     

A Structured Avian Influenza Model with Imperfect Vaccination and Vaccine-Induced Asymptomatic Infection

We introduce a model of avian influenza in domestic birds with imperfect vaccination and age-since-vaccination structure. The model has four components: susceptible birds, vaccinated birds (stratified by vaccination age), asymptomatically infected birds, and infected birds. The model includes reduction in the probability of infection, decreasing severity of disease of vaccinated birds and vaccine waning. The basic reproduction number, \(\mathcal R_0\) , is calculated. The disease-free equilibrium is found to be globally stable under certain conditions when \({\mathcal {R}}_0<1\) . When \({\mathcal {R}}_0>1\) , existence of an endemic equilibrium is proved (with uniqueness for the ODE case and local stability under stricter conditions) and uniform persistence of the disease is established. The inclusion of reduction in susceptibility of vaccinated birds, reduction in infectiousness of asymptomatically infected birds and vaccine waning can have important implications for disease control. We analytically and numerically demonstrate that vaccination can paradoxically increase the total number of infected, resulting in the “silent spread” of the disease. We also study the effects of vaccine efficacy on disease prevalence and the minimum critical vaccination coverage, a threshold value for vaccination coverage to avoid an increase in total disease prevalence due to asymptomatic infection.     

Modeling the Interruption of the Transmission of Soil-Transmitted Helminths by Repeated Mass Chemotherapy of School-Age Children

Background

The control or elimination of neglected tropical diseases has recently become the focus of increased interest and funding from international agencies through the donation of drugs. Resources are becoming available for the treatment of soil-transmitted helminth (STH) infection through school-based deworming strategies. However, little research has been conducted to assess the impact of STH treatment that could be used to guide the design of efficient elimination programs.

Methodology

We construct and analyse an age-structured model of STH population dynamics under regular treatment. We investigate the potential for elimination with finite rounds of treatment, and how this depends on the value of the basic reproductive number R₀ and treatment frequency.

Principal findings

Analysis of the model indicates that its behaviour is determined by key parameter groupings describing the basic reproduction number and the fraction of it attributable to the treated group, the timescale of material in the environment and the frequency and efficacy of treatment. Mechanisms of sexual reproduction and persistence of infectious material in the environment are found to be much more important in the context of elimination than in the undisturbed baseline scenario. For a given rate of drug use, sexual reproduction dictates that less frequent, higher coverage treatment is more effective. For a given treatment coverage level, the lifespan of infectious material in the environment places a limit on the effectiveness of increased treatment frequency.

Conclusions

Our work suggests that for models to capture the dynamics of parasite burdens in populations under regular treatment as elimination is approached, they need to include the effects of sexual reproduction among parasites and the dynamics infectious material in the reservoir. The interaction of these two mechanisms has a strong effect on optimum treatment strategies, both in terms of how frequently to treat and for how long.     

A conceptual model for optimizing vaccine coverage to reduce vector-borne infections in the presence of antibody-dependent enhancement

Background

Many vector-borne diseases co-circulate, as the viruses from the same family are also transmitted by the same vector species. For example, Zika and dengue viruses belong to the same Flavivirus family and are primarily transmitted by a common mosquito species Aedes aegypti. Zika outbreaks have also commonly occurred in dengue-endemic areas, and co-circulation and co-infection of both viruses have been reported. As recent immunological cross-reactivity studies have confirmed that convalescent plasma following dengue infection can enhance Zika infection, and as global efforts of developing dengue and Zika vaccines are intensified, it is important to examine whether and how vaccination against one disease in a large population may affect infection dynamics of another disease due to antibody-dependent enhancement.

Methods

Through a conceptual co-infection dynamics model parametrized by reported dengue and Zika epidemic and immunological cross-reactivity characteristics, we evaluate impact of a hypothetical dengue vaccination program on Zika infection dynamics in a single season when only one particular dengue serotype is involved.

Results

We show that an appropriately designed and optimized dengue vaccination program can not only help control the dengue spread but also, counter-intuitively, reduce Zika infections. We identify optimal dengue vaccination coverages for controlling dengue and simultaneously reducing Zika infections, as well as the critical coverages exceeding which dengue vaccination will increase Zika infections.

Conclusion

This study based on a conceptual model shows the promise of an integrative vector-borne disease control strategy involving optimal vaccination programs, in regions where different viruses or different serotypes of the same virus co-circulate, and convalescent plasma following infection from one virus (serotype) can enhance infection against another virus (serotype). The conceptual model provides a first step towards well-designed regional and global vector-borne disease immunization programs.

     

How Levins’ dynamics emerges from a Ricker metapopulation model

Understanding the dynamics of metapopulations close to extinction is of vital importance for management. Levins-like models, in which local patches are treated as either occupied or empty, have been used extensively to explore the extinction dynamics of metapopulations, but they ignore the important role of local population dynamics. In this paper, we consider a stochastic metapopulation model where local populations follow a stochastic, density-dependent dynamics (the Ricker model), and use this framework to investigate the behaviour of the metapopulation on the brink of extinction. We determine under which circumstances the metapopulation follows a time evolution consistent with Levins’ dynamics. We derive analytical expressions for the colonisation and extinction rates (c and e) in Levins-type models in terms of reproduction, survival and dispersal parameters of the local populations, providing an avenue to parameterising Levins-like models from the type of information on local demography that is available for a number of species. To facilitate applying our results, we provide a numerical algorithm for computing c and e.     

Deciphering the complex nature of bolting time regulation in <Emphasis Type="Italic">Beta vulgaris</Emphasis>

Key message

Only few genetic loci are sufficient to increase the variation of bolting time in Beta vulgaris dramatically, regarding vernalization requirement, seasonal bolting time and reproduction type.

Abstract

Beta species show a wide variation of bolting time regarding the year of first reproduction, seasonal bolting time and the number of reproduction cycles. To elucidate the genetics of bolting time control, we used three F₃ mapping populations that were produced by crossing a semelparous, annual sugar beet with iteroparous, vernalization-requiring wild beet genotypes. The semelparous plants died after reproduction, whereas iteroparous plants reproduced at least twice. All populations segregated for vernalization requirement, seasonal bolting time and the number of reproduction cycles. We found that vernalization requirement co-segregated with the bolting locus B on chromosome 2 and was inherited independently from semel- or iteroparous reproduction. Furthermore, we found that seasonal bolting time is a highly heritable trait (h ² > 0.84), which is primarily controlled by two major QTL located on chromosome 4 and 9. Late bolting alleles of both loci act in a partially recessive manner and were identified in both iteroparous pollinators. We observed an additive interaction of both loci for bolting delay. The QTL region on chromosome 4 encompasses the floral promoter gene BvFT2, whereas the QTL on chromosome 9 co-localizes with the BR ₁ locus, which controls post-winter bolting resistance. Our findings are applicable for marker-assisted sugar beet breeding regarding early bolting to accelerate generation cycles and late bolting to develop bolting-resistant spring and winter beets. Unexpectedly, one population segregated also for dwarf growth that was found to be controlled by a single locus on chromosome 9.

     

Vaccination Strategies for SIR Vector-Transmitted Diseases

Vector-borne diseases are one of the major public health problems in the world with the fastest spreading rate. Control measures have been focused on vector control, with poor results in most cases. Vaccines should help to reduce the diseases incidence, but vaccination strategies should also be defined. In this work, we propose a vector-transmitted SIR disease model with age-structured population subject to a vaccination program. We find an expression for the age-dependent basic reproductive number \(R_0\) , and we show that the disease-free equilibrium is locally stable for \(R_0 \le 1\) , and a unique endemic equilibrium exists for \(R_0 >1\) . We apply the theoretical results to public data to evaluate vaccination strategies, immunization levels, and optimal age of vaccination for dengue disease.     

Stability of a Stochastic Model of an SIR Epidemic with Vaccination

We prove almost sure exponential stability for the disease-free equilibrium of a stochastic differential equations model of an SIR epidemic with vaccination. The model allows for vertical transmission. The stochastic perturbation is associated with the force of infection and is such that the total population size remains constant in time. We prove almost sure positivity of solutions. The main result concerns especially the smaller values of the diffusion parameter, and describes the stability in terms of an analogue \(\mathcal{R}_\sigma\) of the basic reproduction number \(\mathcal{R}_0\) of the underlying deterministic model, with \(\mathcal{R}_\sigma \le \mathcal{R}_0\). We prove that the disease-free equilibrium is almost sure exponentially stable if \(\mathcal{R}_\sigma <1\).