A spatially stochastic epidemic model with partial immunization shows in mean field approximation the reinfection threshold

Recently, the notion of a reinfection threshold in epidemiological models of only partial immunity has been debated in the literature. We present a rigorous analysis of a model of reinfection which shows a clear threshold behaviour at the parameter point where the reinfection threshold was originally described. Furthermore, we demonstrate that this threshold is the mean field version of a transition in corresponding spatial models of immunization. The reinfection threshold corresponds to the transition between annular growth of an epidemics spreading into a susceptible area leaving recovered behind and compact growth of a susceptible-infected-susceptible region growing into a susceptible area. This transition between annular growth and compact growth was described in the physics literature long before the reinfection threshold debate broke out in the theoretical biology literature.     

The reinfection threshold

Thresholds in transmission are responsible for critical changes in infectious disease epidemiology. The epidemic threshold indicates whether infection invades a totally susceptible population. The reinfection threshold indicates whether self-sustained transmission occurs in a population that has developed a degree of partial immunity to the pathogen (by previous infection or vaccination). In models that combine susceptible and partially immune individuals, the reinfection threshold is technically not a bifurcation of equilibria as correctly pointed out by Breban and Blower. However, we show that a branch of equilibria to a reinfection submodel bifurcates from the disease-free equilibrium as transmission crosses this threshold. Consequently, the full model indicates that levels of infection increase by two orders of magnitude and the effect of mass vaccination becomes negligible as transmission increases across the reinfection threshold.     

Infection, reinfection, and vaccination under suboptimal immune protection: epidemiological perspectives

The SIR (susceptible-infectious-resistant) and SIS (susceptible-infectious-susceptible) frameworks for infectious disease have been extensively studied and successfully applied. They implicitly assume the upper and lower limits of the range of possibilities for host immune response. However, the majority of infections do not fall into either of these extreme categories. We combine two general avenues that straddle this range: temporary immune protection (immunity wanes over time since infection), and partial immune protection (immunity is not fully protective but reduces the risk of reinfection). We present a systematic analysis of the dynamics and equilibrium properties of these models in comparison to SIR and SIS, and analyse the outcome of vaccination programmes. We describe how the waning of immunity shortens inter-epidemic periods, and poses major difficulties to disease eradication. We identify a "reinfection threshold" in transmission when partial immunity is included. Below the reinfection threshold primary infection dominates, levels of infection are low, and vaccination is highly effective (approximately an SIR model). Above the reinfection threshold reinfection dominates, levels of infection are high, and vaccination fails to protect (approximately an SIS situation). This association between high prevalence of infection and vaccine failure emphasizes the problems of controlling recurrent infections in high-burden regions. However, vaccines that induce a better protection than natural infection have the potential to increase the reinfection threshold, and therefore constitute interventions with a surprisingly high capacity to reduce infection where reduction is most needed.     

Vaccination based control of infections in SIRS models with reinfection: special reference to pertussis

The aim of this paper is to study the impact of introducing a partially protective vaccine on the dynamics of infection in SIRS models where primary and secondary infections are distinguished. We investigate whether a public health strategy based solely on vaccinating a proportion of newborns can lead to an effective control of the disease. In addition to carrying out the qualitative analysis, the findings are further explained by numerical simulations. The model exhibits backward bifurcation for certain values of the parameters. In these cases the standard basic reproduction number (obtained by inspection of the uninfected state) is not significant. The key threshold is the reinfection level which depends on the relative transmissibility (susceptibility) of secondary, with respect to primary, infected (susceptible) individuals and the relative loss of immunity of vaccinated, with respect to recovered, individuals. If one or all of these ratios decrease, then the threshold increases which increases the possibility to contain the infection by vaccination. The analysis shows further that symptomatic infections can be eliminated by vaccination solely.     

Acquired Resistance to Babesia divergens in Experimental Calves

SYNOPSIS. One intact and 2 splenectomized calves were infected with Babesia divergens and the persistence of the parasites in the blood was followed by periodic subinoculations into susceptible splenectomized calves. After periods varying from 3–7 years the parasites failed to be demonstrable by this method. When the immunity of the animals was challenged by the intravenous injection of blood containing Babesia divergens , they were all resistant to reinfection. These observations add support to the suggestion that sterile immunity may play a part in the resistance of cattle to reinfection with B. divergens.     

Reinfection induced disease in a spatial SIRI model

Spatial models are widely used in epidemiology to investigate persistence and extinction of disease as well as their spatial patterns. One of the most important issues in studying epidemic models is the role of infection on the persistence and extinction of the disease. In this paper, we investigate a spatial susceptible–infected–recovered–infected model using cellular automata. We show that, in the regime where disease disappears in the susceptible–infected–recovered–susceptible model, spiral and target waves will emerge in the two-dimensional space due to the reinfection. The obtained results may point out that reinfection has great influence on the epidemic spreading, which enriches the findings of spatiotemporal dynamics in epidemic models.     

Reinfection after rubella and congenital polymalformation syndrome

A case is reported of a term newborn with intra uterine growth retardation and numerous malformations such as complex heart disease, abnormalities of distal limbs, cleft palate. Death occurred after two days. The diagnosis of rubella embryopathy was confirmed by the following criteria: a high level of rubella antibodies in mother and newborn (1/1000) an isolation of rubella virus from the infant's urine. Diagnosis of rubella after reinfection was documented by a high level of antibodies in the mother three years before this pregnancy. Other observations reported in literature confirm the extreme rarity of congenital rubella after reinfection.     

The position of localized soil compaction determines root and subsequent shoot growth responses

Plants growing in soils typically experience a mixture of loose and compact soil. The hypothesis that the proportion of a root system exposed to compact soil and/or the timing at which this exposure occurs determines shoot growth responses was tested. Broccoli (Brassica oleracea var. italica cv. Greenbelt) seedlings were grown in pot experiments with compact, loose and localized soil compaction created by either horizontal (compact subsoils 75 or 150 mm below loose topsoil) or vertical (adjacent compact and loose columns of soil) configurations of loose (1.2 Mg m(-3)) and compact (1.8 Mg m(-3)) soil. Entirely compact soil reduced leaf area by up to 54%, relative to loose soil. When compaction was localized, only the vertical columns of compact and loose soil reduced leaf area (by 30%). Neither the proportion of roots in compact soil nor the timing of exposure could explain the differing shoot growth responses to localized soil compaction. Instead, the strong relationship between total root length and leaf area (r(2)=0.92) indicated that localized soil compaction reduced shoot growth only when it suppressed total root length. This occurred when isolated root axes of the same plant were exposed to vertical columns of compact and loose soil. When a single root axis grew through loose soil into either a shallow or deep compact subsoil, compensatory root growth in the loose soil maintained total root length and thus shoot growth was unaffected. These contrasting root systems responses to localized soil compaction may explain the variable shoot growth responses observed under heterogeneous conditions.     

From elaborate to compact seasonal plant epidemic models and back: is competitive exclusion in the details?

Seasonality, or periodic host absence, is a central feature in plant epidemiology. In this respect, seasonal plant epidemic models take into account the way the parasite overwinters and generates new infections. These are termed primary infections. In the literature, one finds two classes of models: high-dimensional elaborate models and low-dimensional compact models, where primary infection dynamics are explicit and implicit, respectively. Investigating a compact model allowed previous authors to show the existence of a competitive exclusion principle. However, the way compact models derive from elaborate models has not been made explicit yet. This makes it unclear whether results such as competitive exclusion extend to elaborate models as well. Here, we show that assuming primary infection dynamics are fast in a standard elaborate model translates into a compact form. Yet, it is not that usually found in the literature. Moreover, we numerically show that coexistence is possible in this original compact form. Reversing the question, we show that the usual compact form approximates an alternate elaborate model, which differs from the earlier one in that primary infection dynamics are density dependent. We discuss to which extent these results shed light on coexistence within soil- and air-borne plant parasites, such as within the take-all disease of wheat and the grapevine powdery mildew cryptic species complexes, respectively.     

Distinct Annular Oligomers Captured along the Assembly and Disassembly Pathways of Transthyretin Amyloid Protofibrils

Background

Defects in protein folding may lead to severe degenerative diseases characterized by the appearance of amyloid fibril deposits. Cytotoxicity in amyloidoses has been linked to poration of the cell membrane that may involve interactions with amyloid intermediates of annular shape. Although annular oligomers have been detected in many amyloidogenic systems, their universality, function and molecular mechanisms of appearance are debated.

Methodology/Principal Findings

We investigated with high-resolution in situ atomic force microscopy the assembly and disassembly of transthyretin (TTR) amyloid protofibrils formed of the native protein by pH shift. Annular oligomers were the first morphologically distinct intermediates observed in the TTR aggregation pathway. Morphological analysis suggests that they can assemble into a double-stack of octameric rings with a 16±2 nm diameter, and displaying the tendency to form linear structures. According to light scattering data coupled to AFM imaging, annular oligomers appeared to undergo a collapse type of structural transition into spheroid oligomers containing 8–16 monomers. Disassembly of TTR amyloid protofibrils also resulted in the rapid appearance of annular oligomers but with a morphology quite distinct from that observed in the assembly pathway.

Conclusions/Significance

Our observations indicate that annular oligomers are key dynamic intermediates not only in the assembly but also in the disassembly of TTR protofibrils. The balance between annular and more compact forms of aggregation could be relevant for cytotoxicity in amyloidogenic disorders.     

How does the resistance threshold in spatially explicit epidemic dynamics depend on the basic reproductive ratio and spatial correlation of crop genotypes?

We examined the fraction of resistant cultivars necessary to prevent a global pathogen outbreak (the resistance threshold) using a spatially explicit epidemiological model (SIR model) in a finite, two-dimensional, lattice-structured host population. Infectious diseases in our model could be transmitted to susceptible nearest-neighbour sites, and the infected site either recovered or died after an exponentially distributed infectious period. Threshold behaviour of this spatially explicit SIR model cannot be reduced to that of bond percolation, as was previously noted in the literature, unless extreme assumptions (synchronized infection events with a fixed lag) are imposed on infection process. The resistance threshold is significantly lower than that of conventional mean-field epidemic models, and is even lower if the spatial configuration of resistant and susceptible crops are negatively correlated. Finite size scaling applied to the resistance threshold for a finite basic reproductive ratio ρ of pathogen reveals that its difference from static percolation threshold (0.41) is inversely proportional to ρ. Our formula for the basic reproductive ratio dependency of the resistance threshold produced an estimate for the critical basic reproductive ratio (4.7) in a universally susceptible population, which is much larger than the corresponding critical value (1) in the mean-field model and nearly three times larger than the critical growth rate of a basic contact process (SIS model). Pair approximation reveals that the resistance threshold for preventing a global epidemic is factor 1/(1−η) greater with spatially correlated planting than with random planting, where η is initial correlation in host genotypes between nearest-neighbour sites. Thus the eradication is harder with a positive spatial correlation (η>0) in mixed susceptible/resistant plantings, and is easier with a negative correlation (η<0). The effect of finite field size (L), which corresponded to the mean distance between sources of infections, is given by the increased resistance threshold (by the amount L^−0.75) from its infinite size limit. Implications of these results on effective planting strategies in multi-line control plans are discussed.     

A model for tuberculosis with exogenous reinfection

Following primary tuberculosis (TB) infection, only approximately 10% of individuals develop active T.B. Most people are assumed to mount an effective immune response to the initial infection that limits proliferation of the bacilli and leads to long-lasting partial immunity both to further infection and to reactivation of latent bacilli remaining from the original infection. Infected individuals may develop active TB as a consequence of exogenous reinfection, i.e., acquiring a new infection from another infectious individual. Our results in this paper suggest that exogenous reinfection has a drastic effect on the qualitative dynamics of TB. The incorporation of exogenous reinfection into our TB model allows the possibility of a subcritical bifurcation at the critical value of the basic reproductive number R(0)=1, and hence the existence of multiple endemic equilibria for R(0)<1 and the exogenous reinfection rate larger than a threshold. Our results suggest that reducing R(0) to be smaller than one may not be sufficient to eradicate the disease. An additional reduction in reinfection rate may be required. These results may also partially explain the recently observed resurgence of TB.     

Failure of Echinostoma lindoense to reinfect snails already harboring that species.

In Biomphalaria glabrata snails already harboring Echinostoma lindoense, attempts at reinfection with the same trematode species failed when the interval between exposures was 6 days or more. With 2 days between exposures, the experimental snails were as susceptible to reinfection as the control snails. Between these limits, the reinfection failure rate increased with the exposure interval. Failure to reinfect the snails was probably due to cannibalism, young sporocysts from the second infection being swallowed by predatory rediae from the first infection. Development of sporocysts in successfully reinfected snails was normal.     

Dynamics of Epidemiological Models

We study the SIS and SIRI epidemic models discussing different approaches to compute the thresholds that determine the appearance of an epidemic disease. The stochastic SIS model is a well known mathematical model, studied in several contexts. Here, we present recursively derivations of the dynamic equations for all the moments and we derive the stationary states of the state variables using the moment closure method. We observe that the steady states give a good approximation of the quasi-stationary states of the SIS model. We present the relation between the SIS stochastic model and the contact process introducing creation and annihilation operators. For the spatial stochastic epidemic reinfection model SIRI, where susceptibles S can become infected I, then recover and remain only partial immune against reinfection R, we present the phase transition lines using the mean field and the pair approximation for the moments. We use a scaling argument that allow us to determine analytically an explicit formula for the phase transition lines in pair approximation.     

Model of white spot syndrome virus (WSSV) epidemics in Litopenaeus vannamei

White spot syndrome virus (WSSV) is devastating shrimp aquaculture throughout the world, but despite its economic importance no work has been done on modeling epidemics of this pathogen. Therefore we developed a Reed-Frost epidemic model for WSSV in Litopenaeus vannamei. The model includes uninfected susceptible, latently infected, acutely infected, and dead infected shrimp. The source of new infections during an outbreak is considered to be dead infected shrimp. The transmission coefficient, patency coefficient, virulence coefficient, and removal coefficient (disappearance of dead infected shrimp) control the dynamics of the model. In addition, an explicit area parameter is included to help to clarify the distinction between density and absolute shrimp population size. An analysis of the model finds that as number of shrimp, initial dose, transmission coefficient, patency coefficient, virulence coefficient, or removal coefficient changes, the speed of the epidemic changes. The model predicts that a threshold density of susceptible shrimp exists below which an outbreak of WSSV will not occur. Only initial dose, transmission coefficient, removal coefficient, and area coefficient affect the predicted threshold density. Increases in the transmission coefficient reduce the threshold value, whereas increases in the other factors cause the threshold value to increase. Epidemic models may prove useful to the shrimp aquaculture industry by suggesting testable hypotheses, some of which may contribute to the eventual control of WSSV outbreaks.     

A scaling analysis in the SIRI epidemiological model

For the spatial stochastic epidemic reinfection model SIRI, where susceptibles S can become infected I, then recover and remain only partial immune against reinfection R, we determine the phase transition lines using pair approximation for the moments derived from the master equation. We introduce a scaling argument that allows us to determine analytically an explicit formula for these phase transition lines and prove rigorously the heuristic results obtained previously.     

Effective degree network disease models

An effective degree approach to modeling the spread of infectious diseases on a network is introduced and applied to a disease that confers no immunity (a Susceptible-Infectious-Susceptible model, abbreviated as SIS) and to a disease that confers permanent immunity (a Susceptible-Infectious-Recovered model, abbreviated as SIR). Each model is formulated as a large system of ordinary differential equations that keeps track of the number of susceptible and infectious neighbors of an individual. From numerical simulations, these effective degree models are found to be in excellent agreement with the corresponding stochastic processes of the network on a random graph, in that they capture the initial exponential growth rates, the endemic equilibrium of an invading disease for the SIS model, and the epidemic peak for the SIR model. For each of these effective degree models, a formula for the disease threshold condition is derived. The threshold parameter for the SIS model is shown to be larger than that derived from percolation theory for a model with the same disease and network parameters, and consequently a disease may be able to invade with lower transmission than predicted by percolation theory. For the SIR model, the threshold condition is equal to that predicted by percolation theory. Thus unlike the classical homogeneous mixing disease models, the SIS and SIR effective degree models have different disease threshold conditions.     

Atomic force microscopy and force spectroscopy study of Langmuir-Blodgett films formed by heteroacid phospholipids of biological interest

Langmuir-Blodgett (LB) films of two heteroacid phospholipids of biological interest 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphoethanolamine (POPE) and 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC), as well as a mixed monolayer with chi(POPC)=0.4, were transferred onto mica in order to investigate by a combination of atomic force microscopy (AFM) and force spectroscopy (FS) their height, and particularly, their nanomechanical properties. AFM images of such monolayers extracted at 30 mN m(-1) revealed a smooth and defect-free topography except for the POPE monolayer. Since scratching such soft monolayers in contact mode was proved unsuccessful, their molecular height was measured by means of the width of the jump present in the respective force-extension curves. While for pure POPC a small jump occurs near zero force, for the mixed monolayer with chi(POPC)=0.4 the jump occurs at approximately 800 pN. Widths of approximately 2 nm could be established for POPC and chi(POPC)=0.4, but not for POPE monolayer at this extracting pressure. Such different mechanical stability allowed us to directly measure the threshold area/lipid range value needed to induce mechanical stability to the monolayers. AFM imaging and FS were next applied to get further structural and mechanical insight into the POPE phase transition (LC-LC') occurring at pressures >36.5 mN m(-1). This phase transition was intimately related to a sudden decrease in the area/molecule value, resulting in a jump in the force curve occurring at high force ( approximately 1.72 nN). FS reveals to be the unique experimental technique able to unveil structural and nanomechanical properties for such soft phospholipid monolayers. The biological implications of the nanomechanical properties of the systems under investigation are discussed considering that the annular phospholipids region of some transmembrane proteins is enriched in POPE.