Dynamics of an epidemic model with non-local infections for diseases with latency over a patchy environment

In this paper, with the assumptions that an infectious disease in a population has a fixed latent period and the latent individuals of the population may disperse, we formulate an SIR model with a simple demographic structure for the population living in an n-patch environment (cities, towns, or countries, etc.). The model is given by a system of delay differential equations with a fixed delay accounting for the latency and a non-local term caused by the mobility of the individuals during the latent period. Assuming irreducibility of the travel matrices of the infection related classes, an expression for the basic reproduction number R₀{\mathcal{R}_0} is derived, and it is shown that the disease free equilibrium is globally asymptotically stable if R₀ < 1{\mathcal{R}_0 < 1} , and becomes unstable if ${\mathcal{R}_0 > 1}${\mathcal{R}_0 > 1} . In the latter case, there is at least one endemic equilibrium and the disease will be uniformly persistent. When n = 2, two special cases allowing reducible travel matrices are considered to illustrate joint impact of the disease latency and population mobility on the disease dynamics. In addition to the existence of the disease free equilibrium and interior endemic equilibrium, the existence of a boundary equilibrium and its stability are discussed for these two special cases.     

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.     

Modeling Optimal Age-Specific Vaccination Strategies Against Pandemic Influenza

In the context of pandemic influenza, the prompt and effective implementation of control measures is of great concern for public health officials around the world. In particular, the role of vaccination should be considered as part of any pandemic preparedness plan. The timely production and efficient distribution of pandemic influenza vaccines are important factors to consider in mitigating the morbidity and mortality impact of an influenza pandemic, particularly for those individuals at highest risk of developing severe disease. In this paper, we use a mathematical model that incorporates age-structured transmission dynamics of influenza to evaluate optimal vaccination strategies in the epidemiological context of the Spring 2009 A (H1N1) pandemic in Mexico. We extend previous work on age-specific vaccination strategies to time-dependent optimal vaccination policies by solving an optimal control problem with the aim of minimizing the number of infected individuals over the course of a single pandemic wave. Optimal vaccination policies are computed and analyzed under different vaccination coverages (21%–77%) and different transmissibility levels (R₀\mathcal{R}_{0} in the range of 1.8–3). The results suggest that the optimal vaccination can be achieved by allocating most vaccines to young adults (20–39 yr) followed by school age children (6–12 yr) when the vaccination coverage does not exceed 30%. For higher R₀\mathcal{R}_{0} levels ($\mathcal{R}_{0}>=2.4$\mathcal{R}_{0}>=2.4), or a time delay in the implementation of vaccination (>90 days), a quick and substantial decrease in the pool of susceptibles would require the implementation of an intensive vaccination protocol within a shorter period of time. Our results indicate that optimal age-specific vaccination rates are significantly associated with R₀\mathcal{R}_{0}, the amount of vaccines available and the timing of vaccination.     

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 \(\mathcal{R}_\sigma <1\).     

A Mathematical Model of Anthrax Transmission in Animal Populations

A general mathematical model of anthrax (caused by Bacillus anthracis) transmission is formulated that includes live animals, infected carcasses and spores in the environment. The basic reproduction number \(\mathcal {R}_0\) is calculated, and existence of a unique endemic equilibrium is established for \(\mathcal {R}_0\) above the threshold value 1. Using data from the literature, elasticity indices for \(\mathcal {R}_0\) and type reproduction numbers are computed to quantify anthrax control measures. Including only herbivorous animals, anthrax is eradicated if \(\mathcal {R}_0 < 1\). For these animals, oscillatory solutions arising from Hopf bifurcations are numerically shown to exist for certain parameter values with \(\mathcal {R}_0>1\) and to have periodicity as observed from anthrax data. Including carnivores and assuming no disease-related death, anthrax again goes extinct below the threshold. Local stability of the endemic equilibrium is established above the threshold; thus, periodic solutions are not possible for these populations. It is shown numerically that oscillations in spore growth may drive oscillations in animal populations; however, the total number of infected animals remains about the same as with constant spore growth.     

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 ${\mathcal{R}_0}$ .     

A Within-Host Virus Model with Periodic Multidrug Therapy

This paper is devoted to the investigation of the effects of periodic drug treatment on a standard within-host virus model. We first introduce the basic reproduction ratio for the model, and then show that the infection free equilibrium is globally asymptotically stable, and the disease eventually disappears if $\mathcal{R}_{0} < 1$ , while there exists at least one positive periodic state and the disease persists when $\mathcal{R}_{0}>1$ . We also consider an optimization problem by shifting the phase of these drug efficacy functions. It turns out that shifting the phase can certainly affect the stability of the infection free steady state. A numerical study is performed to illustrate our analytic results.     

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.     

Epidemic models with uncertainty in the reproduction number

One of the first quantities to be estimated at the start of an epidemic is the basic reproduction number, ${\mathcal{R}_0}$ . The progress of an epidemic is sensitive to the value of ${\mathcal{R}_0}$ , hence we need methods for exploring the consequences of uncertainty in the estimate. We begin with an analysis of the SIR model, with ${\mathcal{R}_0}$ specified by a probability distribution instead of a single value. We derive probability distributions for the prevalence and incidence of infection during the initial exponential phase, the peaks in prevalence and incidence and their timing, and the final size of the epidemic. Then, by expanding the state variables in orthogonal polynomials in uncertainty space, we construct a set of deterministic equations for the distribution of the solution throughout the time-course of the epidemic. The resulting dynamical system need only be solved once to produce a deterministic stochastic solution. The method is illustrated with ${\mathcal{R}_0}$ specified by uniform, beta and normal distributions. We then apply the method to data from the New Zealand epidemic of H1N1 influenza in 2009. We apply the polynomial expansion method to a Kermack–McKendrick model, to simulate a forecasting system that could be used in real time. The results demonstrate the level of uncertainty when making parameter estimates and projections based on a limited amount of data, as would be the case during the initial stages of an epidemic. In solving both problems we demonstrate how the dynamical system is derived automatically via recurrence relationships, then solved numerically.     

A mathematical model for assessing control strategies against West Nile virus

Since its incursion into North America in 1999, West Nile virus (WNV) has spread rapidly across the continent resulting in numerous human infections and deaths. Owing to the absence of an effective diagnostic test and therapeutic treatment against WNV, public health officials have focussed on the use of preventive measures in an attempt to halt the spread of WNV in humans. The aim of this paper is to use mathematical modelling and analysis to assess two main anti-WNV preventive strategies, namely: mosquito reduction strategies and personal protection. We propose a single-season ordinary differential equation model for the transmission dynamics of WNV in a mosquito-bird-human community, with birds as reservoir hosts and culicine mosquitoes as vectors. The model exhibits two equilibria; namely the disease-free equilibrium and a unique endemic equilibrium. Stability analysis of the model shows that the disease-free equilibrium is globally asymptotically stable if a certain threshold quantity , which depends solely on parameters associated with the mosquito-bird cycle, is less than unity. The public health implication of this is that WNV can be eradicated from the mosquito-bird cycle (and, consequently, from the human population) if the adopted mosquito reduction strategy (or strategies) can make . On the other hand, it is shown, using a novel and robust technique that is based on the theory of monotone dynamical systems coupled with a regular perturbation argument and a Liapunov function, that if , then the unique endemic equilibrium is globally stable for small WNV-induced avian mortality. Thus, in this case, WNV persists in the mosquito-bird population.     

Topography of nucleic acid helices in solutions. 3. Interactions of spermine and spermidine derivatives with polyadenylic-polyuridylic and polyinosinic-polycytidylic acid helices

The synthesis of spermine derivatives (II), \documentclass{article}\pagestyle{empty}\begin{document}$ {\rm R}_1 {\rm R}_{\rm 2} {\rm R}_{\rm 3} \mathop {\rm N}\limits^ + \left( {{\rm CH}_2 } \right)_3 \mathop {\rm N}\limits^ + {\rm R}_{\rm 1} {\rm R}_{\rm 2} \left( {{\rm CH}_2 } \right)_2 ]_2 \cdot 4{\rm X}^ - $\end{document}, and spermidine derivatives (III), \documentclass{article}\pagestyle{empty}\begin{document}$ {\rm R}_1 {\rm R}_{\rm 2} {\rm R}_{\rm 3} \mathop {\rm N}\limits^ + \left( {{\rm CH}_2 } \right)_4 \mathop {\rm N}\limits^ + {\rm R}_{\rm 1} {\rm R}_{\rm 2} \left( {{\rm CH}_2 } \right)_3 \mathop {\rm N}\limits^ + {\rm R}_{\rm 1} {\rm R}_{\rm 2} {\rm R}_3 \cdot 3{\rm X}^ - $\end{document}, are reported. The effects of these salts on the helix–coil transition of rA–rU and rI–rC helices were examined. Increasing the size of the hydrophobic substituents, R¹, R², and R³ lowers the degree of stabilization of the helical structure. The disproportionation reaction, 2rA–rU→rA–rU₂ + rA occurs readily with salts II and III, especially when the substituents, R₁, R₂, and R₃ are small, i.e., H or Me. Spermine is found to stabilize the rA–rU² and rI–rC helices to approximately the same extent; however, large differences between the degree of stabilization of rA–rU₂ and rI-rC helices are observed when the substituents R₁, R₂, and R₃ are large hydrophobic groups. Similar results are also obtained for the spermidine series. Finally, differences in the interactions of the salts II and III with rA–rU₂ and rI–rC helices suggest that the latter helix is denser.     

Transmission dynamics for vector-borne diseases in a patchy environment

In this paper, a mathematical model is derived to describe the transmission and spread of vector-borne diseases over a patchy environment. The model incorporates into the classic Ross–MacDonald model two factors: disease latencies in both hosts and vectors, and dispersal of hosts between patches. The basic reproduction number \(\mathcal{R }_0\) is identified by the theory of the next generation operator for structured disease models. The dynamics of the model is investigated in terms of \(\mathcal{R }_0\) . It is shown that the disease free equilibrium is asymptotically stable if \(\mathcal{R }_0<1\) , and it is unstable if \(\mathcal{R }_0>1\) ; in the latter case, the disease is endemic in the sense that the variables for the infected compartments are uniformly persistent. For the case of two patches, more explicit formulas for \(\mathcal{R }_0\) are derived by which, impacts of the dispersal rates on disease dynamics are also explored. Some numerical computations for \(\mathcal{R }_0\) in terms of dispersal rates are performed which show visually that the impacts could be very complicated: in certain range of the parameters, \(\mathcal{R }_0\) is increasing with respect to a dispersal rate while in some other range, it can be decreasing with respect to the same dispersal rate. The results can be useful to health organizations at various levels for setting guidelines or making policies for travels, as far as malaria epidemics is concerned.     

Nodal distances for rooted phylogenetic trees

Dissimilarity measures for (possibly weighted) phylogenetic trees based on the comparison of their vectors of path lengths between pairs of taxa, have been present in the systematics literature since the early seventies. For rooted phylogenetic trees, however, these vectors can only separate non-weighted binary trees, and therefore these dissimilarity measures are metrics only on this class of rooted phylogenetic trees. In this paper we overcome this problem, by splitting in a suitable way each path length between two taxa into two lengths. We prove that the resulting splitted path lengths matrices single out arbitrary rooted phylogenetic trees with nested taxa and arcs weighted in the set of positive real numbers. This allows the definition of metrics on this general class of rooted phylogenetic trees by comparing these matrices through metrics in spaces M_n(\mathbb R){\mathcal{M}_n(\mathbb {R})} of real-valued n × n matrices. We conclude this paper by establishing some basic facts about the metrics for non-weighted phylogenetic trees defined in this way using L ^p metrics on M_n(\mathbb R){\mathcal{M}_n(\mathbb {R})}, with ${p \in \mathbb {R}_{ >0 }}${p \in \mathbb {R}_{ >0 }}.     

A graph-theoretic method for the basic reproduction number in continuous time epidemiological models

In epidemiological models of infectious diseases the basic reproduction number is used as a threshold parameter to determine the threshold between disease extinction and outbreak. A graph-theoretic form of Gaussian elimination using digraph reduction is derived and an algorithm given for calculating the basic reproduction number in continuous time epidemiological models. Examples illustrate how this method can be applied to compartmental models of infectious diseases modelled by a system of ordinary differential equations. We also show with these examples how lower bounds for can be obtained from the digraphs in the reduction process.     

The Impact of the Allee Effect in Dispersal and Patch-Occupancy Age on the Dynamics of Metapopulations

In this paper, we introduce a Levins-type metapopulation model with empty and occupied patches, and dispersing population. We structure the proportion of occupied patches according to the patch-occupancy age. We observe that patch-occupancy age may destabilize the metapopulation, leading to persistent oscillations. We also allow for the dispersal rate to vary with the proportion of empty patches in a monotone or unimodal way. The unimodal dependence leads to multiple non-trivial equilibria and bistability when the reproduction number of the metapopulation < 1 but greater than a lower critical value ^*. We show that the metapopulation will persist independently of its initial status if > 1.     

Analytic Calculation of Finite-Population Reproductive Numbers for Direct- and Vector-Transmitted Diseases with Homogeneous Mixing

The basic reproductive number, $\mathcal {R}_{0}$ , provides a foundation for evaluating how various factors affect the incidence of infectious diseases. Recently, it has been suggested that, particularly for vector-transmitted diseases, $\mathcal {R}_{0}$ should be modified to account for the effects of finite host population within a single disease transmission generation. Here, we use a transmission factor approach to calculate such “finite-population reproductive numbers,” under the assumption of homogeneous mixing, for both vector-borne and directly transmitted diseases. In the case of vector-borne diseases, we estimate finitepopulation reproductive numbers for both host-to-host and vector-to-vector generations, assuming that the vector population is effectively infinite. We find simple, interpretable formulas for all three of these quantities. In the direct case, we find that finite-population reproductive numbers diverge from $\mathcal {R}_{0}$ before $\mathcal {R}_{0}$ reaches half of the population size. In the vector-transmitted case, we find that the host-to-host number diverges at even lower values of $\mathcal {R}_{0}$ , while the vector-to-vector number diverges very little over realistic parameter ranges.     

Interaction-induced electric (hyper)polarizability in the dihydrogen–neon pair: basis set and electron correlation effects

We investigated the interaction (hyper)polarizability of neon–dihydrogen pairs by performing high-level ab initio calculations with atom/molecule-specific, purpose-oriented Gaussian basis sets. We obtained interaction-induced electric properties at the SCF, MP2, and CCSD levels of theory. At the CCSD level, for the T-shaped configuration, around the respective potential minimum of 6.437 a₀, the interaction-induced mean first hyperpolarizability varies for 5?<? R/a₀?<?10 as

$$ \left[{\overline{\beta}}_{\mathrm{int}}(R)\hbox{-} {\overline{\beta}}_{\mathrm{int}}\left({R}_{\mathrm{e}}\right)\right]/{e}^3{a_0}^3{E_{\mathrm{h}}}^{-2}=-0.91\left(R\hbox{-} {R}_{\mathrm{e}}\right)+0.50{\left(R\hbox{-} {R}_{\mathrm{e}}\right)}^2\hbox{--} 0.13{\left(R\hbox{-} {R}_{\mathrm{e}}\right)}^3+0.01{\left(R\hbox{-} {R}_{\mathrm{e}}\right)}^4. $$

Again, at the CCSD level, but for the L-shaped configuration around the respective potential minimum of 6.572 a₀, this property varies for 5?<? R/a₀?<?10 as

$$ \left[{\overline{\beta}}_{\mathrm{int}}(R)\hbox{-} {\overline{\beta}}_{\mathrm{int}}\left({R}_{\mathrm{e}}\right)\right]/{e}^3{a_0}^3{E_{\mathrm{h}}}^{-2}=-1.33\left(R\hbox{-} {R}_{\mathrm{e}}\right)+0.75{\left(R\hbox{-} {R}_{\mathrm{e}}\right)}^2-0.20{\left(R\hbox{-} {R}_{\mathrm{e}}\right)}^3+0.02{\left(R\hbox{-} {R}_{\mathrm{e}}\right)}^4. $$

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Graphical Abstract Interaction-induced mean dipole polarizability (\( \overline{a} \)) for the T-shaped configuration of H₂–Ne calculated at the SCF, MP2, and CCSD levels of theory

     

The Final Size of an Epidemic and Its Relation to the Basic Reproduction Number

We study the final size equation for an epidemic in a subdivided population with general mixing patterns among subgroups. The equation is determined by a matrix with the same spectrum as the next generation matrix and it exhibits a threshold controlled by the common dominant eigenvalue, the basic reproduction number R₀{\mathcal{R}_{0}}: There is a unique positive solution giving the size of the epidemic if and only if R₀{\mathcal{R}_{0}} exceeds unity. When mixing heterogeneities arise only from variation in contact rates and proportionate mixing, the final size of the epidemic in a heterogeneously mixing population is always smaller than that in a homogeneously mixing population with the same basic reproduction number R₀{\mathcal{R}_{0}}. For other mixing patterns, the relation may be reversed.     

Asymptotic Enumeration of RNA Structures with Pseudoknots

In this paper, we present the asymptotic enumeration of RNA structures with pseudoknots. We develop a general framework for the computation of exponential growth rate and the asymptotic expansion for the numbers of k-noncrossing RNA structures. Our results are based on the generating function for the number of k-noncrossing RNA pseudoknot structures, , derived in Bull. Math. Biol. (2008), where k−1 denotes the maximal size of sets of mutually intersecting bonds. We prove a functional equation for the generating function and obtain for k=2 and k=3, the analytic continuation and singular expansions, respectively. It is implicit in our results that for arbitrary k singular expansions exist and via transfer theorems of analytic combinatorics, we obtain asymptotic expression for the coefficients. We explicitly derive the asymptotic expressions for 2- and 3-noncrossing RNA structures. Our main result is the derivation of the formula .