Polymorphism in sexual versus non-sexual disease transmission

Pathogens causing sexually transmitted diseases (STDs) often consist of related strains that cause non-sexually transmitted, or ''ordinary infectious'', diseases (OIDs). We use differential equation models of single populations to derive conditions under which a genetic variant with one (e.g. sexual) transmission mode can invade and successfully displace a genetic variant with a different (e.g. non-sexual) transmission mode. Invasion by an STD is easier if the equilibrium population size in the presence of an OID is smaller; conversely an OID can invade more easily if the equilibrium size of the population with the STD is larger. Invasion of an STD does not depend on the degree of sterility caused by the infection, but does depend on the added mortality caused by a resident OID. In contrast, the ability of an OID to invade a population at equilibrium with an STD decreases as the degree of sterility caused by the STD increases. When equilibrium population sizes for a population infected with an STD are above the point at which non-sexual contacts exceed sexual contacts (the sexual–social crossover point) and when equilibrium population sizes for an OID are below this point, there can be a stable genetic polymorphism for transmission mode. This is most likely when the STD is mildly sterilizing, and the OID causes low or intermediate levels of added mortality. Because we assume the strains are competitively equivalent and there are no heterogeneities associated with the transmission process, the polymorphism is maintained by density-dependent selection brought about by pathogen effects on population size.     

Determining Important Parameters in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical Model

We perform sensitivity analyses on a mathematical model of malaria transmission to determine the relative importance of model parameters to disease transmission and prevalence. We compile two sets of baseline parameter values: one for areas of high transmission and one for low transmission. We compute sensitivity indices of the reproductive number (which measures initial disease transmission) and the endemic equilibrium point (which measures disease prevalence) to the parameters at the baseline values. We find that in areas of low transmission, the reproductive number and the equilibrium proportion of infectious humans are most sensitive to the mosquito biting rate. In areas of high transmission, the reproductive number is again most sensitive to the mosquito biting rate, but the equilibrium proportion of infectious humans is most sensitive to the human recovery rate. This suggests strategies that target the mosquito biting rate (such as the use of insecticide-treated bed nets and indoor residual spraying) and those that target the human recovery rate (such as the prompt diagnosis and treatment of infectious individuals) can be successful in controlling malaria.     

A mathematical model for the spread of Strepotococcus pneumoniae with transmission dependent on serotype

We examine a mathematical model for the transmission of Streptococcus Pneumoniae amongst young children when the carriage transmission coefficient depends on the serotype. Carriage means pneumococcal colonization. There are two sequence types (STs) spreading in a population each of which can be expressed as one of two serotypes. We derive the differential equation model for the carriage spread and perform an equilibrium and global stability analysis on it. A key parameter is the effective reproduction number R (e). For R (e) ≤ 1, there is only the carriage-free equilibrium (CFE) and the carriage will die out whatever be the starting values. For R (e) > 1, unless the effective reproduction numbers of the two STs are equal, in addition to the CFE there are two carriage equilibria, one for each ST. If the ST with the largest effective reproduction number is initially present, then in the long-term the carriage will tend to the corresponding equilibrium.     

The effects of human movement on the persistence of vector-borne diseases

With the recent resurgence of vector-borne diseases due to urbanization and development there is an urgent need to understand the dynamics of vector-borne diseases in rapidly changing urban environments. For example, many empirical studies have produced the disturbing finding that diseases continue to persist in modern city centers with zero or low rates of transmission. We develop spatial models of vector-borne disease dynamics on a network of patches to examine how the movement of humans in heterogeneous environments affects transmission. We show that the movement of humans between patches is sufficient to maintain disease persistence in patches with zero transmission. We construct two classes of models using different approaches: (i) Lagrangian models that mimic human commuting behavior and (ii) Eulerian models that mimic human migration. We determine the basic reproduction number R₀ for both modeling approaches. We show that for both approaches that if the disease-free equilibrium is stable (R₀<1) then it is globally stable and if the disease-free equilibrium is unstable (R₀>1) then there exists a unique positive (endemic) equilibrium that is globally stable among positive solutions. Finally, we prove in general that Lagrangian and Eulerian modeling approaches are not equivalent. The modeling approaches presented provide a framework to explore spatial vector-borne disease dynamics and control in heterogeneous environments. As an example, we consider two patches in which the disease dies out in both patches when there is no movement between them. Numerical simulations demonstrate that the disease becomes endemic in both patches when humans move between the two patches.     

An epidemiological model for direct and indirect transmission of typhoid fever

The modified SIS epidemiological model considers the usual direct transmission (short cycle) and indirect transmission (long cycle) of typhoid fever. Thresholds are determined, and the equilibrium points are shown to be globally stable. Local stability of the equilibrium points is shown in the corresponding model with vaccines. After estimating parameters using current statistical data for typhoid fever in Chile, computer simulations are used to obtain the numerical behavior of this disease and to estimate the effect of several control policies.     

Dynamical behavior of a hepatitis B virus transmission model with vaccination

Hepatitis B virus (HBV) infection is a globally health problem. In 2005, the WHO Western Pacific Regional Office set a goal of reducing chronic HBV infection rate to less than 2% among children five years of age by 2012, as an interim milestone towards the final goal of less than 1%. Many countries made some plans (such as free HBV vaccination program for all neonates in China now) to control the transmission HBV. We develop a model to explore the impact of vaccination and other controlling measures of HBV infection. The model has simple dynamical behavior which has a globally asymptotically stable disease-free equilibrium when the basic reproduction number R₀≤1, and a globally asymptotically stable endemic equilibrium when R₀>1. Numerical simulation results show that the vaccination is a very effective measure to control the infection and they also give some useful comments on controlling the transmission of HBV.     

Presumptive horizontal symbiont transmission in the fungus-growing termite Macrotermes natalensis

All colonies of the fungus-growing termite Macrotermes natalensis studied so far are associated with a single genetically variable lineage of Termitomyces symbionts. Such limited genetic variation of symbionts and the absence of sexual fruiting bodies (mushrooms) on M. natalensis mounds would be compatible with clonal vertical transmission, as is known to occur in Macrotermes bellicosus. We investigated this hypothesis by analysing DNA sequence polymorphisms as codominant SNP markers of four single-copy gene fragments of Termitomyces isolates from 31 colonies of M. natalensis. A signature of free recombination was found, indicative of frequent sexual horizontal transmission. First, all 31 strains had unique multilocus genotypes. Second, SNP markers (n = 55) were largely in Hardy-Weinberg equilibrium (90.9%) and almost all possible pairs of SNPs between genetically unlinked loci were in linkage equilibrium (96.7%). Finally, extensive intragenic recombination was found, especially in the EF1alpha fragment. Substantial genetic variation and a freely recombining population structure can only be explained by frequent horizontal and sexual transmission of Termitomyces. The apparent variation in symbiont transmission mode among Macrotermes species implies that vertical symbiont transmission can evolve rapidly. The unexpected finding of horizontal transmission makes the apparent absence of Termitomyces mushrooms on M. natalensis mounds puzzling. To our knowledge, this is the first detailed study of the genetic population structure of a single lineage of Termitomyces.     

Effect of pathogen-resistant vectors on the transmission dynamics of a vector-borne disease

A model is introduced for the transmission dynamics of a vector-borne disease with two vector strains, one wild and one pathogen-resistant; resistance comes at the cost of reduced reproductive fitness. The model, which assumes that vector reproduction can lead to the transmission or loss of resistance (reversion), is analyzed in a particular case with specified forms for the birth and force of infection functions. The vector component can have, in the absence of disease, a coexistence equilibrium where both strains survive. In the case where reversion is possible, this coexistence equilibrium is globally asymptotically stable when it exists. This equilibrium is still present in the full vector-host system, leading to a reduction of the associated reproduction number, thereby making elimination of the disease more feasible. When reversion is not possible, there can exist an additional equilibrium with only resistant vectors.     

Modeling and Analyzing the Transmission Dynamics of HBV Epidemic in Xinjiang,China

Hepatitis B is an infectious disease caused by the hepatitis B virus (HBV) which affects livers. In this paper, we formulate a hepatitis B model to study the transmission dynamics of hepatitis B in Xinjiang, China. The epidemic model involves an exponential birth rate and vertical transmission. For a better understanding of HBV transmission dynamics, we analyze the dynamic behavior of the model. The modified reproductive number σ is obtained. When σ < 1, the disease-free equilibrium is locally asymptotically stable, when σ > 1, the disease-free equilibrium is unstable and the disease is uniformly persistent. In the simulation, parameters are chosen to fit public data in Xinjiang. The simulation indicates that the cumulated HBV infection number in Xinjiang will attain about 600,000 cases unless there are stronger or more effective control measures by the end of 2017. Sensitive analysis results show that enhancing the vaccination rate for newborns in Xinjiang is very effective to stop the transmission of HBV. Hence, we recommend that all infants in Xinjiang receive the hepatitis B vaccine as soon as possible after birth.     

Models of multifactorial inheritance: II, the covariance structure for a scalar phenotype under selective assortative mating and sex-dependent symmetric parental-transmission

This study concentrates on a scalar trait for a two-sex population entailing symmetric sex-dependent parent-offspring transmission, a nonlinear (selective) assortative mating pattern, and random perturbation factors. A number of qualitative and comparative properties of the equilibrium covariance structure are set forth. In particular, it is shown that the phenotype variance generally increases under attenuated assortative mating tendencies and for an increased level of parent-offspring transmission. Moreover, with more contrasting sex-dependent transmission the sibling covariances diverge from the parent-offspring covariance. Also, shared residual-environmental covariance enhances the sibling covariances. The absolute product moment of the deviation of a child phenotype from the separate parental phenotypes exhibits a minimum for a mixed assortative random mating scheme rather than under extreme assortative mating or pure random mating. We establish that the equilibrium variance of a married individual is reduced compared to the variance of the population at large. Interpretations and implications are discussed.     

Malaria Drug Resistance: The Impact of Human Movement and Spatial Heterogeneity

Human habitat connectivity, movement rates, and spatial heterogeneity have tremendous impact on malaria transmission. In this paper, a deterministic system of differential equations for malaria transmission incorporating human movements and the development of drug resistance malaria in an \(n\) patch system is presented. The disease-free equilibrium of the model is globally asymptotically stable when the associated reproduction number is less than unity. For a two patch case, the boundary equilibria (drug sensitive-only and drug resistance-only boundary equilibria) when there is no movement between the patches are shown to be locally asymptotically stable when they exist; the co-existence equilibrium is locally asymptotically stable whenever the reproduction number for the drug sensitive malaria is greater than the reproduction number for the resistance malaria. Furthermore, numerical simulations of the connected two patch model (when there is movement between the patches) suggest that co-existence or competitive exclusion of the two strains can occur when the respective reproduction numbers of the two strains exceed unity. With slow movement (or low migration) between the patches, the drug sensitive strain dominates the drug resistance strain. However, with fast movement (or high migration) between the patches, the drug resistance strain dominates the drug sensitive strain.     

Static behavioral effects on gonorrhea transmission dynamics in a MSM population

An SIS/SAS model of gonorrhea transmission in a population of highly active men-having-sex-with-men (MSM) is presented in this paper to study the impact of safe behavior on the dynamics of gonorrhea prevalence. Safe behaviors may fall into two categories—prevention and self-awareness. Prevention will be modeled via consistent condom use and self-awareness via STD testing frequency. Stability conditions for the disease free equilibrium and endemic equilibrium are determined along with a complete analysis of global dynamics. The control reproductive number is used as a means for measuring the effect of changes to model parameters on the prevalence of the disease. We also find that appropriate intervention would be in the form of a multifaceted approach at overall risk reduction rather than tackling one specific control individually.     

Allocation to sexual versus nonsexual disease transmission

Many diseases have both sexual and nonsexual transmission routes, and closely related diseases often differ in their degree of sexual transmission. We investigate the evolution of transmission mode as a function of host social and mating structure using a model in which disease transmission is explicitly dependent on the numbers of sexual and nonsexual contacts (which are themselves a function of population density) and per-contact infection probabilities. Most generally, and in the absence of trade-offs between the degree of sexual transmission and effects on host fecundity and mortality, nonsexual transmission is favored above the social-sexual crossover point (the host density at which the number of nonsexual contacts exceeds the number of sexual contacts), while sexual transmission is favored below this point. When changes in allocation to the two transmission modes are accompanied by changes in mortality or fecundity, both mixed and pure transmission strategies can be favored. If invading genotypes differ substantially from resident genotypes, genetic polymorphism in transmission mode is possible. The evolutionary outcomes are predictable from a knowledge of the equilibrium population sizes in relation to the social-sexual crossover point. Our results also show that predictions about dynamic outcomes, based on rates of invasion for single pathogens into healthy populations, do not adequately describe the resulting disease prevalence nor predict the subsequent evolutionary dynamics; once invasion of a pathogen has occurred, the conditions for spread of a second pathogen are themselves altered. If the host is considered as a single resource, our results show that two pathogens may coexist on a single resource if they use that resource differentially and have differential feedbacks on resource abundance; such resource feedback effects may be present in other biological systems.     

Coupled population dynamics of endosymbionts within and between hosts

Many insect species are infected with maternally transmitted endosymbionts, the most widely documented being Wolbachia . The rate of spread and equilibrium of prevalence of these infections depend on two parameters – maternal transmission fidelity and relative fitness of infected cytoplasmic lineages. Both transmission fidelity and the phenotypic effect of endosymbionts often increase with endosymbiont density within hosts. Thus, the dynamics of infection prevalence in host populations depends on processes affecting within-host density of endosymbionts. In theory, the equilibrium prevalence of infection by male-killing endosymbionts is much more sensitive to changes in transmission fidelity and relative fitness than is that of endosymbionts that cause cytoplasmic incompatibility. In natural populations, male-killers exhibit much greater temporal and spatial variation in the prevalence of infection than do endosymbionts that cause cytoplasmic incompatibility. Thus, the population dynamics of endosymbiont infections, especially those that cause male-killing, is likely to be governed by environmental and genetic variables that affect within-host density of these infections.     

Global stability of an SIR epidemic model with time delays

An SIR disease transmission model is formulated under the assumption that the force of infection at the present time depends on the number of infectives at the past. It is shown that a disease free equilibrium point is globally stable if no endemic equilibrium point exists. Further the endemic point (if it exists) is globally stable with respect to the whole state space except the neighborhood of the disease free state.Research partly supported by the Ministry of Education, Science and Culture, Japan, Grant 05640256     

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.     

Analysis of a Dengue Model with Vertical Transmission and Application to the 2014 Dengue Outbreak in Guangdong Province,China

There is evidence showing that vertical transmission of dengue virus exists in Aedes mosquitoes. In this paper, we propose a deterministic dengue model with vertical transmission in mosquitoes by including aquatic mosquitoes (eggs, larvae and pupae), adult mosquitoes (susceptible, exposed and infectious) and human hosts (susceptible, exposed, infectious and recovered). We first analyze the existence and stability of disease-free equilibria, calculate the basic reproduction number and discuss the existence of the disease-endemic equilibrium. Then, we study the impact of vertical transmission of the virus in mosquitoes on the spread dynamics of dengue. We also use the model to simulate the reported infected human data from the 2014 dengue outbreak in Guangdong Province, China, carry out sensitivity analysis of the basic reproduction number in terms of the model parameters, and seek for effective control measures for the transmission of dengue virus.     

Spatial patterns in a discrete-time SIS patch model

How do spatial heterogeneity, habitat connectivity, and different movement rates among subpopulations combine to influence the observed spatial patterns of an infectious disease? To find out, we formulated and analyzed a discrete-time SIS patch model. Patch differences in local disease transmission and recovery rates characterize whether patches are low-risk or high-risk, and these differences collectively determine whether the spatial domain, or habitat, is low-risk or high-risk. In low-risk habitats, the disease persists only when the mobility of infected individuals lies below some threshold value, but for high-risk habitats, the disease always persists. When the disease does persist, then there exists an endemic equilibrium (EE) which is unique and positive everywhere. This EE tends to a spatially inhomogeneous disease-free equilibrium (DFE) as the mobility of susceptible individuals tends to zero. The limiting DFE is nonempty on all low-risk patches and it is empty on at least one high-risk patch. Sufficient conditions for the limiting DFE to be empty on other high-risk patches are given in terms of disease transmission and recovery rates, habitat connectivity, and the infected movement rate. These conditions are also illustrated using numerical examples.     

Backward bifurcations in dengue transmission dynamics

A deterministic model for the transmission dynamics of a strain of dengue disease, which allows transmission by exposed humans and mosquitoes, is developed and rigorously analysed. The model, consisting of seven mutually-exclusive compartments representing the human and vector dynamics, has a locally-asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number(R(0)) is less than unity. Further, the model exhibits the phenomenon of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium. The epidemiological consequence of this phenomenon is that the classical epidemiological requirement of making R(0) less than unity is no longer sufficient, although necessary, for effectively controlling the spread of dengue in a community. The model is extended to incorporate an imperfect vaccine against the strain of dengue. Using the theory of centre manifold, the extended model is also shown to undergo backward bifurcation. In both the original and the extended models, it is shown, using Lyapunov function theory and LaSalle Invariance Principle, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. In other words, in addition to establishing the presence of backward bifurcation in models of dengue transmission, this study shows that the use of standard incidence in modelling dengue disease causes the backward bifurcation phenomenon of dengue disease.     

Costs,benefits, and loss of vertically transmitted symbionts affect host population dynamics

The costs and benefits of symbiotic interactions may vary with host and symbiont ontogeny. Effects of symbionts at different stages of host development or on different host demographic rates do not contribute equally to fitness. Although rarely applied, a population dynamics approach that integrates over the host life cycle is therefore necessary for capturing the net costs or benefits and, thus, the mutualistic or parasitic nature of symbioses. Using the native, disturbance‐specialist grass Agrostis hyemalis, we asked how a symbiotic endophyte affected the population dynamics of its host and how imperfect vertical transmission influenced symbiont frequency in a late successional environment. A size‐structured integral projection model (IPM) parameterized with experimental field data showed that greater rates of individual growth and reproduction for endophyte‐symbiotic (E+) hosts outweighed their lower rates of survival, leading to a net positive effect of symbiosis on equilibrium plant population growth (slower rate of extinction). Given that populations under going successional transitions are unlikely to be at an equilibrium size structure, we also conducted transient analysis that showed an initial short‐term cost to endophyte symbiosis. We used a megamatrix approach to link E? and E+ IPMs via imperfect vertical transmission and found that this parameter strongly influenced the frequency of symbiosis via complex interactions with host demographic rates. Overall, our population dynamics approach improves the ability to characterize the outcome of symbiotic interactions, and results suggest that particular attention should be paid to interactions between the rate of vertical transmission and host demography.