A Mathematical Model of the Transmission Dynamics of Bovine Schistosomiasis with Contaminated Environment.

Schistosomiasis, a vector-borne chronically debilitating infectious disease, is a serious public health concern for humans and animals in the affected tropical and sub-tropical regions. We formulate and theoretically analyze a deterministic mathematical model with snail and bovine hosts. The basic reproduction number [Formula: see text] is computed and used to investigate the local stability of the model's steady states. Global stability of the endemic equilibrium is carried out by constructing a suitable Lyapunov function. Sensitivity analysis shows that the basic reproduction number is most sensitive to the model parameters related to the contaminated environment, namely: shedding rate of cercariae by snails, cercariae to miracidia survival probability, snails-miracidia effective contact rate and natural death rate of miracidia and cercariae. Numerical results show that when no intervention measures are implemented, there is an increase of the infected classes, and a rapid decline of the number of susceptible and exposed bovines and snails. Effects of the variation of some of the key sensitive model parameters on the schistosomiasis dynamics as well as on the initial disease transmission threshold parameter [Formula: see text] are graphically depicted.

A Human-Bovine Schistosomiasis Mathematical Model with Treatment and Mollusciciding.

To mitigate the spread of schistosomiasis, a deterministic human-bovine mathematical model of its transmission dynamics accounting for contaminated water reservoirs, including treatment of bovines and humans and mollusciciding is formulated and theoretically analyzed. The disease-free equilibrium is locally and globally asymptotically stable whenever the basic reproduction number [Formula: see text], while global stability of the endemic equilibrium is investigated by constructing a suitable Lyapunov function. To support the analytical results, parameter values from published literature are used for numerical simulations and where applicable, uncertainty analysis on the non-dimensional system parameters is performed using the Latin Hypercube Sampling and Partial Rank Correlation Coefficient techniques. Sensitivity analysis to determine the relative importance of model parameters to disease transmission shows that the environment-related parameters namely, [Formula: see text] (snails shedding rate of cercariae), [Formula: see text] (probability that cercariae shed by snails survive), c (fraction of the contaminated environment sprayed by molluscicides) and [Formula: see text] (mortality rate of cercariae) are the most significant to mitigate the spread of schistosomiasis. Mollusciciding, which directly impacts the contaminated environment as a single control strategy is more effective compared to treatment. However, concurrently applying mollusciciding and treatment will yield a better outcome.

Mathematical analysis of a lymphatic filariasis model with quarantine and treatment.

BACKGROUND: Lymphatic filariasis is a globally neglected tropical parasitic disease which affects individuals of all ages and leads to an altered lymphatic system and abnormal enlargement of body parts. METHODS: A mathematical model of lymphatic filariaris with intervention strategies is developed and analyzed. Control of infections is analyzed within the model through medical treatment of infected-acute individuals and quarantine of infected-chronic individuals. RESULTS: We derive the effective reproduction number, [Formula: see text] and its interpretation/investigation suggests that treatment contributes to a reduction in lymphatic filariasis cases faster than quarantine. However, this reduction is greater when the two intervention approaches are applied concurrently. CONCLUSIONS: Numerical simulations are carried out to monitor the dynamics of the filariasis model sub-populations for various parameter values of the associated reproduction threshold. Lastly, sensitivity analysis on key parameters that drive the disease dynamics is performed in order to identify their relative importance on the disease transmission.

The Role of Hyalomma Truncatum on the Dynamics of Rift Valley Fever: Insights from a Mathematical Epidemic Model.

To date, our knowledge of Rift Valley fever (RVF) disease spread and maintenance is still limited, as flooding, humid weather and presence of biting insects such as mosquitoes, have not completely explained RVF outbreaks. We propose a model that includes livestock, mosquitoes and ticks compartments structured according to their questing and feeding behaviour in order to study the possible role of ticks on the dynamics of RVF. To quantify disease transmission at the initial stage of the epidemic, we derive an explicit formula of the basic reproductive number, [Formula: see text]. Using the concept of Metzler matrix, we state necessary conditions for global asymptotic stability of the disease-free equilibrium. Results suggest that although host-ticks interactions may serve as disease reservoirs or disease amplifiers, the Aedes reproductive number should be kept under unity if disease post-epizootics activities are to be controlled. Results of both local and global sensitivity analysis of selected model parameters indicate that [Formula: see text] is more sensitive to the ticks attachment and detachment rates, probability of transmission from ticks to host and from host to ticks, length of infection in livestock and ticks death rate. Furthermore, when comparing the mean value of [Formula: see text] with that from previous studies which did not include ticks we found that our [Formula: see text] is very much larger resulting in an increase in the exponential phase of an outbreak. These findings suggest that if ticks are capable of transmitting the virus, they may be contributing to disease outbreaks and endemicity.

Predicting Rift Valley Fever Inter-epidemic Activities and Outbreak Patterns: Insights from a Stochastic Host-Vector Model.

Rift Valley fever (RVF) outbreaks are recurrent, occurring at irregular intervals of up to 15 years at least in East Africa. Between outbreaks disease inter-epidemic activities exist and occur at low levels and are maintained by female Aedes mcintoshi mosquitoes which transmit the virus to their eggs leading to disease persistence during unfavourable seasons. Here we formulate and analyse a full stochastic host-vector model with two routes of transmission: vertical and horizontal. By applying branching process theory we establish novel relationships between the basic reproduction number, R0, vertical transmission and the invasion and extinction probabilities. Optimum climatic conditions and presence of mosquitoes have not fully explained the irregular oscillatory behaviour of RVF outbreaks. Using our model without seasonality and applying van Kampen system-size expansion techniques, we provide an analytical expression for the spectrum of stochastic fluctuations, revealing how outbreaks multi-year periodicity varies with the vertical transmission. Our theory predicts complex fluctuations with a dominant period of 1 to 10 years which essentially depends on the efficiency of vertical transmission. Our predictions are then compared to temporal patterns of disease outbreaks in Tanzania, Kenya and South Africa. Our analyses show that interaction between nonlinearity, stochasticity and vertical transmission provides a simple but plausible explanation for the irregular oscillatory nature of RVF outbreaks. Therefore, we argue that while rainfall might be the major determinant for the onset and switch-off of an outbreak, the occurrence of a particular outbreak is also a result of a build up phenomena that is correlated to vertical transmission efficiency.

Tolerance and nature of residual refraction in symmetric power space as principal lens powers and meridians change.

Unacceptable principal powers in well-centred lenses may require a toric over-refraction which differs in nature from the one where correct powers have misplaced meridians. This paper calculates residual (over) refractions and their natures. The magnitude of the power of the over-refraction serves as a general, reliable, real scalar criterion for acceptance or tolerance of lenses whose surface relative curvatures change or whose meridians are rotated and cause powers to differ. Principal powers and meridians of lenses are analogous to eigenvalues and eigenvectors of symmetric matrices, which facilitates the calculation of powers and their residuals. Geometric paths in symmetric power space link intended refractive correction and these carefully chosen, undue refractive corrections. Principal meridians alone vary along an arc of a circle centred at the origin and corresponding powers vary autonomously along select diameters of that circle in symmetric power space. Depending on the path of the power change, residual lenses different from their prescription in principal powers and meridians are pure cross-cylindrical or spherocylindrical in nature. The location of residual power in symmetric dioptric power space and its optical cross-representation characterize the lens that must be added to the compensation to attain the power in the prescription.

Double diffusive magnetohydrodynamic (MHD) mixed convective slip flow along a radiating moving vertical flat plate with convective boundary condition.

In this study combined heat and mass transfer by mixed convective flow along a moving vertical flat plate with hydrodynamic slip and thermal convective boundary condition is investigated. Using similarity variables, the governing nonlinear partial differential equations are converted into a system of coupled nonlinear ordinary differential equations. The transformed equations are then solved using a semi-numerical/analytical method called the differential transform method and results are compared with numerical results. Close agreement is found between the present method and the numerical method. Effects of the controlling parameters, including convective heat transfer, magnetic field, buoyancy ratio, hydrodynamic slip, mixed convective, Prandtl number and Schmidt number are investigated on the dimensionless velocity, temperature and concentration profiles. In addition effects of different parameters on the skin friction factor, [Formula: see text], local Nusselt number, [Formula: see text], and local Sherwood number [Formula: see text] are shown and explained through tables.

Stability, bifurcation and chaos analysis of vector-borne disease model with application to Rift Valley fever.

This paper investigates a RVF epidemic model by qualitative analysis and numerical simulations. Qualitative analysis have been used to explore the stability dynamics of the equilibrium points while visualization techniques such as bifurcation diagrams, Poincaré maps, maxima return maps and largest Lyapunov exponents are numerically computed to confirm further complexity of these dynamics induced by the seasonal forcing on the mosquitoes oviposition rates. The obtained results show that ordinary differential equation models with external forcing can have rich dynamic behaviour, ranging from bifurcation to strange attractors which may explain the observed fluctuations found in RVF empiric outbreak data, as well as the non deterministic nature of RVF inter-epidemic activities. Furthermore, the coexistence of the endemic equilibrium is subjected to existence of certain number of infected Aedes mosquitoes, suggesting that Aedes have potential to initiate RVF epidemics through transovarial transmission and to sustain low levels of the disease during post epidemic periods. Therefore we argue that locations that may serve as RVF virus reservoirs should be eliminated or kept under control to prevent multi-periodic outbreaks and consequent chains of infections. The epidemiological significance of this study is: (1) low levels of birth rate (in both Aedes and Culex) can trigger unpredictable outbreaks; (2) Aedes mosquitoes are more likely capable of inducing unpredictable behaviour compared to the Culex; (3) higher oviposition rates on mosquitoes do not in general imply manifestation of irregular behaviour on the dynamics of the disease. Finally, our model with external seasonal forcing on vector oviposition rates is able to mimic the linear increase in livestock seroprevalence during inter-epidemic period showing a constant exposure and presence of active transmission foci. This suggests that RVF outbreaks partly build upon RVF inter-epidemic activities. Therefore, active RVF surveillance in livestock is recommended.

Paraxial ocular measurements and entries in spectral and modal matrices: analogy and application.

Lensometers and keratometers yield powers along perpendicular meridians even if the principal meridians of the lens and the cornea are oblique. From each such instrument, multiple raw data represented on optical crosses require conversion to determine elementary statistics. Calculations for research decisions need to be authentic. Principles common to meridians generalize formulaic methods for oblique meridians. Like a lens or a cornea, matrix latent quantities are represented on a matrix cross. Our problem is to determine the matrix whose cross represents quantities on the optical cross. All measurements on an optical cross that include corneal and lens powers and oblique meridians can be considered. Once determined, a portfolio of matrix calculations applies and is justified for ophthalmic calculation. Matrices can be unique and, like a cornea before it is measured, contain latent observations. Asymmetric power component matrices quantify a deviation of a corneal surface from smoothness and toricity. Entries may identify those measurements causing irregular astigmatism that may stem from surgical or other external intervention. Irregular astigmatism is detected primarily from significant measurements in the paraxial range. Measurements are assimilated with matrix factors in a holistic way in order to support choices with calculations and statistics.

Bounds and intervals around nonzero cylinder powers in symmetric dioptric power space.

We seek to analyze the geometry and explain how bounds and intervals of nonzero purely cylindrical powers are obtained and applied in symmetric dioptric power space and envisaged in the clinic. The principal powers at zero and at the focus at the cylinder power of a lens are subject to the same uncertainty when measured. Accompanying these uncertainties is an error in axis position. Error cells are constructed for typical cylinder axes and an associated power. The geometry contains an elegant clinical determination for cross-cylinder compensation of astigmatism in terms of calculation friendly quantities. The extreme positions in the error cells define bounds for the cross-cylinder powers and their meridians. When clinical powers in a chosen error cell are transposed, the new powers are within a different cell. This ambiguous cell pair maps to a single cell in an antistigmatic plane around cross-cylinder powers.

Isomorphism and possible invariance of error cells under spherocylindrical transposition.

PURPOSE: The purpose of this article is to investigate the invariance and isomorphism of error cells with cylinder power measured positively and negatively in the cylinder-sphere plane. Cells with cylinder power measured positively and negatively are mapped to symmetric power space to see whether isomorphism and the invariance are conserved under the mapping and spherocylindrical transposition. METHOD: Principal powers of refraction are measured and are rounded off to multiples of 0.25 D and 1 degrees or 5 degrees , say, for the principal meridians. Geometric regions in the cylinder-sphere plane with positive and negative cylinder represent uncertainty surrounding the powers and are respectively transformed to a plane containing the axis of scalar powers in symmetric power space. RESULTS: Cells from principal powers become error cells surrounding readings of cylinder and sphere as well as error regions about astigmatic powers. These are presented in the plane for scalar powers and semipowers of positive and negative cylinders. In symmetric power space, planes containing the axis of scalar powers are distinguished from one another by the cylinder axis of the lens power. CONCLUSION: Although error cells in clinical measure are not invariant under spherocylindrical transposition, cells represented in positive cylinder form have the same shape as cells for which the power is expressed in negative cylinder form. Error cells in symmetric power space about powers in negative cylinders can be rotated about the axis of scalar powers to coincide perfectly with cells about powers in positive cylinder form for near spherical and astigmatic powers. The error regions in symmetric power space do not depend on the spherocylindrical form in which the original measurements are made and their isomorphism is conserved in the mapping.