Table 1.
The parameters for the RVF model for high rainfall and moderate temperature (wet season) for model in Table 2 with values, range and references.
Note that all parameter units are days. The parameter α1 is a function of the mosquito’s gonotrophic cycle (the amount of time a mosquito requires to produce eggs) and its preference for livestock blood, while α2 is a function of the ruminant’s exposed surface area, the efforts it takes to prevent mosquito bites (such as swishing its tail), and any vector control interventions in place to kill mosquitoes encountering cows or prevent bites [24].
Fig 1.
Flow diagram of RVF model with both vertical and horizontal transmission.
Susceptible livestock, S2, acquire infection and move to compartment I2 when they are bitten by an Aedes infectious mosquito I1. They then recover with a constant per capita recovery rate to enter the recovered compartment, R2, class. Susceptible mosquito vectors, S1, become infected when they bite infectious livestock and progress to class I1. The solid lines represent the transition between compartments and the dashed lines represent the transmission between different species.
Table 2.
Stochastic model for vector-host disease system.
The parameter m0 = N1/N2 is the ratio mosquitoes to hosts, and is for general forces of infections λ21 and λ12, and α′ = α is for standard forces of infections
and
.
Fig 2.
Realization of the RVF host-vector stochastic model and its deterministic counterpart.
The trajectories of the deterministic counterpart are generated by integrating the mean field Eq (10). The values of the parameters in years are as follows: q1 = 0.2, μ1 = (1/20) ∗ 360, μ2 = 1/8, β12 = 0.194, β21 = 0.128, ϵ2 = (1/4) ∗ 360, α′ = α = 256, m0 = 1.5 and R0 = 1.8809, and their description and sources is given in Table 1.
Fig 3.
Solution of Eq (14) when the product R12 × R21 is greater than unity.
The curves in (a) and (b) are contours in the plane (R12, R21), along which the probabilities of extinction and invasion respectively, after an introduction of a single vector is constant. In (c) and (d) we plot probabilities of extinction and invasion respectively, when varying parameters α1 and α2. The values of the remaining parameters in days are as follows: μ1 = 1/30, μ2 = 0.00046, β12 = 0.676, β21 = 0.28, ϵ2 = 0.25, m0 = 10.
Fig 4.
The curves represent contours in the plane (R12, R21), with varying vertical transmission efficiency, along which the probability of invasion after an introduction of a single vector is constant.
These probabilities are obtained from the solutions of Eq (18).
Fig 5.
(a) and (b) A surface plot for the invasion probability when varying both vertical transmission, q1 and mosquitoes to hosts ratio, m0. (c) Numerical and analytical solution of the extinction probability Eq (18) when varying vertical transmission efficiency. (d) Analytical solutions of the extinction probability Eq (18) when varying vertical transmission efficiency for different values of the ratio female mosquitoes to hosts. The values of the remaining parameters in days are as follows: μ1 = 1/20, μ2 = 1/(8 ∗ 360), β12 = 0.55, β21 = 0.22, ϵ2 = 0.25 and α1 = 0.33, α2 = 19, m0 = 1.5 as baseline values.
Fig 6.
Temporal history of RVF outbreaks in some countries of Sub-Saharan Africa.
In (a) and (b) the circles represent years of outbreaks occurrence in Kenya and South Africa [3, 9] and the prevalence indicated in the figure is not real, it is just for representation only since data on prevalence is not available. In (c) the circles represent the prevalence of disease outbreaks in Tanzania [7].
Fig 7.
Theoretical prediction of the power spectrum density (PSD) (Eq (13)) for fluctutions of the total number of susceptible livestock, infected livestock and infected mosquitoes.
(First Row) The theoretical prediction using the simplified force of infection. The values of the parameters used in years are as follows: q1 = 0.05, μ1 = (1/16) ∗ 360, μ2 = 1/8, β12 = 0.170, β21 = 0.116, ϵ2 = (1/4) ∗ 360, α′ = α = 256 and m0 = 1.5. This gives R0 = 1.0066. (Second Row) Comparison between theoretical predictions of PSD under the simplified and complex versions of the forces of infection. For the complex force of infection the new parameters are α1 = 0.33, α2 = 19, m0 = 9.45 and , and R0 = 1.0074. Note that description and sources of all model parameters are given in Table 1.
Fig 8.
Power Spectra Density (PSD) for the variable I2 (Eq (13)).
a) Effects of vertical transmission efficiency on the PSD. Three-dimensional representation of the PSD when varying R0 and the frequency for q1 = 0.05 and q5 = 0.5 in b) and c) respectively. Model parameter values used are as follows: β12 = 0.170, β21 = 0.116, ϵ2 = (1/4) ∗ 360, α′ = α = 256, μ2 = 1/8, m0 = 1.5, μ1 = (1/16) ∗ 360.