Mathematical analysis and application of a cholera transmission model with waning vaccine-induced immunity

Considering the effect of stochasticity including white noise and colored noise, this paper aims to study a hybrid stochastic cholera epidemic model with waning vaccine‐induced immunity and nonlinear telegraph perturbations. First, we derive a critical value ℛ0C related to the basic reproduction number ℛ0 of the deterministic model. The key aim of this paper is to generalize the θ‐stochastic criterion method proposed by the recent work (Han et al. in Chaos Solit Fract 140:110238, 2020) to eliminate nonlinear telegraph perturbations. Next, via constructing several θ‐stochastic Lyapunov functions and using the generalized method, we further prove that the stochastic model have a unique ergodic stationary distribution under ℛ0C>1. Results show that the prevention and control of cholera epidemic depend on low transmission rate and small telegraph perturbations. Finally, the corresponding numerical simulations are performed to illustrate our analytical results and a practical application on the Somalia cholera outbreak is shown at the end of this paper.

“…In their study, Bai et al 17 defined the basic reproduction number of system () as

ℛ_{0} = \frac{A false(μ + q ϕ false)}{μ false(μ + ϕ false) false(μ + d + γ false)} ((), β_{h} + \frac{β_{H} ξ}{m_{H} δ_{H}} + \frac{β_{L} ξ}{m_{L} δ_{L}})

. They proved that system () has the following dynamical properties: If

ℛ_{0} < 1

, the unique disease‐free equilibrium

E_{0} = false(S^{0}, V^{0}, I^{0}, B_{H}^{0}, B_{L}^{0} false) = ((), \frac{A}{μ + ϕ}, \frac{A ϕ}{μ false(μ + ϕ false)}, 0,0, 0)

is globally asymptotically stable (GAS), which implies cholera extinction in a long term. If

ℛ_{0} > 1

, the unique endemic equilibrium

E^{*} = false(S^{*}, V^{*}, I^{*}, B_{H}^{*}, B_{L}^{*} false)

is GAS, which means that cholera will be persistent in population. …”

Section: Introductionmentioning

confidence: 91%

“…In particular,

ℛ_{0}^{C}

will coincide with the basic reproduction number

ℛ_{0}

Section: Ergodic Property and Stationary Distributionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

({, \begin{matrix} array \frac{d S (t)}{d t} = A - (β_{h} I + \frac{β_{H} B_{H}}{m_{H} + B_{H}} + \frac{β_{L} B_{L}}{m_{L} + B_{L}}) S - (μ + ϕ) S, \\ array \frac{d V (t)}{d t} = ϕ S - q (β_{h} I + \frac{β_{H} B_{H}}{m_{H} + B_{H}} + \frac{β_{L} B_{L}}{m_{L} + B_{L}}) V - μ V, \\ array \frac{d I (t)}{d t} = (β_{h} I + \frac{β_{H} B_{H}}{m_{H} + B_{H}} + \frac{β_{L} B_{L}}{m_{L} + B_{L}}) (S + q V) - (μ + d + γ) I, \\ array \frac{d R (t)}{d t} = γ I - μ R, \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Ergodic stationary distribution and practical application of a hybrid stochastic cholera transmission model with waning vaccine‐induced immunity under nonlinear regime switching

Zhou

Jiang

Han

et al. 2021

Dynamical analysis and optimal control of an age‐structured epidemic model with asymptomatic infection and multiple transmission pathways

Kang,

Nie

2024

The diversity of transmission modes and the heterogeneity of populations in the transmission of infectious diseases are issues that have to be faced in the current disease protection. In this paper, an infectious disease model incorporating age structure and horizontal and environmental spread, along with asymptomatic infection, is proposed to describe diversification of disease transmission routes and population heterogeneity. The expression of the basic reproduction number is derived by a linear approximation method, and it is concluded that if , the disease‐free steady state is globally asymptotically stable; if , the model has a unique endemic steady state which is locally asymptotically stable under certain conditions. Further, the uniform persistence of disease is proved using the persistence theory of the infinite‐dimensional dynamical system when . In additional, the issue of optimal control problem according to this model is investigated, and the representation of optimal control with respect to state variables and adjoint variables is obtained which also imply the existence and uniqueness of optimal control. Finally, numerical simulations are carried out to interpret the theoretical results and to discuss the impact of age and control measures in disease transmission.

Stationary distribution and probability density function of a stochastic waterborne pathogen model with saturated direct and indirect transmissions

Liu

2023

We investigate the dynamical behavior of deterministic and stochastic waterborne pathogen models with saturated incidence rates. In particular, the indirect transmission via person–water–person contact is half‐saturated and the direct transmission by person–person contact takes the saturated form. Possible equilibrium points of the model are investigated, and their stability criterion is discussed. Basic reproduction number of the model is obtained through the next‐generation matrix method. It has been shown that the disease‐free equilibrium is locally stable when and unstable for . Furthermore, a stochastic Lyapunov method is adopted to establish sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of the proposed stochastic model, which reveals that the infection will persist if . We also obtain an exact expression of the probability density function near the quasi‐endemic equilibrium of stochastic system. Finally, numerical simulations are carried out to support our analytical findings. Results suggest that saturation constants in both direct and indirect transmissions play a positive role in controlling disease. Our results may provide some new insights for the elimination of waterborne disease.