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Foot-and-mouth disease virus remains one of the most important livestock diseases in sub-Saharan Africa and several Southeast Asian countries. Vaccination of livestock has been recognized as an important tool for the control of foot-and-mouth disease virus. However, this intervention strategy has some limitations. Generally, vaccine production is a complex multistep process which involves development, manufacturing, and delivery processes, and through this extensive process, some challenges such as poor vaccine storage often arise. More often, these challenges alter the validity of the vaccination. Foot-and-mouth disease virus epidemic dynamics have been extensively explored, but understanding the role of vaccination validity on virus endemicity is lacking. We present a time-delayed foot-and-mouth disease model that incorporates relevant biological and ecological factors, vaccination effects, and disease carriers. We determined the basic reproduction number and demonstrated that it is an important metric for persistence and extinction of the disease in the community. Numerical illustrations were utilised to support some of the analytical results.

Foot-and-mouth disease (FMD) is one of the most important livestock diseases in sub-Saharan Africa and several Southeast Asian countries. It affects cloven-hoofed animals and is caused by an RNA virus commonly known as foot-and-mouth disease virus (FMDV). In many developed nations, the disease often occurs as epidemic (outbreak followed by extinction), whereas in many developing nations, such as Zimbabwe and Zambia and several Southeast Asian countries including Cambodia, Lao PDR, Malaysia, Myanmar, Thailand, and Vietnam, it is epidemic (persistent disease’s presence) [

Among several other control strategies such as movement restrictions, public education, and culling, vaccination of animals has been recognized as the most important strategy to control the spread of the disease [

Since the 2001 FMD outbreak in UK, several mathematical models have been proposed to study the transmission and control of FMD (see, e.g., [

Despite all these efforts, however, mathematical models that aim to explore the role of vaccine waning in FMD endemic regions are still lacking. Therefore, in this study, we developed a time-delayed FMD model to explore the effects of vaccine waning on the dynamics of the disease. The proposed model incorporates all the relevant biological and ecological factors, vaccination, and disease carriers. We are aware that the role of FMD carriers in disease transmission remains a debatable issue [

We organize this paper as follows. In Section

In this section, we propose and analyze a time-delayed foot-and-mouth disease model incorporating direct and indirect disease transmission pathways, vaccination effects, and virus excreting carriers. We will investigate the qualitative behaviours of the proposed framework through studying the stability of the model steady states.

Let the variables

All model parameters are assumed to be positive and are defined as follows:

FMD is rarely fatal and infected cattle generally clear the systemic infection within 8–15 days [

The following theorem shows that the model proposed in this study is biologically meaningful. Precisely, the theorem demonstrates that all the solutions of the proposed model are nonnegative and bounded for all

There exists a unique solution for the FMD model (

In order to demonstrate that the solutions of model (

Since

Similarly, by the standard comparison theorem [

In this section, we investigate the qualitative behaviours of the delay differential model (

The threshold quantity

Next, we investigate the global stability of the disease-free equilibrium, using a Lyapunov functional. We define a continuous and differentiable function

We note that

Let

For system (

To establish the global stability of the DFE, we consider the following Lyapunov functional:

Taking the derivative of

At disease-free equilibrium, we have the identity

Since the arithmetic mean is greater than or equal to the geometric mean, we have

Furthermore, we define a continuous and differentiable function

From the earlier discussion on initial conditions (

Hence, if

In what follows, we shall investigate the global stability of the endemic equilibrium

Let

If

We consider a Lyapunov functional

From Section

Taking the derivative of

At the endemic equilibrium of model system (

Utilising the identities leads to

Once again, since the arithmetic mean is greater than or equal to the geometric mean, we have

Furthermore, as already illustrated on demonstrating the global stability of the disease-free equilibrium,

Moreover, using the function

Furthermore,

Hence, from the illustrations (

To explore the behaviour of system (

Model parameters and their interpretations.

Symbol | Definition | Baseline value | Units | Source |
---|---|---|---|---|

Rate of progression from latent to infectious | 0.25 | Reference [ | ||

Environmental pathogen decay rate | 0.07 | Reference [ | ||

Indirect disease transmission rate | ||||

Direct disease transmission rate | ||||

Proportion of infectious animals which progress to carrier population | 0.5 | Dimensionless | Reference [ | |

Modification factor | Varied | Dimensionless | ||

Proportion of vaccinated animals | 0.01 | Dimensionless | ||

Average infectious period of clinically infected animals | 7 | Days | Reference [ | |

Recovery rate of FMD carriers | Varied | Days | ||

Pathogen shedding rate | Pathogen | Reference [ | ||

Natural mortality rate | 0.001 | Reference [ |

Simulation results in Figures

Numerical results of system (

Numerical results of system (

Numerical results of system (

Graphical results in Figures

Numerical results of system (

Numerical results of system (

Numerical results of system (

Foot-and-mouth disease, a highly contagious, acute viral disease of cloven-hoofed domestic and wild ruminants including cattle, buffalo, swine, goats, and sheep, remains a major challenge in many developing nations in the sub-Sahara Africa, such as Zimbabwe and Zambia. Vaccination is regarded as the most effective means of foot-and-mouth disease mitigation. Nevertheless, this important intervention strategy has some limitations. Generally, vaccine production is a complex multistep process which involves development, manufacturing, and delivery processes, and through this extensive process, challenges such as poor vaccine storage often arise. Such challenges are known to affect the validity period of the vaccines. In this work, we proposed and analyzed a time-delayed foot-and-mouth disease model that incorporates vaccine waning and foot-and-mouth disease carriers. We are aware that the role of FMD carriers in disease transmission remains a debatable issue [

This work demonstrated that understanding the role of vaccine waning and FMD carriers is an important topic that requires multidimensional research. Nevertheless, this work is not exhaustive; it can be extended by adding diffusion terms to the proposed model so as to represent the movement and dispersal of the animals and the pathogen and by adding convection terms to represent the migration of animals.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.