Transmission Dynamics of Peste des petit ruminant (PPR) in sheep and goats: A Mathematical Modelling Approach

Transmission Dynamics of Peste des petit ruminant (PPR) in sheep and goats: A Mathematical Modelling Approach

 *Bashir S., Muhammad A., and Garba I. D.
Department of Mathematics, Federal University Dutsinma, Katsina State, Nigeria.
*Corresponding author
DOI: https://doi.org/10.51584/IJRIAS.2023.8723
Received: 29 June 2023; Accepted: 11 July 2023; Published: 12 August 2023

Abstract: The study is mainly concerned with mathematical modelling of peste des petit ruminant (PPR) disease using deterministic approach. A system of differential equations was formulated. Disease free and epidemic equilibria were calculated and used Jacobian Matrix to carried out stability analysis of the model. We then perform numerical simulations using Euler’s method. Sensitivity analysis with basic reproduction number were finally conducted to identify the most important parameters in the model. It was finally recommended that animal suffering from peste des petit ruminant (PPR) diseases should be immediately quarantined so as to reduce the contact rate between the infected and the susceptible and other items that have been in contact with the sick animals must be disinfected with common disinfectants.

Keyword: small ruminants, mathematical model, deterministic approach, sensitivity analysis.

I. Introduction

Livestock are very important for both the subsistence and economic development of the African continent. They provide a flow of essential food products throughout the year. In some countries, like Sudan, they are a major source of government revenue and export earnings. They also sustain the employment and income of millions of people in rural areas. Contribute to energy and manure for crop production and are the only food and cash security available to many Africans.
In many African countries, small ruminants animal (sheep and goats) constitute an important proportion of the meat supply.Peste des Petits Ruminants (PPR) is an infectious disease affecting goats and sheep (Michael, et al., 2017).Researches show that a large number of fatalities may be resulted in a herd due to quick spread of PPR disease(Mbyuzi,et al., 2014).

For identifying strategies to mitigate the spread of PPR disease propagation in ruminant herds, a mathematical model is found very vital for understanding how the disease spreads (Schloeder & Jacobs, 2010). The need for a model to address uncertainty in parameter values arises due to lack of firsthand observations on how PPR spreads through a herd. This model allows us to perform Sensitivity Analysis (SA) on environment and disease parameters for which we do not have empirical data.
A system of differential equations has benn used in modelling problems in the field of electrical circuits, mechanics, vibrations, chemical reactions, kinetic and population growth (Muhammad, Bashir and Mustapha, 2021). Several modelling techniques and methods were used to model infectious diseases. Ordinary Differential Equation (ODE) modeling and stochastic lattice modeling are the two common techniques (Li, et al., 1999). In deterministic (ODE) SI, SIS, or SIR model, the number of individuals in each infection-related class is calculated through a set of ODEs that stand for average transition rates between the classes (Boccara and Cheong, 1992). Magal and Ruan (2014) design traditional SEIR models in which they ignore demographic changes, such as births and natural deaths. The performance of different models can be compared to select the best model that fits a given data set (Chinenye and Bashir, 2021).

In this paper, we describe a PSIR epidemic mathematical model that has compartments; Pre-susceptible, Susceptible, Infectious and Recovered. We prefer this compartmental model over others because it generalise some of the models such as SI SIS and SIR as it takes care of the P class which is left in those models. It does not also make things too complicated as in the models with more and more compartments. For many infections, the new born animals are not born into the susceptible compartment but are immune to the disease for the first four months of life due to protection from maternal antibodies. This new detail can be shown by including a P (pre-susceptible) class, for maternally derived immunity at the beginning of the model.