As quest for solution to addressing social problems increased by way of applied mathematics, a notable field called Mathematical Sociology has emerged. Very recently, a good number of scholars have investigated a wide range of social phenomenon using mathematical and computational models; the outcome resulted into formulation of various theories about rumour and other social science related issues by both applied mathematicians and social scientists [1]. Rumour is a generally circulated story, report or statement without facts to confirm its truth; it is like a virus, once it is transmitted to an individual, an outbreak of events happen in a short period of time. People who diffuse rumours may have different purposes, such as slandering others, increasing awareness, diverting attention, and so on [2]. It is thus thought that spreading a rumour is like a contagious disease that does not need more than a few infected individuals to infect the entire population [3]. If a rumour starts in a population, sooner or later everyone will know it.

More often than not the concept of mathematical epidemiology is being employed for addressing sociological problems, most especially, the issues patterning to crime, corruption and rumour. Wang and Wood [4] adopted this approach when attempting to model viral meme propagation. So also did Zhao et al. [5] for exploring the interplay of authorities, information, rumour dissemination and the evolution of emergency, aiming to give more insights into the effectiveness of strategies for rumour management, hence the need to explore more about it. The significant of studying the dynamics of rumours spread and meme ideas lies in the understanding of ways to change perception, to persuade others, to generate predictions regarding social issues, to predict what the public may find interesting in future, and to influence decision making and public opinion. Thus, in this article, we aimed to find a qualitative behaviour of a simple rumour model as well as the dynamics of rumour spread with moral etiquette class.

Even though it appears in the time past that the use of epidemiological modelling in addressing rumour spreading has yielded good results, yet reviewing the fundamental assumptions concerning spreading of rumour, in the light of present-day reality, will proffer better results. That is why we seek to modify a model proposed by Al-Amoudi et al. [3] by bringing into cognisance an essential but seemly unaccounted state variable - moral etiquette class. Particularly we noted that, under normal circumstances, not all individuals who are aware of a rumour that will keep spreading it before becoming stiflers; this is so perhaps due to religious belief, psychosocial status or moral etiquette. That is the reason why we seek to equally extend the Daley and Kendall (D-K) as reported in Cintron-Arias [6] by adding moral etiquette class, especially within the African context. But unlike D-K model, we first consider a simple rumour model and later build up a deterministic model of a system of nonlinear ordinary differential equations. Therefore, just like [7], the rumour spreading problem we considered is more similar to the “hearsay” which is inconvenient or not allowed to be spread openly, but can only be transmitted privately through human to human transmission.

The initial condition $U(t_0),S(t_0),V(t_0),M(t_0)=0.80{,}^{ }0.02{,}^{ }0.08{,}^{ }0.10$ was used for the result computation, and the parameters estimated value for the simulation of Figure 2 and Figure 3 are $\theta =0.8,$ ${\sigma }_1=0.05,$ ${\sigma }_2=0.02,$ $\mu =0.01,$ $\kappa =0.01,$ $\gamma =0.5,$ and $\mathsf{\Lambda }=0.05$ in that order. We considered only two principal classes that seem to matter most in the dynamics of rumour spread - that is the spreaders’ class and newly introduced compartment called the moral etiquette class. The results of the model simulation could be seen on Figure 2 and Figure 3 respectively.

Figure 1 points out to the fact that rumour spread is somewhat similar to a transmission dynamics of an infectious disease, once it starts in a population, unless or otherwise controlled, it will get to everyone. The plots further show the speediness of how fast or slow a rumour circulates a population depending on the rate $\lambda ,$ that is rumour spread intrinsic rate as well as the initial condition$y(t_0)$, which is the fraction of individuals who are aware of the rumour at the early stage.

On the other hand, the dynamics of the spread of rumour become controllable by Figure 2 plots; it asserts that, the spreaders could be wipe away if $R_0<1$ is maintained. Since the plots on Figure 2 indicate a good possibility of controlling rumour provided $\beta ,$ the probability of spread is reduced, it implies therefore that the lower the value of $\beta$ the easier it is tract rumour spread. On Figure 3 the plots suggest that, as rumour is being controlled, moral etiquette class will tend to a steady state depending on the rate of contact between uninformed class and the spreader’ class. It gives an idea about the moral etiquette class that they will be affected in the course of rumour spread by the rate contacts between uninformed individuals and the spreaders’ class.

Rumour spread behaves similar with a dynamics of a contagious disease; however, the rate of rumour spread depends largely on intrinsic rumour spread rate and the fraction of individuals who are aware of the rumour at the early stage. In addition, the result the modified deterministic model of a system of nonlinear ordinary deferential equations reveals that by lowering the value of $\beta$ (i.e the probability communicating a rumour on average number of contacts per day), the easier it will be to control rumour spread provided $R_0<1$is maintained.