Citation: Pei F, Du Y (2026) Numerical investigation of a reaction-diffusion model used for rumor spreading in a ‘street’. PLoS One 21(2):
e0339059.
https://doi.org/10.1371/journal.pone.0339059
Editor: Pablo Martin Rodriguez, Federal University of Pernambuco: Universidade Federal de Pernambuco, BRAZIL
Received: March 24, 2025; Accepted: December 1, 2025; Published: February 20, 2026
Copyright: © 2026 Pei, Du. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper.
Funding: This work was supported by the Research Project of Huainan Normal University (2023XJZD003, 2022XJYB006, 2023XJYB008 and 2020XJZD011) to Feiyun Pei (FP); the Anhui Philosophy and Social Sciences Research Project (AHSKYQ2023D012) to Yamin Du (YD); the Key Project for Training Young and Middle-aged Teachers in Anhui Province’s Universities (YQZD2023072) to Yamin Du (YD); and the Key Project for Scientific Research Planning in Anhui Province (2024AH053240) to Yamin Du (YD). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
1. Introduction
In the past decades, with the development of information technology, the diffusion of the rumor is much more rapid and wide than the truth [1]. Rumor spreading [2] becomes a very common research topic in today’s society, which plays a significant role in a variety of human affairs. The content of rumors can range from gossip to propaganda and marketing material, and has a significant impact on the financial markets [3,4]. Rumor-like mechanisms form the basis for the phenomena of viral marketing, where companies exploit social networks of their customers on the Internet in order to promote their products via the so-called ‘word-of email’ and ‘word-of-web’ [5]. Several methods, such as complex networks [5–7], are used by researchers to understand the dynamics of the rumor spreading. Besides these methods, Reaction-diffusion equations [8,9], a nonlinear evolution equation varying with time and space, can be analyzed by means of analytical and numerical methods from the theory of partial differential equations and dynamical systems. It has been widely used as models for biology and medicine [10], epidemiology [11] and social sciences [12,13], which could be used to describe the process of rumor spreading.
To simplify the model, we assume the rumor spreads on a one-dimensional region representing a ‘street’. Similar model, named crimo–taxis [14], is proposed by Epstein to deal with urban crime dynamics. In this paper, Reaction-diffusion model is proposed to describe the rumor spread among the human population distributed at ‘street’ position 
at time 
. This nonlinear reaction-diffusion model is described in section 2. In section 3, numerical results are given to perform the evolution of rumor for different personnel distribution. On the basis of numerical calculation, we discuss the effects of spreading rate, spreading decay and diffusion coefficients on the spreading of rumor, which is given in section 4. Finally, the conclusion is given in section 5.
2. Nonlinear reaction-diffusion model
Rumor can be viewed as an ‘infection of the mind’, whose spreading could be expressed as social interactions in human population. The human population is divided into 3 groups [15], i.e., ignorant, spreader and stifler. Ignorant is the person who has not heard the rumor and susceptible to be informed. Spreader is the person who spreads rumor. Stifler knows the rumor, but no longer spreads it, who corresponds to death, isolation or immunity in the real world.
The spreading process is assumed through direct contact between the spreaders and others in the population. When a spreader encounters an ignorant, the ignorant has a certain spreading rate α of becoming a new spreader, while the identity of the spreader unchanged. This process corresponds to the increasing of spreader. When a spreader encounters a spreader or a stifler, they will exchange information which may lead them to become aware of rumor and lose the motivation to spread rumor. It leads to a certain rate β that the spreader may transform into a stifler. This process corresponds to the decay of the spreading process.
In a closed system, we define 
, 
and 
as the normalized number of ignorant, spreader and stifler at normalized ‘street’ position 
(
) at time 
, respectively. The normalization condition is shown as equation 1, which indicates that the total number of people remains constant (unit 1) at any time in this closed system.
(1)Based on Ref. [7] and considering diffusion in one-dimensional space, we incorporate diffusion terms into the original model; the evolution of the three densities is then governed by the following set of coupled differential equations:
(2)
(3)
(4)The above equations state that the density of spreaders increases at a rate proportional to the spreading rate 
. On the other hand, the annihilation mechanism considers that spreaders decay into the stifler class at a rate 
times the sum of the density of spreaders and stiflers, i.e., the density of non-ignorants at time 
. 
, 
and 
are self-diffusion coefficients for ignorant, spreader and stifler, respectively. Here, we do not consider the influence of other density distributions on density diffusion, and the cross-diffusion coefficients are set to 0. Equation 2–4 tell us that the densities of ignorant, spreader and stifler at 
are not only determined by the interaction among themselves, but also influenced by their respective diffusion effects.
3. Numerical simulation
In terms of space, the ‘street’ 
has been divided evenly into 100 segments (yielding 101 nodal points). The ignorants, spreaders and stiflers are placed at these grid points according to a certain distribution. The processes of spreading and decay occur at the grid points, and the process of diffusion occurs between grid points. In terms of time, each iteration is equivalent to one time of spreading and/or decay, corresponding to the pass of time. The central difference scheme for 
at point 
for the n-th iteration is given as equation 5.
(5)where 
, and 
is the uniform spatial step size. The same treatment is applied to 
and 
.
In our simulation, assuming the initial conditions 
, 
and 
. the values of spreading rate 
, decay rate 
, and self-diffusion coefficients (
) are assumed to be constants due to simplification. In this section, we assume 
, 
, 
, and investigates the evolution of rumor for different initial population distribution forms.
3.1 Uniform distribution
Firstly, using the simplest distribution, it is assumed that ignorants (80% of the sample) and spreaders (20% of the sample) are evenly distributed in the 101 divided grid points, shown as equation 6.
(6)Due to the uniform distribution, 
. The values of 
are assumed to be constants for different grids, the set of equation 2–4 is converted into equation 7, which is similar to the equation 2–4 in reference [7].
(7)Fig 1 shows the temporal evolution of number of ignorants (black line), spreaders (blue dashed line) and stiflers (magenta dotted line). The spreaders increase to maximum value of 
at 
, and then reduce to 
at 
, and finally approach 0 slowly and smoothly. Because of rumor spreading, the ignorants decrease to 
at about 
. Due to the decay mechanism, spreaders convert to the stiflers leading to an increase in stiflers and a decrease in spreaders. Stiflers increase to the stable value of about 
at about 
. We found that, until the end, there will still be about 
of ignorants who have not been infected, about 
of stiflers who are immunized to rumors, and near 0 (
) spreaders who remain active after a sufficiently long time 
. During the evolution, the total number of ignorants, spreaders and stiflers is unit 1 (red dash-dot line).
Fig 2 gives the space distribution of ignorants (left), spreaders (middle) and stiflers (right) at 
, 
, 
and 
, respectively. As mentioned before, the diffusion terms of equations (2)–(4) has disappeared for uniform distribution at 
, and the distributions of ignorants, spreaders and stiflers are flat during the evolution. The distribution of 
and 
basically overlap, and the distribution has basically stabilized during this period.
3.2 Gaussian distribution
In the real world, people always gather together, corresponding to a large number of personnel at the center and a small number of personnel at the edge. In this paper, Gaussian distribution is used to describe this characteristic, shown as equation 8.
(8)where 
and 
. We consider the ignorants gather in the center of the ‘street’ by defining 
. Because the area inside 
is 95.45% for Gaussian distribution, we define 
to describe the initial group of the ignorants. As 
decreases, the Gaussian distribution becomes more concentrated. Assuming the initial spreaders concentrated in a narrow area by defining 
.
For the first case, we consider the initial rumors mainly generate in the center of ‘street’ by defining 
. The temporal evolution of total number of ignorants, spreaders and stiflers are given as Fig 3. The trajectory of spreaders number is interesting. Unlike case of uniform distribution (section 3.1), the number of spreaders decreases quick to 
at 
, then increase relatively slowly to a regional maximum value 
at 
, and finally decrease to close to 0 (
) at 
. The ignorants number decrease to a stable value 
at 
, while the stiflers number increase to a stable value 
at 
. Comparing to uniform distribution, it takes more time to reach a stable value, and almost 5.1% more ignorants do not convert to spreaders, and maintain their identity as ignorant individuals throughout the entire evolutionary process.
The space distributions of ignorants (left), spreaders (middle) and stiflers (right) at 
, 
, 
and 
are shown in Fig 4, respectively. The initial Gaussian distributions of ignorants and spreaders are shown as red line 
. The maximum density of ignorants and spreaders located at the position 
, and intense interactions of spreading and decay occur at the 
. it causes a hole of ignorants density and a peak of stiflers at 
. Then, the diffusion of ignorants, spreaders and stiflers are beginning to dominate. The hole of ignorants density begin to recovery, and the peak of stiflers begin to flat.
For the second case, we consider the initial rumors is mainly located at the edge of ‘street’ by defining 
. The temporal evolution of total number of ignorants, spreaders and stiflers are given as Fig 5. The number of spreaders drops rapidly before 
, and decrease slowly to the regional minimum value 
at 
. It seems the rumors have been eliminated. But then, the trends changes, the spreaders increase to regional maximum value 
at 
, as if rumors were reigniting. Finally, the number deceases slowly to 
at 
. At the same time, the ignorants number decrease to a stable value 
, while the stiflers number increase to a stable value 
, which is close to the first case.
The space distributions of ignorants (left), spreaders (middle) and stiflers (right) at 
, 
, 
, 
, 
and 
are shown in Fig 6, respectively. Due to the initial distribution of spreader, the effects of spreading and decay are occurred at the edge of ‘street’ locally. At the edge of the ‘street’, many ignorants change to the spreaders. And then, the original and newly converted spreaders change to the stiflers, which is responsible to the peak of stiflers at 
. If there are no diffusion terms 
, the spreaders will be converted to stiflers locally and eradicated finally. Due to the diffusion terms, the remaining spreaders diffuse to the new place and develop the new spreaders, which explains the increasing of spreaders between 
and 
. At 
, shown as left figure in Fig 6, most ignorants are located on the right side of ‘street’. The density of remaining ignorants, who may be converted to spreaders, start to decrease. It explains the reduce of spreaders since 
. After 
, with the diffusion of spreaders, the right side of ignorants change to the spreaders, and then convert to stiflers. It corresponds to the increasing of stiflers and decreasing of ignorants in the final stage.
4. The effect of the coefficients
For the set of equations 2–4, the coefficients include the spreading rate 
and decay rate 
, which represent the interaction between the ignorants, spreaders and stiflers, while self-diffusion coefficients 
, 
and 
, which represent the diffusion of ignorants, spreaders and stiflers along the ‘street’. In the introduction of the section 3, we found that uniform distribution naturally eliminates the diffusion terms, which are related to self-diffusion coefficients 
, 
and 
. So, we analysis the effect of the spreading rate 
and decay rate 
with uniform distribution at first.
Firstly, we scan the spreading rate 
from 0 to 1.6, while the decay rate 
is constant (0.4), shown as Fig 7. The key results are listed in Table 1. The maximum value of spreaders increases with 
increase. The time, when the spreaders reach its maximum, would increase with the growing of 
for 
, and then decrease with the growing of 
. For a smaller 
, it takes longer time to reach the stable value. It means the course of rumor spreading will last long time. At the same time, fewer ignorants convert to spreaders, and change to the stiflers finally. The number of ignorants is relative higher, while the number of stiflers is relative lower. It means the course of rumor spreading is of low intensity.
Then, we scan the decay rate 
from 0 to 1.6, while the spreading rate 
is constant (0.8), shown as Fig 8. The key results are listed in Table 2. When 
, it means there is no decay effect which convert the spreaders into stiflers. As a result, all ignorants change to the spreaders. There is no ignorants and stiflers in the ‘street’ at last. As 
increases, the time which the spreaders required to reach the maximum value reduce, and the maximum value of spreaders decreases. It means the peak of rumor spreading comes early and the degree is low for large 
. At 
, due to the suppressive effect of 
, the higher value of 
leads to the more ignorants and the less stiflers. It means higher 
protects more ignorants from rumors. Thereby, it reduces the stiflers who have been ‘infected’ by rumors but no longer spread it. These stiflers may be the victims of rumors.
The effects of self-diffusion coefficients 
, 
and 
are given by the case that initial rumors spreaders are mainly located at the edge. For comparison purpose, the spreading rate 
and the decay rate 
. We change the self-diffusion coefficients 
, 
and 
one by one, and observe the difference between their evolutions, shown as Table 3. Case 1, as a reference, corresponds to Fig 5 and 6 in section 3.
Only increasing the self-diffusion coefficient of ignorants 
(case 2 in Table 3), we could find that the key results, including the regional minimum & maximum and numbet of each group at last time, are similar to the case 1. The evolutions of total number for three groups are given in Fig 9 (left), which is almost same to Fig 5 for case 1. The effect of 
on the evolutions of spreaders and stiflers is not very significant.
Case 3 only considers the effect of self-diffusion coefficient 
. Shown as Fig 9 (middle), it directly affects the evolution of spreaders, which causes the spreaders number rapidly ramp down to a regional minimum value at first stage, then ramp up quickly to a regional maximum value, and slowly reduces to approach a fixed value. The regional minimum value and maximum value are higher than the data of case 1, which means an intense spreading for case 3. Compared to case 1, it approaches the fixed value faster, which means the time required to spread the rumor is much shorter. Correspondingly, the process of ignorants becoming stiflers is also faster, while the decrease in remain ignorant individuals is not very significant.
In case 4, we only enhance the mobility of stiflers by increasing 
. Shown as Fig 9 (right), the spreaders reduce quickly at the early stage, Later, it enters a process of slow reduction and then recovery, which is very smooth and lasts for a very long time. This process corresponds to the low intensity spread of rumors, rather than the end of the rumor. Since 
, the growth of spreader has reached a small peak, and then trends towards a fixed value. We found that the remaining ignorants are much larger than other cases.
5. Conclusion
In this paper, a reaction-diffusion model, which consist of three coupled reaction-diffusion equations involving self-diffusion of ignorants, spreaders and stiflers, is proposed to simulate the rumor spreading on a one-dimensional ‘street’. By this model, we have analyzed the dynamics of rumors spreading for population in Uniform distribution form and Gaussian distribution form. Besides temporal evolution of ignorants, spreaders and stiflers, we also give their space distributions along the ‘street’ at some important time-slices. The results of numerical simulation provide an intuitive process of rumor propagation.
Based on the Numerical investigation of reaction-diffusion model, we explore the effect of spreading rate 
, decay rate 
and self-diffusion coefficients 
on the dynamics of rumors spreading. (1) Larger spreading rate 
causes a shorter rumor spreading process and faster propagation process. The number of ignorants who are not ‘infected’ by rumors is smaller finally. (2) larger decay rate 
results in a shorter rumor spreading process and more moderate propagation process. The number of ignorants who are not ‘infected’ by rumors is larger at last. (3) Self-diffusion coefficient of ignorants 
seems to have little impact on the evolutions of spreaders and stiflers. It mainly changes the spatial distribution of ignorants. (4) Self-diffusion coefficient of spreaders 
, corresponding to diffusion of spreaders, makes the propagation process fast. Larger 
leads to more spreaders at each time-slice in the early stage of rumors spread. The remaining ignorants reduce slightly from 7.85% to 6.05% when 
changes from 1.0 × 10−6 to 5.0 × 10−6. (5) Self-diffusion coefficient of stiflers 
causes long-term low-density rumor spread. As 
increases, more ignorants can avoid the influence of rumors and thus maintain their identity as ignorants.
The more interesting work is to make more careful improvements to the model to adapt to the spread of rumors in reality. For example, spreading rate 
and decay rate 
could be modelled as a function of 
to indicate that the interaction rate varies in different environments. Some enlightening results have been found in reference 5–7. In some real situations, the diffusions of ignorants, spreaders and stiflers are not only influenced by their own distribution, but also by other distributions. The cross-diffusion terms can be considered in our future research. Our research aims to have the ability to explain more phenomena in financial markets, information dissemination, communication networks, replication database maintenance, and disease transmission.
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