11footnotetext: Guiyun Li and Ting Wang contributed equivalently to this
work.22footnotetext<EMAIL_ADDRESS>Corresponding author.

# To Simulate the Spread of Infectious Diseases by the Random Matrix

Ting Wang  Gui-Yun Li  Xin-Hui Li  Chi-Chun Zhou  Yuan-Yuan Wang  Li-Juan Li
and Yan-Ting Yang

###### Abstract

The main aim to build models capable of simulating the spreading of infectious
diseases is to control them. And along this way, the key to find the optimal
strategy for disease control is to obtain a large number of simulations of
disease transitions under different scenarios. Therefore, the models that can
simulate the spreading of diseases under scenarios closer to the reality and
are with high efficiency are preferred. In the realistic social networks, the
random contact, including contacts between people in the public places and the
public transits, becomes the important access for the spreading of infectious
diseases. In this paper, a model can efficiently simulate the spreading of
infectious diseases under random contacts is proposed. In this approach, the
random contact between people is characterized by the random matrix with
elements randomly generated and the spread of the diseases is simulated by the
Markov process. We report an interesting property of the proposed model: the
main indicators of the spreading of the diseases such as the death rate are
invariant of the size of the population. Therefore, representative simulations
can be conducted on models consist of small number of populations. The main
advantage of this model is that it can easily simulate the spreading of
diseases under more realistic scenarios and thus is able to give a large
number of simulations needed for the searching of the optimal control
strategy. Based on this work, the reinforcement learning will be introduced to
give the optimal control strategy in the following work.

##  1 Introduction

Infectious diseases, especially malignant infectious diseases that can spread
on a large scale and cause large-scale deaths, seriously threaten the safety
of individual lives and social orders [1], and are major problems in modern
public health events. For example, the Black Death in the mid-14th century
swept across Europe, killing about $1/3$ of the European population at that
time [2]. The SARS, outbreaks at 2003, is estimated to have cost the global
economy more than $50$ billion dollars worth of damages [3]. The 2019 novel
coronavirus pneumonia (COVID-19) has killed more than $6$ million people, the
epidemic is still happening locally, and the death toll will continue to rise.
Moreover, they cause a lot of economic losses making the world economy fall
into a downturn.

Establishing a model capable of simulating the spreading of infectious
diseases is important to predicting the trend of infectious diseases and
providing theoretical guidance for adopting the disease prevention measures.
Along this way, the key to find the optimal strategy for disease control is to
obtain a large number of simulations of disease transitions under different
scenarios closer to the reality. Therefore, we need models that can
efficiently simulate the spreading of diseases under realistic scenarios.

Researches on infectious disease models can be traced back to 1760, when Dutch
physicist Bernoull uses mathematical models to study the effects of vaccinia
vaccines on the spreading of smallpox [4]. In 1906, Hamer used a discrete
model to study the recurrent epidemic of measles [5]. In 1911, Ross used
differential equations to study the transmission of malaria between mosquitoes
and people, proving that when the number of mosquitoes is reduced below a
critical value, the outbreak of malaria can be controlled [6].

Recently, the infectious disease models can be divided into three types. (1)
The methods based on differential equations. In this approach, the dynamic
process of disease transmission is simulated by setting different compartments
to represent people in different disease states [7, 8]. For example, the SIR
model with $S$, $I$, and $R$ the susceptible, infectious and removed
populations is a typical compartment model. The disease transition is
simulated by introducing the ordinary equations, $dS/dt=-\beta SI$,
$dI/dt=\beta SI-\gamma I$, and $dR/dt=\gamma I$, where parameters $\beta$ and
$\gamma$ stand for the infection rate and the recovery rate respectively [9].
In order to simulate the spreading of diseases under more realistic scenarios,
various compartment models are proposed. For examples, the SEIR model [10]
(with $S$, $E$, $I$, and $R$ the susceptible, exposed, infected, and recover
respectively), the SIRS model [10] (with $S$, $I$, $R$, and $S$ the
susceptible, infectious, recovered, and susceptible respectively), and the
model considers asymptomatic carriers [11] are proposed, where more classes or
compartments according to the epidemiological status are considered. Beyond
the deterministic compartment models, the stochastic compartment models are
introduced to simulate the stochastic factors [12, 13, 9, 14] and other types
of differential equations, such as the delay differential equation [15, 10],
nonlinear Volterra integral equations [16], and the fractional differential
equation [14], are introduced into the compartment models. The differential
equation based compartment models are indeed able to simulate various kinds of
infectious diseases [17, 18, 19, 20] and evaluate the effectiveness of the
control measures [21, 22] however, they ignore the individual differences
[14], are unable to account for disease’s true infectious period distribution
[16], and are inflexible to simulate realistic scenarios such as random
contact in the public places.

(2) The methods based on the cellular automata. Unlike the compartment models,
the cellular automata is individual–based–model, where each individual of the
population is represented by a cell of the automata, and the transmission
dynamic is simulated by setting a set of simple rules that encode the
individual behaviour [23]. The model based on cellular automata can simulate
more realistic scenarios. For example, it can simulate the impact of lockdown,
migration and vaccination on COVID-19 dynamics [24], consider social community
with varying sex ratio, age structure, population movement, incubation and
treatment period, immunity [25], and address the population heterogeneity and
distribution in epidemics models [23, 26]. Beyond the conventional cellular
automata models, the probabilistic cellular automata models [27], the models
considers stochastic modeling [28], and the models combine the differential
equation and the cellular automata [29] are proposed. The models based on the
cellular automata considers individual differences and can simulate more
realistic scenarios. However, there are limitations in its definition of cell
morphology and neighbor rules.

(3) The methods based on the complex network. In this approach, the contact
between people based on the real social networks is considered [30, 31] and
thus are able to simulate the epidemic spreading under scenarios closest to
the real-world. In the complex network, nodes represent individuals and the
connection between individuals are characterized by links between nodes [30,
31]. To simulate the epidemic spreading, one, for example, has to solve the
ordinary differential equations on that network [32]. However, to solve the
epidemic models on the underlying topology of complex network is difficult.
For example, the Monte Carlo simulations are useless when study the network
with a large system size due to memory needs [33]. Therefore, unlike the
compartment models and the cellular automata models, the models based on the
complex network focus on the profound impact of the complex properties of
real-world networks on the behavior of equilibrium and nonequilibrium
phenomena [31], or thresholds [34, 35] of epidemic spreading. For example, an
epidemic threshold that is inversely proportional to the largest eigenvalue of
the connectivity matrix is proposed [36] and Immunization and epidemic
threshold of an SIS model in complex networks is investigated [37, 38].
Moreover, various kinds of complex networks are investigated, such as the
correlated complex networks [33] and the scale-free complex networks [32, 38].
The models based on the complex network are indeed the model closest to the
realistic, however, it is difficult to obtain a large simulations on those
model due to the complexity in a given time.

In the realistic social networks, on the one hand, the random contact,
including contacts between people in the public places and the public
transits, becomes the important access for the spreading of infectious
diseases. On the other hand, the situations are complicated. For example,
individuals vary from one to another: those with good physical condition
perhaps become the asymptomatic carriers and those with strong sense of
prevention might become the vaccine recipients or the mask wearers. Therefore,
in order to obtain a large number of simulations of disease spreading
supporting the searching of the optimal strategy of disease control, a model
can simulate the disease spreading under various conditions and quickly return
a simulation result is needed.

In this paper, a model can efficiently simulate the spreading of infectious
diseases under random contacts is proposed. In this approach, the random
contact between people is characterized by the random matrix with elements
randomly generated and the spreading of the diseases is simulated by the
Markov-like process, a process has slightly differences between the Markov
process. We report an interesting property of the proposed model: the main
indicators of the spreading of the diseases such as the death rate are
invariant of the size of the population. Therefore, representative simulations
can be conducted on models consist of small number of populations. The main
advantage of this model is that it can easily simulate the spreading of
diseases under more realistic scenarios and thus is able to give a large
number of simulations needed for the searching of the optimal control
strategy. For example, we simulate over $7$ scenarios where the spreading of
different infectious diseases, characterized by their infections and deaths,
under different control measures are given. This work is the first work in the
series studies aimed at giving the optimal disease control strategy. Based on
this work, the reinforcement learning will be introduced to give the optimal
control strategy in the following work.

The paper is organized as follows: In Sec. 2, we introduce the random matrix
and establish the connection between the random matrix and random contact. In
Sec. 3, the main method is introduced. In Sec. 4, simulations under different
scenarios are conducted. Conclusions and discussions are given in Sec. 5.

##  2 The main method

In this section, we introduce the main method, including the approach of
simulate the random contact by the random matrix and the rule a disease spread
on the random matrix.

###  2.1 The Random Matrix and the Random Contact

In this section, we introduce the approach of simulate the random contact by
the random matrix.

Random matrix is also called probability transition matrix in mathematics,
which is a basic tool for characterizing Markov processes [39], It can be used
to describe complex systems with random interactions. For example, physical
system in non-equilibrium state [40], complex communication system [41], and
complex social network of infectious disease population.

In this approach, we consider a group of individuals indexed from $1$ to $N$.
The social connection of these individuals are characterized by a random
matrix $A$ with order $N$. Here are two basic assumptions of the proposed
model. (1) The element $A_{i,j}$ at $ith$ row and $jth$ column represents the
connection between individuals $i$ and $j$. For example, if there is
connections, including the fixed connection and random contact, between
individuals $i$ and $j$, the matrix element at position $i$ and $j$ will be
assigned with a non-negative value. Otherwise the element will be $0$. The
larger the value, the closer the contact. For example, a random contact with a
stranger at a bus is characterized by $0.01$ and a contact with a family
member is $0.5$. In order to simulate both the fixed connection, say the
contact between families and colleague, and random contact, say the contact
between strangers at public places, we assign randomly generated values, under
certain constraint, to the elements of the matrix. As a result, a symmetric
random matrix is generated. (2) The element at $ith$ row and $jth$ column also
represents the exposure of spreading the diseases from the $ith$ individual to
the $jth$ individual and vice versa. Therefore, the diagonal element represent
the recover coefficient of the diseases for each individuals. For the sake of
clarity, we give an example where $100$ individuals are considered, as shown
in Fig. (1). In this example, each individual averagely has connection with
$4$ other individuals. The number of contacts for each individual is rounded
from a value sampled from a normal distribution with mean $4$ and variance
$4$. The matrix elements are either $0$ or sampled from a normal distribution
with mean $0.4$ and variance $0.2$. For the sake of convenience, the matrix
element are named the contact coefficient. To summarize, the average number of
contacts and the average exposure coefficient are two constraint applied on
the generation of the random matrix revealing the social activity.

Figure 1: An example of simulate the fixed connection and random contact
between $100$ individuals by the random matrix: the matrix and the
corresponding network. Individual $2$ has closer connection with individual
$100$ than $1$. The recover coefficient for each individuals is $0.9$.

###  2.2 The Rule of Diseases Spreading

In this section, we introduce the rule where a disease spread on the random
matrix.

After the construction of the random matrix characterizing the social
connection between a given group of individuals, we set up the following rule
to simulate the spreading of the diseases. (1) A vector $x$, named the
criterion vector, of size $N\times 1$ is given characterizing the disease
infection status of each individual. The $ith$ component of the criterion
vector evaluates the exposure to the disease. (2) If the exposure to the
disease is below a certain value, the pathogenic threshold, the individual is
health. Otherwise, he becomes infected. If the exposure beyond the given
value, the lethal threshold, the individual will die. If the exposure return
to a value below the pathogenic threshold, he is recovered. The diseases are
mainly characterized by the recover coefficient and the pathogenic and lethal
thresholds. (3) The spreading of the disease is simulated by considering the
random matrix $A$ as the transition probability matrix and the criterion
vector $x$ as the vector of state in a Markov process. In a transition
probability matrix, the summation of a row is $1$ making the total probability
conserved. However, here, the elements of the matrix is not regarded as a
probability and the summation of the row of the random matrix is not $1$.
Therefore, we name it as the Markov-like process. The spreading is described
by the following equation,

$x_{n+1}=Ax_{n},$ (2.1)

where $x_{n+1}$ is the criterion vector of next round. For the sake of
clarity, we give an example where $10$ individuals are considered, as shown in
Fig. (2).

Figure 2: An example illustrating the spreading of the disease. The pathogenic
threshold is $0.70$ and the lethal threshold is $3.00$.

To summarize, in this approach, the diseases are characterized by the recover
coefficient and the pathogenic and lethal thresholds. And the social
connections are described by the random matrix generated under the constraint
characterized by the average number of contacts and the average exposure
coefficient. By applying the rules, we can simulate the spreading of different
diseases under various realistic situations, as show in Fig. (3).

Figure 3: An example illustrating the spreading of the disease. The pathogenic
threshold is $0.70$ and the lethal threshold is $3.00$.

##  3 The Model’s Parameter and Some Discussions

In this section, before conducting various simulations by the proposed models,
we give explanations to the model’s parameters since the estimation of model’s
parameters is always important. Beside, we report an interesting property of
the proposed model and give a simple discussion on the outbreak threshold of
the epidemic spreading.

###  3.1 The Model’s Parameter

Usually, the model’s parameters should be estimated and verified based on the
real data first, and then the model could be used to simulate the spreading.
In this work, the main aim is to propose a model that can give a large amount
of simulations in a given time supporting the searching of the optimal disease
control strategy. Therefore, we focus on testing the flexibility of the
proposed model to simulate various realistic situations by setting the model’s
parameters manually. The estimating of the parameter will be considered in
further researches.

In this section, we consider three types of diseases with low, medium, and
high pathogenic and lethal thresholds. These three types of diseases represent
the general and malignant infectious diseases. Given that the social
connections between individuals are described by the random matrix generated
under the constraint characterized by the average number of contacts and the
average exposure coefficient and the matrix will be altered during the
simulation according to different control measures, we set the average number
of contacts $4$ and the average exposure coefficient $0.13$. An overview of
the setting of model’s parameter is given in Table. (1)

Table 1: The setting of model’s parameter. Av-cont-coe, Av-cont-num, Path-thre, Leth-thre, and Rec-coe stand for the average exposure coefficient, the average number of contacts, the pathogenic threshold, the lethal threshold, and the recover coefficient respectively. Diseases | Av-exp-coe | Av-cont-num | Path-thre | Leth-thre | Rec-coe
---|---|---|---|---|---
Malignant | $0.13$ | $4.0$ | $0.4$ | $4.0$ | $0.9$
Moderate | $0.13$ | $4.0$ | $0.7$ | $6.0$ | $0.7$
General | $0.13$ | $4.0$ | $0.9$ | $8.0$ | $0.5$

Here is a diagram, Fig. (4), showing the spreading of these three types of
diseases. The curve trends and the infected and death populations show the
differences.

###  3.2 An Interesting Property: the Model is Invariant of the Size of the
Model

Usually, to simulate the spreading of diseases closer to the realistic
situations, one needs to simulate a system with large number of individuals.
For example, a small town consists of $100$ thousand populations. In this
section, we report an interesting property of the proposed model: the model is
invariant of the size. That is, the main indicators of disease transmission,
such as the death rate and the infection rate of simulations, remain stable
and are invariant of the population sizes, as shown in Table. (2).

To investigate this interesting property, we also repeat the experiments on
community networks, a kind of networks describing the social connection of
individuals in realistic world. The random matrix describing the community
networks can be easily constructed. For example, for the network proposed in
this work, the elements of the matrix are sampled from a Gauss distribution
under given constraint, see section 2.1. For the community networks, between
individuals in a community, the random matrix is generated in the same way,
however, between individuals in different communities, only selected
individuals have connections, say $5$. Here is the results of another sets of
experiments, as shown in Table. (3).

Figure 4: A diagram showing the spreading of these three types of diseases. We
consider $100$ individuals with average number of contacts $4$ and the average
exposure coefficient $0.13$.

Tables. (2) and (3) shows that the main indicators of disease transmission is
invariant of the size of the population. It means that, without loss of
generality, the following simulations can be conducted in a model of small
number of population, say $100$, which saves computer resources to a desirable
extent. Moreover, Fig. (5) shows the difference between simulations of the
spreading of diseases on the network proposed in this work and the community
networks. Although, the construction of those two random matrices are
different, to construct random matrices describing those networks is easy. It
shows in Fig. (5) that the spreading on the community network can be divided
into different stages while that on the networks provided in this work has
only one stage. It is because the spreading of diseases on different
communities is not synchronous.

Table 2: The main indicators of disease transmission in different simulations with different population size. The average exposure coefficients is $0.13$, the average number of contacts is $4$, the pathogenic threshold is $0.7$, the lethal threshold is $6.0$, and the recover coefficient is $0.7$. N | Infection rate | Death rate
---|---|---
$100$ | $0.910$ | $0.260$
$500$ | $0.894$ | $0.260$
$1000$ | $0.895$ | $0.243$
$2000$ | $0.900$ | $0.249$
Table 3: The main indicators of disease transmission in different simulations on community networks. Con-of-com stands for the number of connections between communities. N | Con-of-com | Infection rate | Death rate
---|---|---|---
$100$ | $1$ | $0.660$ | $0.280$
$500$ | $1$ | $0.664$ | $0.286$
$1000$ | $1$ | $0.672$ | $0.314$
$2000$ | $1$ | $0.665$ | $0.297$
$2000$ | $2$ | $0.699$ | $0.305$
$2000$ | $5$ | $0.686$ | $0.301$
$2000$ | $10$ | $0.747$ | $0.309$
Figure 5: A comparision between the simulations of the spreading of diseases
on the network proposed in this work and the community networks.

In the following simulations, we consider $100$ individuals. Here we only
report this interesting property and further research on this phenomenon will
be carried out in later works.

###  3.3 A Simple Discussion on Whether the Infectious Disease Will Break Our
or Not

In this section, we investigate the outbreak threshold of the epidemic
spreading. Instead of a rigorous discussion, we give a simple discussion on
whether the infectious disease will outbreaks or not. Further investigation on
the equilibrium and nonequilibrium phenomena and thresholds of epidemic
spreading will be given in later work.

To explore the outbreak threshold of the epidemic spreading, in this approach,
we take the advantages of the proposed method and obtain a large number of
simulations where the spreading of different diseases on various situations
are considered. Here, the outbreak of the epidemic spreading is defined as
over $70\%$ populations are infected. It shows in Fig. (6) that there is a
clear boundary between the outbreak and no outbreak cases. For example, the
diseases with low pathogenic threshold won’t break out regardless of the
average number of contacts of the social network.

Figure 6: A diagram showing the result of simulations where the spreading of
different diseases on various situations are considered. The orange triangle
represents that the epidemic dose not break out and the blue dot represents
breakout. The recover coefficient is $0.7$ and the average exposure rate is
$0.13$.

##  4 The Simulation of the Spreading of Different Diseases under Different
Realistic Scenarios

In this section, we simulation of the spreading of different diseases under
different realistic scenarios. The diseases are mainly characterized by the
pathogenic and lethal thresholds and the realistic scenarios are simulated by
the random matrix. For instance, to simulate the spreading of the diseases
under the scenario where the control measures of isolating the infected is
adopted, the random matrix is altered after each round. The elements
corresponding to the infected individuals are all set to $0$.

###  4.1 Scenarios 1: the Effectiveness of Passive Quarantine

In this section, we consider the effectiveness of passive quarantine. The
quarantine can be simply simulated by setting the matrix elements
corresponding to the individual $0$ if we find this individual need to be in
quarantine. In this simulation only those who go to hospital seeking for a
treatment themselves are considered to be in quarantine. Usually, not all
infected individuals go to hospital seeking for medical treatment and not all
be diagnosed due to wrong diagnosis. Therefore, in this simulation, we set the
probability of being diagnosed $0.3$. That is for those who are infected, only
$30\%$ will be diagnosed and quarantined. It shows in Fig. (7) and Table. (4)
that the quarantine contributes greatly to reduce the infected and death cases
in the spreading of the diseases.

Figure 7: The simulations of spreading under passive quarantine and no actions. The curve trends show the effectiveness of passive quarantine. For example, the infected and death are obviously reduced. Table 4: The main indicators of spreading under passive quarantine and no actions. The infection rate and the death rate (the ratio between death and all population) show the effectiveness of passive quarantine. Diseases | Scenarios | Infected | Death
---|---|---|---
$Malignant$ | $Noquarantine$ | $94$ | $53$
| $Quarantine$ | $77$ | $0$
$Moderate$ | $Noquarantine$ | $91$ | $26$
| $Quarantine$ | $53$ | $0$
$General$ | $Noquarantine$ | $82$ | $9$
| $Quarantine$ | $20$ | $0$

Here are another simulations where the probability of being diagnosed $0.4$,
$0.2$, and $0.1$ respectively, as shown in Fig. (8). The simulations suggest
that the probability of being diagnosed to be an infected is crucial for the
deaths. For example, when the probability is below $0.1$, there are deaths.

Figure 8: The simulations of spreading under passive quarantine with different
diagnose probabilities.

###  4.2 Scenarios 2: the Effectiveness of Active Quarantine

In this section, we consider the effectiveness of active quarantine. Unlike
the passive quarantine, in the active quarantine, we conduct a disease
detection for all individuals and those found to be infected are in
quarantine. The active quarantine considers those who don’t go to hospital for
treatment themselves. Given that the method we use for disease detection is
imperfect, that is, the method can’t find all infected individuals. We
simulate the cases where the detection recall is $99\%$, $90\%$, and $80\%$
respectively. In realistic life, conducting an overall disease detection costs
considerable human and financial costs, therefore, in the simulation, the
overall disease detection is conducted every $5$ and $10$ rounds. Figs. (9)
and (10) show the effectiveness of active quarantine. The simulations of
proposed model show that, the overall disease detection helps little with the
control of spreading of the diseases. Increasing the frequency and accuracy of
the overall disease detection contributs little to the control of spreading of
the diseases too.

Figure 9: The simulations of spreading under active quarantine with different
overall disease detection accuracies. The overall disease detection frequency
is every $5$ round. Figure 10: The simulations of spreading under active
quarantine with different overall disease detection accuracies. The overall
disease detection frequency is every $10$ round.

###  4.3 Scenarios 3: the Effectiveness of the Green Code

In this section, we consider the effectiveness of the green code, a measure
used in China to control the COVID-19. In this control measure, not only those
who are diagnosed are quarantined but also those who have direct contact with
the infected, namely close contacts, are quarantined too. In this simulations,
those who go to hospital seeking for a medical treatment themselves are
possibly be diagnosed as infected. The result of simulations are shown in
Figs. (11) and (12). It shows that the green code measure contributes to the
control of the spreading of the diseases obviously. Especially from the curve
trends in Fig. (12), the infected population is reduced obviously.

Figure 11: The simulations of spreading under the green code measure with
different diagnose probabilities. Figure 12: The simulations of spreading
under the passive quarantine with and without green code measure.

###  4.4 Scenarios 4: the Effectiveness of the Green Code and Overall
Detection

In this section, we consider the effectiveness of the green code and overall
detection. That is, the overall detection is conducted to find the infected
population such as those who never go to hospital. Moreover, the infected and
the close contacts are quarantined.

The result of simulations are shown in Fig. (13). It shows that the green code
measure and overall detection together contribute to the control of the
spreading of the diseases to some degree. For the malignant infectious disease
the control measure is with higher effectiveness. The simulation suggests that
to effectively control the spreading of the disease, to find the infected and
their close contacts and make them quarantined are the keys.

Figure 13: The simulations of spreading under the green code measure with and
without overall detection.

###  4.5 Scenarios 5: Asymptomatic Carriers

In this section, we consider the asymptomatic carriers. The asymptomatic
carriers are individuals who are infected and are contagious but have no
uncomfortable signs and symptoms. Usually, the asymptomatic carriers are hard
to be found. In the following simulations, we consider the spreading of
diseases with asymptomatic carriers under the most strict control measures,
the green code and overall detection. Here, we make the assumption that the
asymptomatic carriers can not be found even in the overall detection.

The result of simulations are shown in Fig. (14). It shows that the
asymptomatic carriers are the main group to break down the strict control
measure, the green code and overall detection. Especially when there are over
$10\%$ of the populations are asymptomatic carriers, there are deaths.

Figure 14: The simulations of spreading with asymptomatic carriers under the
green code and overall detection.

###  4.6 Scenarios 6: Vaccination

In this section, we consider the spreading of diseases with vaccination. The
vaccination is a useful method to control the spreading of diseases. In this
simulation, we make the assumption that the vaccination will increase the
resistance of the pathogens. That is the pathogenic and lethal thresholds stay
the same, but the recover coefficient will the lowered. The lower the recover
coefficient the quicker the infected will recover. Here, after vaccination,
the recover coefficient multiplies $0.7$.

The result of simulations are shown in Fig. (15). It shows that the
vaccination can effectively control the spreading of diseases, especially for
the general infectious diseases. However, to avoid deaths, only vaccination is
not enough. That is, we need control measures such as the quarantine and green
code.

Figure 15: The simulations of spreading with and without vaccination.

###  4.7 Scenarios 7: Mask Wears

In this section, we consider diseases spread by air, and to wear a mask can
efficiently prevent the spreading. Here, we consider two kinds of mask, one is
medical mask and the other is the general mask. In this simulation, the
medical mask can reduce the exposure coefficient by multiplying $0.9$ and the
general mask $0.95$. Without loss of generality, we make assumption that all
individuals wear masks.

The result of simulations are shown in Fig. (16). It shows that wearing a mask
can control the spreading of diseases, especially for the general infectious
diseases the medical mask reduces the deaths to $0$.

Figure 16: The simulations of spreading with medical and general masks.

##  5 Conclusions and Discussions

Establishing a model that can simulate the spread of infectious diseases is
the key to studying the law of infectious disease transmission and predicting
the trend of transmission, and it is of great significance to control the
spread of infectious diseases. To find the optimal control strategy, we need a
large number of simulations to support the searching process. Therefore, one
needs the model that can at one hand simulate the spreading of diseases under
realistic conditions and at the other hand quickly return the result of the
simulations. In the realistic social network, on the one hand, the random
contact, including the contact with strangers at public places, becomes more
and more important access for the spreading of diseases. One the other hand,
the individuals are different form one to the other, and the individual
differences will effect the spreading of diseases. For example, the
asymptomatic carriers will cause a wider spread of the disease.

In this work, a model that can simulate the spreading of diseases under
realistic situations with random contact is proposed. Unlike the conventional
approaches, such as the differential equation based compartment models, the
cellular automata and the complex network based models, the social
connections, in this approach, are described by using the random matrix which
is generated under given constraint, the average contact population and the
average exposure coefficient. The spreading of the disease is simulated by
using the Markov-like process, where the random matrix is considered as a
probability transition matrix with the exposure coefficient (the elements of
the random matrix) the transition probability. The diseases are characterized
by the recover coefficient and the pathogenic and lethal thresholds. Instead
of estimating the model’s parameters according to the real data, we firstly
manually choose the parameters, in order to simulate the spreading of diseases
under various realistic scenarios. We report an interesting property of the
proposed method: the major indicators such as the infection and death rates
are almost invariant of the size of the model.

A set of experiments show that the proposed model can efficiently simulate the
spreading of various diseases under various realistic situations. Therefore,
the proposed model can give a large number of simulations supporting the
searching of optimal diseases control strategy in the following researches. In
the next research, the reinforcement learning method, a method that is good at
finding the optimal strategy, will be introduced. Together with the proposed
model, we give a further investigation to the optimal control strategy of the
infectious diseases.

##  6 Acknowledgments

We are very indebted to Prof. Wu-Sheng Dai for his enlightenment and
encouragement. We are very indebted to Prof. Guan-Wen Fang for his
encouragement. This work is supported by National Natural Science Funds of
China (Grant No. 62106033), Yunnan Youth Basic Research Projects
(202001AU070020), and Doctoral Programs of Dali University (KYBS201910).
YuanYuan Wang acknowledges the support from Yunnan Fundamental Research
Projects (grant NO. 2019FD099)

##  7 Conflict of interest statement

We declare that we have no financial and personal relationships with other
people or organizations that can inappropriately influence our work, there is
no professional or other personal interest of any nature or kind in any
product, service and/or company that could be construed as influencing the
position presented in, or the review of, the present manuscript.

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