# The Effect of Seasonal Weather Variation on the Dynamics of the Plague Disease

Authors: Rigobert C. Ngeleja; Livingstone S. Luboobi; Yaw Nkansah-Gyekye
Journal: International Journal of Mathematics and Mathematical Sciences
 (2017)

---

## Abstract

Plague is a historic disease which is also known to be the most devastating disease that ever occurred in human history, caused by gram-negative bacteria known asYersinia pestis. The disease is mostly affected by variations of weather conditions as it disturbs the normal behavior of main plague disease transmission agents, namely, human beings, rodents, fleas, and pathogens, in the environment. This in turn changes the way they interact with each other and ultimately leads to a periodic transmission of plague disease. In this paper, we formulate a periodic epidemic model system by incorporating seasonal transmission rate in order to study the effect of seasonal weather variation on the dynamics of plague disease. We compute the basic reproduction number of a proposed model. We then use numerical simulation to illustrate the effect of different weather dependent parameters on the basic reproduction number. We are able to deduce that infection rate, progression rates from primary forms of plague disease to more severe forms of plague disease, and the infectious flea abundance affect, to a large extent, the number of bubonic, septicemic, and pneumonic plague infective agents. We recommend that it is more reasonable to consider these factors that have been shown to have a significant effect on RT for effective control strategies.


---

## Body

## 1. Introduction

Plague is the ancient disease caused by the bacteriumYersinia pestis and has had significant effects on human societies throughout the history [1]. Dynamics of plague disease are the result of complex interactions between human beings, rodent population, flea population, and pathogens in the environment. Seasonal variation particularly temperature, humidity, rainfall, and precipitation greatly affects the normal transmission capacity of plague disease by either lowering it or raising it. It affects pathogen in the environment, fleas, rodents, and even human behavior by altering their normal immigration rate, death rate, survival rate, and infectious capability [2].


### 1.1. Seasonality in Flea Development Stages and Behavior

Flea’s survival is greatly affected by temperature and relative humidity [3]. The ectothermic characteristics of fleas make them very sensitive to temperature fluctuations.Xenopsylla cheopis is the primary vector flea forYersinia pestis. It is significantly affected by seasonal weather variation as most of its life stages depend on temperature, humidity, and precipitation. The rate of metamorphosis of this kind of flea from egg to adult is also regulated by temperature.Flea larvae feed on almost any organic debris but mostly they feed on adult excreta which consist of relatively undigested blood [4]. This adult fecal matter when dried falls from the host to serve as food for the larvae. Thus the availability of food (dried flea dirt) for larvae to feed depends on the weather condition particularly temperature and humidity. The larvae develop well in areas where the relative humidity is greater than 75 percent and the temperature is between 21°C and 32°C [5, 6]. At constant temperature fleas become most sensitive to air saturation and are massively killed when the air saturation is insufficiency [7]. Considering the fact that all immature flea stages occur outside the host, development rates of flea increase with temperature until it reaches a critical value which makes flea most vulnerable. High temperature combined with low humidity hinders flea’s survival at immature stages [8].The condition where relative humidity is below50% is unfavorable for flea growth. It is at this condition that the biting rate of flea onto the infected human and rodent or of the infected flea onto the susceptible human and rodent is significantly low. But when the relative humidity is 80% the flea becomes very active and as a result the biting rate and infection increase significantly. Moreover when temperature is above 27.5°C the rapid disappearance of plague bacilli from the flea stomach occurs, resulting in reduced rates of plague disease transmission. This in turn reduces the flea’s efficiency in its ability to transmit the plague bacillus to human beings and rodents [9, 10].When fleas are in rodent burrows, their survival of immature stages is affected by soil moisture that is partly controlled by outside precipitation [11]. As a way of getting rid of detrimental moisture losses and temperature swings, rodents normally shift to start living underground [12]. On the other hand, when they are attached with a high organic load, excessively wet conditions in rodent burrows (e.g., relative humidity 95%) can stimulate the growth of destructive fungi that diminish flea’s larval and egg survival [13].Different studies justify the negative correlation between rainfall and plague epidemics. For example, Cavanaugh and Marshall Jr. [3] reported that, in areas where drains are absent, or where drainage is insufficient as a result of soil composition or impoundments of water, flooding unquestionably causes a drop in the flea population. In areas with improved drainage, such as those with sandy soils, the lessening of the flea population is minimal. Precipitation also influences plague infection for it influences the concentration of rodents, fleas, and humans in the same shelter.


### 1.2. Seasonality in Rodents

The direct effect posed on rodent population due to temperature change is minor. This is due to the fact that rodents are homoeothermic and hence do not respond immediately to changes in ambient temperatures [14]. Temperature indirectly affects the spread of plague in rodent population in different ways as follows: at a low mean temperature of 10°C the bacteria within host (rodent) become very active as a result a large number of infected rodents dying before even the plague bacilli appear in their blood. At this particular temperature rodents also lose the ability to infect other susceptible individuals.Rainfall may pose positive or negative effect on the increase of rodent population depending on its intensity [11]. A season of moderate rainfall may be considered to affect positively the increase of rodent abundance but when the amount of rainfall is extremely heavy it results in a tremendous rodent population decline [15]. When it is moderate and upon a proper timing, rainfall may foster the increase of rodent population [3]. This is due to the fact that rodent’s reproduction period normally follows wet seasons [16–18]. That is to say, the increase of rodent population during wet period is expected to be higher than that during the dry seasons. This clearly concurs with the result in the study by Leirs et al. [19], which narrates that, in Tanzania, rodent population densities show clear association with the annual rainfall and its seasonal distribution. However when rainfall is of high intensity, it causes flooding of rodent burrows. Large number of rodents population dies and the remaining ones normally move from forest to the households where they can protect themselves [3, 8, 20]. In other cases, increased precipitation or drought stalwartly disturbs rodent population dynamics, as it deters food availability.


### 1.3. Seasonality in Pathogens in the Environment

When the bacteria are in lungs, the transmission ofYersinia pestis is possible through various ways: contact transmission, in which one may be infected through physical contact with respiratory particles on the infected surface; airborne transmission, which occurs through inhaling the bacteria causing the disease through successive contact with the nose or mouth of an infected individual; respiratory particles, which occurs through respiratory droplets which is through shedding of respiratory particles (i.e., droplets or aerosols) from an infected human or rodent into the environment [21].Extreme temperatures regularly are ruinous to the survival of pathogens causing plague. The changes in temperature may lead to varying effects on the pathogens in the environment and vectors that live in an environment. When the mean temperature approaches the maximum limit that can be endured by the pathogens, a small increase in temperature may be very dangerous to the pathogen survival. Conversely when pathogens are in the environment characterized by low mean temperature, a small increase in temperature may result in increased development, incubation, and replication of the pathogen in the environment [22, 23].Davis [24] compared the seasonal incidence of plague with usual atmospheric conditions in particular temperature and rainfall. It was depicted that human plague is more frequent in warm moist weather between 15°C and 27°C than in hot dry (over 27°C) or cold weather (under 15°C). Mitscherlich and Marth [25] narrate that the solar exerts a detrimental effect on bacterial aerosol and the decay rate ofYersinia pestis is proportional to the increase of UV light.The reports by Ayyadurai et al. [26] and Mollaret [27] justify the ability of theYersinia pestis to culture the organism from deep within contaminated soil. Eisen et al. [28] were able to show the great potential durability ofYersinia pestis in the soil substrate. The long duration of their survival in the soil supports indirectly the virulence maintenance.Yersinia pestis exhibit a very slow growth at the temperature between 35°C and 37°C but they grow very fast at the temperature 28°C. They die very rapid if exposed to a UV light or temperature exceeding 40°C or when exposed to intensive desiccation [29–31]. Bacteria decrease their sensitivity when the level of humidity drops below 76% [25].When an infected individual coughs or sneezes, thousands of the bacteria are released in air [32]. The released respiratory particles may be large and heavy that they cannot remain suspended in the air. When respiratory particles are large the transmission can only occur when these particles are expelled directly onto another close susceptible individual. In some cases the release of smaller respiratory particles may occur; this is when the airborne transmission is possible. The smaller released particles are easily suspended in the air respired (i.e., passed to the lower respiratory tract) [33].Relative humidity and temperature affect the transmission ofYersinia pestis from one individual to the other. Humidity affects the size of the respiratory particle [34]. When humidity is low the large drops partially evaporate to create smaller, lighter drops that are more likely to remain airborne for extended periods of time [35]. That is to say, when the air is sufficiently dry the large sized particles shrink to a size that favors long-range transport which in turn leads to increased infection.


### 1.4. Seasonality in Human Behavior

Human activities and behavior in plague-infected areas are also to be considered as important determinants of plague transmission to and by humans [42]. When occurrences of plague are due to human intrusions in natural plague areas, it is thus important to consider season variation as a second-order variable that influences disease incidence through human behavior. In Tanzania drought and famine which are the result of lack of rainfall and temperature fluctuation have a great impact on the farmers and pastoralists as they force them to move from one area to another searching for food for themselves and their cattle. These human intrusions from one place to another may lead to the increase of plague disease transmission in rodents, fleas, human population, and pathogens in the environment.


## 1.1. Seasonality in Flea Development Stages and Behavior

Flea’s survival is greatly affected by temperature and relative humidity [3]. The ectothermic characteristics of fleas make them very sensitive to temperature fluctuations.Xenopsylla cheopis is the primary vector flea forYersinia pestis. It is significantly affected by seasonal weather variation as most of its life stages depend on temperature, humidity, and precipitation. The rate of metamorphosis of this kind of flea from egg to adult is also regulated by temperature.Flea larvae feed on almost any organic debris but mostly they feed on adult excreta which consist of relatively undigested blood [4]. This adult fecal matter when dried falls from the host to serve as food for the larvae. Thus the availability of food (dried flea dirt) for larvae to feed depends on the weather condition particularly temperature and humidity. The larvae develop well in areas where the relative humidity is greater than 75 percent and the temperature is between 21°C and 32°C [5, 6]. At constant temperature fleas become most sensitive to air saturation and are massively killed when the air saturation is insufficiency [7]. Considering the fact that all immature flea stages occur outside the host, development rates of flea increase with temperature until it reaches a critical value which makes flea most vulnerable. High temperature combined with low humidity hinders flea’s survival at immature stages [8].The condition where relative humidity is below50% is unfavorable for flea growth. It is at this condition that the biting rate of flea onto the infected human and rodent or of the infected flea onto the susceptible human and rodent is significantly low. But when the relative humidity is 80% the flea becomes very active and as a result the biting rate and infection increase significantly. Moreover when temperature is above 27.5°C the rapid disappearance of plague bacilli from the flea stomach occurs, resulting in reduced rates of plague disease transmission. This in turn reduces the flea’s efficiency in its ability to transmit the plague bacillus to human beings and rodents [9, 10].When fleas are in rodent burrows, their survival of immature stages is affected by soil moisture that is partly controlled by outside precipitation [11]. As a way of getting rid of detrimental moisture losses and temperature swings, rodents normally shift to start living underground [12]. On the other hand, when they are attached with a high organic load, excessively wet conditions in rodent burrows (e.g., relative humidity 95%) can stimulate the growth of destructive fungi that diminish flea’s larval and egg survival [13].Different studies justify the negative correlation between rainfall and plague epidemics. For example, Cavanaugh and Marshall Jr. [3] reported that, in areas where drains are absent, or where drainage is insufficient as a result of soil composition or impoundments of water, flooding unquestionably causes a drop in the flea population. In areas with improved drainage, such as those with sandy soils, the lessening of the flea population is minimal. Precipitation also influences plague infection for it influences the concentration of rodents, fleas, and humans in the same shelter.


## 1.2. Seasonality in Rodents

The direct effect posed on rodent population due to temperature change is minor. This is due to the fact that rodents are homoeothermic and hence do not respond immediately to changes in ambient temperatures [14]. Temperature indirectly affects the spread of plague in rodent population in different ways as follows: at a low mean temperature of 10°C the bacteria within host (rodent) become very active as a result a large number of infected rodents dying before even the plague bacilli appear in their blood. At this particular temperature rodents also lose the ability to infect other susceptible individuals.Rainfall may pose positive or negative effect on the increase of rodent population depending on its intensity [11]. A season of moderate rainfall may be considered to affect positively the increase of rodent abundance but when the amount of rainfall is extremely heavy it results in a tremendous rodent population decline [15]. When it is moderate and upon a proper timing, rainfall may foster the increase of rodent population [3]. This is due to the fact that rodent’s reproduction period normally follows wet seasons [16–18]. That is to say, the increase of rodent population during wet period is expected to be higher than that during the dry seasons. This clearly concurs with the result in the study by Leirs et al. [19], which narrates that, in Tanzania, rodent population densities show clear association with the annual rainfall and its seasonal distribution. However when rainfall is of high intensity, it causes flooding of rodent burrows. Large number of rodents population dies and the remaining ones normally move from forest to the households where they can protect themselves [3, 8, 20]. In other cases, increased precipitation or drought stalwartly disturbs rodent population dynamics, as it deters food availability.


## 1.3. Seasonality in Pathogens in the Environment

When the bacteria are in lungs, the transmission ofYersinia pestis is possible through various ways: contact transmission, in which one may be infected through physical contact with respiratory particles on the infected surface; airborne transmission, which occurs through inhaling the bacteria causing the disease through successive contact with the nose or mouth of an infected individual; respiratory particles, which occurs through respiratory droplets which is through shedding of respiratory particles (i.e., droplets or aerosols) from an infected human or rodent into the environment [21].Extreme temperatures regularly are ruinous to the survival of pathogens causing plague. The changes in temperature may lead to varying effects on the pathogens in the environment and vectors that live in an environment. When the mean temperature approaches the maximum limit that can be endured by the pathogens, a small increase in temperature may be very dangerous to the pathogen survival. Conversely when pathogens are in the environment characterized by low mean temperature, a small increase in temperature may result in increased development, incubation, and replication of the pathogen in the environment [22, 23].Davis [24] compared the seasonal incidence of plague with usual atmospheric conditions in particular temperature and rainfall. It was depicted that human plague is more frequent in warm moist weather between 15°C and 27°C than in hot dry (over 27°C) or cold weather (under 15°C). Mitscherlich and Marth [25] narrate that the solar exerts a detrimental effect on bacterial aerosol and the decay rate ofYersinia pestis is proportional to the increase of UV light.The reports by Ayyadurai et al. [26] and Mollaret [27] justify the ability of theYersinia pestis to culture the organism from deep within contaminated soil. Eisen et al. [28] were able to show the great potential durability ofYersinia pestis in the soil substrate. The long duration of their survival in the soil supports indirectly the virulence maintenance.Yersinia pestis exhibit a very slow growth at the temperature between 35°C and 37°C but they grow very fast at the temperature 28°C. They die very rapid if exposed to a UV light or temperature exceeding 40°C or when exposed to intensive desiccation [29–31]. Bacteria decrease their sensitivity when the level of humidity drops below 76% [25].When an infected individual coughs or sneezes, thousands of the bacteria are released in air [32]. The released respiratory particles may be large and heavy that they cannot remain suspended in the air. When respiratory particles are large the transmission can only occur when these particles are expelled directly onto another close susceptible individual. In some cases the release of smaller respiratory particles may occur; this is when the airborne transmission is possible. The smaller released particles are easily suspended in the air respired (i.e., passed to the lower respiratory tract) [33].Relative humidity and temperature affect the transmission ofYersinia pestis from one individual to the other. Humidity affects the size of the respiratory particle [34]. When humidity is low the large drops partially evaporate to create smaller, lighter drops that are more likely to remain airborne for extended periods of time [35]. That is to say, when the air is sufficiently dry the large sized particles shrink to a size that favors long-range transport which in turn leads to increased infection.


## 1.4. Seasonality in Human Behavior

Human activities and behavior in plague-infected areas are also to be considered as important determinants of plague transmission to and by humans [42]. When occurrences of plague are due to human intrusions in natural plague areas, it is thus important to consider season variation as a second-order variable that influences disease incidence through human behavior. In Tanzania drought and famine which are the result of lack of rainfall and temperature fluctuation have a great impact on the farmers and pastoralists as they force them to move from one area to another searching for food for themselves and their cattle. These human intrusions from one place to another may lead to the increase of plague disease transmission in rodents, fleas, human population, and pathogens in the environment.


## 2. Model Formulation

We describe the complex interaction that leads to plague disease transmission and use it to formulate a model for the dynamics of the plague disease coupled with the effect of seasonal weather variation in its transmission. The model includes four populations, namely, human beings, rodents, fleas, and pathogens, in the environment. We generally assume that all individuals from each population are susceptible to the disease, the recovered individuals confer temporary immunity and return to be susceptible again, and the infectious are all individuals with either bubonic plague or pneumonic or septicemic plague.


### 2.1. Variables and Parameters Used in the Model

In Notations and Table1 we present variables and parameters, their description, and their values as used in the model. We have obtained the parameter values from the literature that relate to this study and the present information on plague disease and through estimation.Table 1
Parameters and their description.
Parameters
Description
Value
Reference/source
Γ
r
b
f
(
t
)
Adequate contact rate: betweenIRB and flea
0.1
Eisen et al. [36]
Γ
r
s
f
(
t
)
Adequate contact rate: betweenIRS and flea
0.1
Eisen et al. [36]
Γ
f
h
(
t
)
Adequate contact rate: betweenIF and human
0.0641
Eisen et al. [36]
Γ
f
r
(
t
)
Adequate contact rate: betweenIF and rodent
0.0641
Eisen et al. [36]
Γ
h
p
h
(
t
)
Adequate contact rate: betweenIHP and SH
0.39
Estimated
Γ
h
s
h
(
t
)
Adequate contact rate: betweenIHS and SH
0.12
Estimated
Γ
r
b
h
(
t
)
Adequate contact rate: betweenIRB and SH
Γ
r
p
h
(
t
)
Adequate contact rate: betweenIRP and SH
0.19
Estimated
Γ
r
s
h
(
t
)
Adequate contact rate: betweenIRS and SH
0.21
Estimated
α
1
Progression rate ofSH to EH population
0.99
Estimated
α
2
Progression rate out ofEH to infectious state
0.23
Gani and Leach [37]
ρ
1
α
3
Progression rate out ofIHB to IHP
ρ
2
α
3
Progression rate out ofIHB to RH
ρ
3
α
3
Progression rate out ofIHB to IHS
δ
1
b
Disease induced death rate ofIHB
0.04
Keeling and Gilligan [38]
α
4
Progression rate out ofIHS to IHP and RH
0.06
Estimated
δ
1
s
Disease induced death rate ofIHS
0.04
Estimated
α
5
Progression rate out ofIHP to RH
0.4
Gani and Leach [37]
δ
1
p
Disease induced death rate ofIHP
0.63
Kugeler et al. [39]
γ
1
Progression rate ofSR to ER
0.92
Estimated
Γ
h
b
f
(
t
)
Adequate contact rate: betweenIHB and flea
0.1
Eisen et al. [36]
Γ
h
s
f
(
t
)
Adequate contact rate: betweenIHS and flea
0.1
Eisen et al. [36]
Γ
r
p
r
(
t
)
Adequate contact rate: betweenIRP and SR
0.9
Estimated
Γ
r
s
r
(
t
)
Adequate contact rate: betweenIRS and SR
0.9
Estimated
Γ
h
p
r
(
t
)
Adequate contact rate: betweenIHP and SR
0.00005
Estimated
Γ
h
s
r
(
t
)
Adequate contact rate: betweenIHS and SR
0.00008
Estimated
γ
2
The rate at which rodent becomes infectious
0.98
Estimated
γ
3
Progression rate out ofIRB to IRS and IRP
0.194
Tollenaere et al. [40]
δ
3
b
Disease induced death rate ofIRB
0.1
Estimated
γ
4
Progression rate out ofIRS to IRP
0.05
Estimated
δ
3
s
Disease induced death rate ofIRS
73
Tollenaere et al. [40]
δ
3
p
Disease induced death rate ofIRP
0.14
Estimated
ϖ
Progression rate ofRH to SH
0.33
Kugeler et al. [39]
μ
1
Natural death rate for human being
0.04
Keeling and Gilligan [38]
μ
2
Natural death rate for flea
0.2
Bacot and Martin [7]
μ
3
Natural death rate for rodent
1
Morand and Harvey [41]
ω
1
(
t
)
Adequate contact rate:A and human being
ω
2
(
t
)
Adequate contact rate:A and rodent
η
1
(
t
)
Recruitment rate ofA by IHP
0.2
Estimated
η
2
(
t
)
Recruitment rate ofA by IRP
0.4
Estimated
μ
4
Natural death rate for pathogens
0.1
Estimated
ψ
1
Recruitment rate of human beings
0.09
Estimated
ψ
Recruitment rate of fleas
ψ
3
Recruitment rate of rodents


### 2.2. Model Description

The human population is divided into six subgroups: the subgroup of people who have not contracted the disease, to be referred to as susceptible and denoted bySH, but may get it if they come into contact with IHS, IHP, IRS, IHP, IF, or A; people who have the disease but have not shown any symptom and are incapable of transmitting the disease to be referred to as exposed and denoted by EH; those who are infected and capable of transmitting the disease are divided into three subgroups: there are those who have bubonic plague denoted by IHB, those with septicemic plague denoted by IHS, and those who have pneumonic plague disease denoted by IHP. The fraction of population in IHB if treated or through strong body immunity may recover and move to subgroup RH; otherwise they progress either to a septicemic disease infective agent IHS or to pneumonic plague disease infective agent IHP or else they die. The population in the subgroup IHS through strong body immunity or if treated recover and progress to the subgroup RH and if not treated they progress and join subgroup IHP; otherwise they die. The population of the subgroup IHP is considered as a very dangerous stage of plague disease; it is a very fatal stage of plague disease with the fatality rate of about 100%; however if treated they recover and join subgroup RH; otherwise they die. So the total human population N1 is as given by(1)N1=SH+EH+IHB+IHS+IHP+RH.Fleas are divided into two subgroups, those who have not contracted the disease but may get it if they get in contact with infectious agent (rodent or human) referred to as susceptible flea and denoted bySF and those who are infected and are capable of transmitting the disease referred to as infective agents and denoted by IF. The total flea population N2 is as given by(2)N2=SF+IF.The rodents are divided into five subgroups; those who have not contracted the disease but may get it if they get in contact withIHS, IHP, IRS, IHP, IF, or A, referred to as susceptible rodents and denoted by SR; those who have the disease but have not shown any symptom and are incapable of transmitting the disease referred to as exposed and denoted by ER; those who are infected and capable of transmitting the disease are divided into three subgroups: those who have bubonic plague denoted by IRB, those with septicemic plague denoted by IRS, and those who have pneumonic plague IRP. The fraction of population in IRB may progress either to a septicemic plague disease infective agent IRS or to pneumonic plague disease infective agent IRP. The rodent population in the subgroup IRS may either progress to pneumonic plague disease infective agent IRP; otherwise they die. The population in the subgroup IRP is considered as a very dangerous stage of plague disease and very fatal so the mortality due to disease in this subgroup is approximated to be 100%. Then the total rodent population N3 is as given by(3)N3=SR+ER+IRB+IRS+IRP.The individuals with pneumonic plague may release pathogens causing plague disease to the environment denoted byA through coughing or sneezing. When the condition in soil/environment is favorable, pathogens may remain infectious in the environment for a long time. When a susceptible individual adequately interacts with the environment infested withYersinia pestis, he/she gets the disease even in the absence of any vector.


### 2.3. Description of Interactions

The susceptible fleas in subgroupSF getYersinia pestis bacteria through biting the infected rodent IRB or IRS who are the primary reservoir for the bacteria and become infected at the rates Γrbf and Γrsf, respectively. Fleas may also get the disease when they bite the infected human being with bubonic plague IHB or septicemic plague IHS at the rates Γhbf and Γhsf, respectively. Thus the flea population gets plague infection with the force of infection given in(4)G3t=ΓhbftIHB+ΓhsftIHSN1+ΓrbftIRB+ΓrsftIRSN3.The human population may get the disease in one of the following ways: when the infected fleaIF bites and infects the susceptible human being SH at a rate Γfh; when they interact with one another; this can be with either a person with pneumonic plague IHP through airborne transmission or septicemic plague IHS through physical or sexual contact at the rates Γhph and Γhsh, respectively. Another infection is through airborne transmission through interaction with rodent infected with pneumonic plague IRP or through touching or eating the infected rodent with septicemic plague IRS at rates of Γrph and Γrsh, respectively. Human beings may also get the infection from the environment when they breath in the bacteria or physically contact the infected material at the rate of ω1. That is to say, human population acquire plague disease following effective contact with infected human, rodent, flea, and the environment with force of infection G1 given by(5)G1t=ΓhphtIHP+ΓhshtIHSN1+ΓfhtIFN2+ΓrphtIRP+ΓrshtIRSN3+ω1tA.The subgroupSH, after the infection, progresses and becomes latent to the disease at a rate α1. After 2 to 7 days the subgroups EH become infected into one of the three infectious classes, IHB, IHS, or IHP (depending on the mode of transmission an individual is exposed to), and are capable of transmitting the disease. The proportion of EH progresses and becomes infected by bubonic plague IHB, septicemic plague IHS, or pneumonic plague IHP at the rate α2 and proportion to ν1, ν2, or ν3, respectively. The compartment IHB either through strong body immunity or if they get treatment they recover and move to subgroup RH at a rate α3; otherwise they either progress to subgroup IHP or IHS at a rate α3 or die either naturally at a rate μ1 or due to the disease at a rate δ1b. The fraction of humans with septicemic plague IHS either through strong body immunity or if treated recover at a rate α4 and join RH; otherwise they either progress to subgroup IHP at a rate α4 or die due to the disease at a rate δ1s or naturally at a rate μ1. The compartments IHP if treated recover at a rate α5; otherwise they die either naturally at a rate μ1 or due to the disease at a rate δ1p. The subgroup RH attain temporary immunity and then return and become susceptible SH at a rate ϖ.The rodent population may get a disease in one of the following ways: when the infected fleaIF bites and infects the susceptible rodent SR at a rate Γfr, through interaction between rodents themselves, which may be with rodent infected by pneumonic plague IRP or septicemic plague IRS at the rates Γrpr and Γrsr, respectively. The other infection may be through interaction with human infected with either pneumonic plague IHP or septicemic plague IHS at rates of Γhpr and Γhsr, respectively. When the susceptible rodent sufficiently interacts with the pathogens in environment through breathing in the bacteria or physically touches the infected material, it gets the infections at the rate of ω2. Rodent also gets the disease through adequate interaction with rodent, human, flea, and pathogens in the environment with force of infection G2 given by(6)G2t=ΓhprtIHP+ΓhsrtIHSN1+ΓfrtIFN2+ΓrprtIRP+ΓrsrtIRSN3+ω2tA.The subgroupSR, after the infection, progress and become latent to the disease at a rate γ1. After 2 to 7 days the subgroup ER become infected and capable of transmitting the disease; the fraction of it progresses and becomes infected by bubonic plague IRB, septicemic plague IRS, or pneumonic plague IRP at the rate γ2 and proportional to τ1, τ2, or τ3 respectively. The rodent in subgroup IRB may either progress to subgroup IRP or IRS at a rate γ3 or die either naturally at a rate μ3 or due to the disease at a rate δ3b. The compartment IRS may either progress to IRP at a rate γ4 or die due to a disease at a rate δ3s or naturally at a rate μ3 and the compartments IRP die either naturally at a rate μ3 or due to the disease at a rate δ3p.With regard to the pathogens in the environment, we assume that the adequate interaction withSH and SR has a negligible effect on the dynamics of pathogens population size in the environment. The pathogens in the environment are populated at a constant rate λ4. The infected human with pneumonic plague IHP and rodent with pneumonic plague IRP also populate the environment A with the bacteria at the rates η1 and η2, respectively. Thus the environment is populated with pathogens causing plague disease with the force of infection G4 given by(7)G4t=λ4t+η1tIHPN1+η2tIRPN3.The pathogens within the environment suffer natural mortality at a rateμ4. Human population in subgroups SH and EH, flea population in subgroup SF, and rodent population in subgroups SR and ER suffer natural mortality at rates μ1,μ2, and μ3, respectively. The compartments IHB, IHS, IHP, IF, IRB, IRS, and IRP suffer both natural death at the rates μ1, μ2, and μ3 and disease induced mortality at rates δ1b, δ1s, δ1p, δ2, δ3b, δ3s, and δ3p respectively. Human, flea, and rodent are recruited at the rates ψ1, ψ2, and ψ3, respectively.


### 2.4. Model Equations for Plague Disease

Now we assume that the variation of infection capability from one individual to the other, migration of individuals from one place to another, and recruitment and death rates of individuals in different stages due to seasonal weather variation affect only the rate at which the disease is transmitted from one infected individual to the other. We now use the variables and parameters and their description given in Notations and Table1 and the description of interactions to drive the system of differential equations given as follows.Human Beings
(8a)
d
S
H
d
t
=
σ
1
ψ
1
+
ϖ
R
H
-
α
1
G
1
t
S
H
-
μ
1
S
H
,
(8b)
d
E
H
d
t
=
1
-
σ
1
ψ
1
+
α
1
G
1
t
S
H
-
α
2
E
H
-
μ
1
E
H
,
(8c)
d
I
H
B
d
t
=
α
2
ν
2
E
H
-
α
3
I
H
B
-
μ
1
+
δ
1
b
I
H
B
,
(8d)
d
I
H
S
d
t
=
α
3
ρ
3
I
H
B
+
α
2
ν
3
E
H
-
α
4
I
H
S
-
μ
1
+
δ
1
s
I
H
S
,
(8e)
d
I
H
P
d
t
=
α
2
ν
1
E
H
+
α
3
ρ
1
I
H
B
+
α
4
ξ
I
H
S
-
α
5
I
H
P
-
μ
1
+
δ
1
p
I
H
P
,
(8f)
d
R
H
d
t
=
α
3
ρ
2
I
H
B
+
α
4
1
-
ξ
I
H
S
+
α
5
I
H
P
-
ϖ
R
H
-
μ
1
R
H
.Rodents
(9a)
d
S
R
d
t
=
σ
2
ψ
3
-
γ
1
G
2
t
S
R
-
μ
3
S
R
,
(9b)
d
E
R
d
t
=
1
-
σ
2
ψ
3
+
γ
1
G
2
t
S
R
-
γ
2
E
R
-
μ
3
E
R
,
(9c)
d
I
R
B
d
t
=
γ
2
τ
3
E
R
-
γ
3
I
R
B
-
μ
3
+
δ
3
b
I
R
B
,
(9d)
d
I
R
S
d
t
=
γ
2
τ
2
E
R
+
γ
3
1
-
ϕ
I
R
B
-
γ
4
I
R
S
-
μ
3
+
δ
3
s
I
R
S
,
(9e)
d
I
R
P
d
t
=
γ
2
τ
1
E
R
+
γ
3
ϕ
I
R
B
+
γ
4
I
R
S
-
μ
3
+
δ
3
p
I
R
P
.Fleas
(10a)
d
S
F
d
t
=
ψ
2
s
-
β
G
3
t
S
F
-
μ
2
S
F
,
(10b)
d
I
F
d
t
=
ψ
2
i
+
β
G
3
t
S
F
-
μ
2
+
δ
2
I
F
.Pathogens
(11)
d
A
d
t
=
λ
4
t
+
η
1
t
I
H
P
N
1
+
η
3
t
I
R
P
N
3
-
μ
4
t
A
.


## 2.1. Variables and Parameters Used in the Model

In Notations and Table1 we present variables and parameters, their description, and their values as used in the model. We have obtained the parameter values from the literature that relate to this study and the present information on plague disease and through estimation.Table 1
Parameters and their description.
Parameters
Description
Value
Reference/source
Γ
r
b
f
(
t
)
Adequate contact rate: betweenIRB and flea
0.1
Eisen et al. [36]
Γ
r
s
f
(
t
)
Adequate contact rate: betweenIRS and flea
0.1
Eisen et al. [36]
Γ
f
h
(
t
)
Adequate contact rate: betweenIF and human
0.0641
Eisen et al. [36]
Γ
f
r
(
t
)
Adequate contact rate: betweenIF and rodent
0.0641
Eisen et al. [36]
Γ
h
p
h
(
t
)
Adequate contact rate: betweenIHP and SH
0.39
Estimated
Γ
h
s
h
(
t
)
Adequate contact rate: betweenIHS and SH
0.12
Estimated
Γ
r
b
h
(
t
)
Adequate contact rate: betweenIRB and SH
Γ
r
p
h
(
t
)
Adequate contact rate: betweenIRP and SH
0.19
Estimated
Γ
r
s
h
(
t
)
Adequate contact rate: betweenIRS and SH
0.21
Estimated
α
1
Progression rate ofSH to EH population
0.99
Estimated
α
2
Progression rate out ofEH to infectious state
0.23
Gani and Leach [37]
ρ
1
α
3
Progression rate out ofIHB to IHP
ρ
2
α
3
Progression rate out ofIHB to RH
ρ
3
α
3
Progression rate out ofIHB to IHS
δ
1
b
Disease induced death rate ofIHB
0.04
Keeling and Gilligan [38]
α
4
Progression rate out ofIHS to IHP and RH
0.06
Estimated
δ
1
s
Disease induced death rate ofIHS
0.04
Estimated
α
5
Progression rate out ofIHP to RH
0.4
Gani and Leach [37]
δ
1
p
Disease induced death rate ofIHP
0.63
Kugeler et al. [39]
γ
1
Progression rate ofSR to ER
0.92
Estimated
Γ
h
b
f
(
t
)
Adequate contact rate: betweenIHB and flea
0.1
Eisen et al. [36]
Γ
h
s
f
(
t
)
Adequate contact rate: betweenIHS and flea
0.1
Eisen et al. [36]
Γ
r
p
r
(
t
)
Adequate contact rate: betweenIRP and SR
0.9
Estimated
Γ
r
s
r
(
t
)
Adequate contact rate: betweenIRS and SR
0.9
Estimated
Γ
h
p
r
(
t
)
Adequate contact rate: betweenIHP and SR
0.00005
Estimated
Γ
h
s
r
(
t
)
Adequate contact rate: betweenIHS and SR
0.00008
Estimated
γ
2
The rate at which rodent becomes infectious
0.98
Estimated
γ
3
Progression rate out ofIRB to IRS and IRP
0.194
Tollenaere et al. [40]
δ
3
b
Disease induced death rate ofIRB
0.1
Estimated
γ
4
Progression rate out ofIRS to IRP
0.05
Estimated
δ
3
s
Disease induced death rate ofIRS
73
Tollenaere et al. [40]
δ
3
p
Disease induced death rate ofIRP
0.14
Estimated
ϖ
Progression rate ofRH to SH
0.33
Kugeler et al. [39]
μ
1
Natural death rate for human being
0.04
Keeling and Gilligan [38]
μ
2
Natural death rate for flea
0.2
Bacot and Martin [7]
μ
3
Natural death rate for rodent
1
Morand and Harvey [41]
ω
1
(
t
)
Adequate contact rate:A and human being
ω
2
(
t
)
Adequate contact rate:A and rodent
η
1
(
t
)
Recruitment rate ofA by IHP
0.2
Estimated
η
2
(
t
)
Recruitment rate ofA by IRP
0.4
Estimated
μ
4
Natural death rate for pathogens
0.1
Estimated
ψ
1
Recruitment rate of human beings
0.09
Estimated
ψ
Recruitment rate of fleas
ψ
3
Recruitment rate of rodents


## 2.2. Model Description

The human population is divided into six subgroups: the subgroup of people who have not contracted the disease, to be referred to as susceptible and denoted bySH, but may get it if they come into contact with IHS, IHP, IRS, IHP, IF, or A; people who have the disease but have not shown any symptom and are incapable of transmitting the disease to be referred to as exposed and denoted by EH; those who are infected and capable of transmitting the disease are divided into three subgroups: there are those who have bubonic plague denoted by IHB, those with septicemic plague denoted by IHS, and those who have pneumonic plague disease denoted by IHP. The fraction of population in IHB if treated or through strong body immunity may recover and move to subgroup RH; otherwise they progress either to a septicemic disease infective agent IHS or to pneumonic plague disease infective agent IHP or else they die. The population in the subgroup IHS through strong body immunity or if treated recover and progress to the subgroup RH and if not treated they progress and join subgroup IHP; otherwise they die. The population of the subgroup IHP is considered as a very dangerous stage of plague disease; it is a very fatal stage of plague disease with the fatality rate of about 100%; however if treated they recover and join subgroup RH; otherwise they die. So the total human population N1 is as given by(1)N1=SH+EH+IHB+IHS+IHP+RH.Fleas are divided into two subgroups, those who have not contracted the disease but may get it if they get in contact with infectious agent (rodent or human) referred to as susceptible flea and denoted bySF and those who are infected and are capable of transmitting the disease referred to as infective agents and denoted by IF. The total flea population N2 is as given by(2)N2=SF+IF.The rodents are divided into five subgroups; those who have not contracted the disease but may get it if they get in contact withIHS, IHP, IRS, IHP, IF, or A, referred to as susceptible rodents and denoted by SR; those who have the disease but have not shown any symptom and are incapable of transmitting the disease referred to as exposed and denoted by ER; those who are infected and capable of transmitting the disease are divided into three subgroups: those who have bubonic plague denoted by IRB, those with septicemic plague denoted by IRS, and those who have pneumonic plague IRP. The fraction of population in IRB may progress either to a septicemic plague disease infective agent IRS or to pneumonic plague disease infective agent IRP. The rodent population in the subgroup IRS may either progress to pneumonic plague disease infective agent IRP; otherwise they die. The population in the subgroup IRP is considered as a very dangerous stage of plague disease and very fatal so the mortality due to disease in this subgroup is approximated to be 100%. Then the total rodent population N3 is as given by(3)N3=SR+ER+IRB+IRS+IRP.The individuals with pneumonic plague may release pathogens causing plague disease to the environment denoted byA through coughing or sneezing. When the condition in soil/environment is favorable, pathogens may remain infectious in the environment for a long time. When a susceptible individual adequately interacts with the environment infested withYersinia pestis, he/she gets the disease even in the absence of any vector.


## 2.3. Description of Interactions

The susceptible fleas in subgroupSF getYersinia pestis bacteria through biting the infected rodent IRB or IRS who are the primary reservoir for the bacteria and become infected at the rates Γrbf and Γrsf, respectively. Fleas may also get the disease when they bite the infected human being with bubonic plague IHB or septicemic plague IHS at the rates Γhbf and Γhsf, respectively. Thus the flea population gets plague infection with the force of infection given in(4)G3t=ΓhbftIHB+ΓhsftIHSN1+ΓrbftIRB+ΓrsftIRSN3.The human population may get the disease in one of the following ways: when the infected fleaIF bites and infects the susceptible human being SH at a rate Γfh; when they interact with one another; this can be with either a person with pneumonic plague IHP through airborne transmission or septicemic plague IHS through physical or sexual contact at the rates Γhph and Γhsh, respectively. Another infection is through airborne transmission through interaction with rodent infected with pneumonic plague IRP or through touching or eating the infected rodent with septicemic plague IRS at rates of Γrph and Γrsh, respectively. Human beings may also get the infection from the environment when they breath in the bacteria or physically contact the infected material at the rate of ω1. That is to say, human population acquire plague disease following effective contact with infected human, rodent, flea, and the environment with force of infection G1 given by(5)G1t=ΓhphtIHP+ΓhshtIHSN1+ΓfhtIFN2+ΓrphtIRP+ΓrshtIRSN3+ω1tA.The subgroupSH, after the infection, progresses and becomes latent to the disease at a rate α1. After 2 to 7 days the subgroups EH become infected into one of the three infectious classes, IHB, IHS, or IHP (depending on the mode of transmission an individual is exposed to), and are capable of transmitting the disease. The proportion of EH progresses and becomes infected by bubonic plague IHB, septicemic plague IHS, or pneumonic plague IHP at the rate α2 and proportion to ν1, ν2, or ν3, respectively. The compartment IHB either through strong body immunity or if they get treatment they recover and move to subgroup RH at a rate α3; otherwise they either progress to subgroup IHP or IHS at a rate α3 or die either naturally at a rate μ1 or due to the disease at a rate δ1b. The fraction of humans with septicemic plague IHS either through strong body immunity or if treated recover at a rate α4 and join RH; otherwise they either progress to subgroup IHP at a rate α4 or die due to the disease at a rate δ1s or naturally at a rate μ1. The compartments IHP if treated recover at a rate α5; otherwise they die either naturally at a rate μ1 or due to the disease at a rate δ1p. The subgroup RH attain temporary immunity and then return and become susceptible SH at a rate ϖ.The rodent population may get a disease in one of the following ways: when the infected fleaIF bites and infects the susceptible rodent SR at a rate Γfr, through interaction between rodents themselves, which may be with rodent infected by pneumonic plague IRP or septicemic plague IRS at the rates Γrpr and Γrsr, respectively. The other infection may be through interaction with human infected with either pneumonic plague IHP or septicemic plague IHS at rates of Γhpr and Γhsr, respectively. When the susceptible rodent sufficiently interacts with the pathogens in environment through breathing in the bacteria or physically touches the infected material, it gets the infections at the rate of ω2. Rodent also gets the disease through adequate interaction with rodent, human, flea, and pathogens in the environment with force of infection G2 given by(6)G2t=ΓhprtIHP+ΓhsrtIHSN1+ΓfrtIFN2+ΓrprtIRP+ΓrsrtIRSN3+ω2tA.The subgroupSR, after the infection, progress and become latent to the disease at a rate γ1. After 2 to 7 days the subgroup ER become infected and capable of transmitting the disease; the fraction of it progresses and becomes infected by bubonic plague IRB, septicemic plague IRS, or pneumonic plague IRP at the rate γ2 and proportional to τ1, τ2, or τ3 respectively. The rodent in subgroup IRB may either progress to subgroup IRP or IRS at a rate γ3 or die either naturally at a rate μ3 or due to the disease at a rate δ3b. The compartment IRS may either progress to IRP at a rate γ4 or die due to a disease at a rate δ3s or naturally at a rate μ3 and the compartments IRP die either naturally at a rate μ3 or due to the disease at a rate δ3p.With regard to the pathogens in the environment, we assume that the adequate interaction withSH and SR has a negligible effect on the dynamics of pathogens population size in the environment. The pathogens in the environment are populated at a constant rate λ4. The infected human with pneumonic plague IHP and rodent with pneumonic plague IRP also populate the environment A with the bacteria at the rates η1 and η2, respectively. Thus the environment is populated with pathogens causing plague disease with the force of infection G4 given by(7)G4t=λ4t+η1tIHPN1+η2tIRPN3.The pathogens within the environment suffer natural mortality at a rateμ4. Human population in subgroups SH and EH, flea population in subgroup SF, and rodent population in subgroups SR and ER suffer natural mortality at rates μ1,μ2, and μ3, respectively. The compartments IHB, IHS, IHP, IF, IRB, IRS, and IRP suffer both natural death at the rates μ1, μ2, and μ3 and disease induced mortality at rates δ1b, δ1s, δ1p, δ2, δ3b, δ3s, and δ3p respectively. Human, flea, and rodent are recruited at the rates ψ1, ψ2, and ψ3, respectively.


## 2.4. Model Equations for Plague Disease

Now we assume that the variation of infection capability from one individual to the other, migration of individuals from one place to another, and recruitment and death rates of individuals in different stages due to seasonal weather variation affect only the rate at which the disease is transmitted from one infected individual to the other. We now use the variables and parameters and their description given in Notations and Table1 and the description of interactions to drive the system of differential equations given as follows.Human Beings
(8a)
d
S
H
d
t
=
σ
1
ψ
1
+
ϖ
R
H
-
α
1
G
1
t
S
H
-
μ
1
S
H
,
(8b)
d
E
H
d
t
=
1
-
σ
1
ψ
1
+
α
1
G
1
t
S
H
-
α
2
E
H
-
μ
1
E
H
,
(8c)
d
I
H
B
d
t
=
α
2
ν
2
E
H
-
α
3
I
H
B
-
μ
1
+
δ
1
b
I
H
B
,
(8d)
d
I
H
S
d
t
=
α
3
ρ
3
I
H
B
+
α
2
ν
3
E
H
-
α
4
I
H
S
-
μ
1
+
δ
1
s
I
H
S
,
(8e)
d
I
H
P
d
t
=
α
2
ν
1
E
H
+
α
3
ρ
1
I
H
B
+
α
4
ξ
I
H
S
-
α
5
I
H
P
-
μ
1
+
δ
1
p
I
H
P
,
(8f)
d
R
H
d
t
=
α
3
ρ
2
I
H
B
+
α
4
1
-
ξ
I
H
S
+
α
5
I
H
P
-
ϖ
R
H
-
μ
1
R
H
.Rodents
(9a)
d
S
R
d
t
=
σ
2
ψ
3
-
γ
1
G
2
t
S
R
-
μ
3
S
R
,
(9b)
d
E
R
d
t
=
1
-
σ
2
ψ
3
+
γ
1
G
2
t
S
R
-
γ
2
E
R
-
μ
3
E
R
,
(9c)
d
I
R
B
d
t
=
γ
2
τ
3
E
R
-
γ
3
I
R
B
-
μ
3
+
δ
3
b
I
R
B
,
(9d)
d
I
R
S
d
t
=
γ
2
τ
2
E
R
+
γ
3
1
-
ϕ
I
R
B
-
γ
4
I
R
S
-
μ
3
+
δ
3
s
I
R
S
,
(9e)
d
I
R
P
d
t
=
γ
2
τ
1
E
R
+
γ
3
ϕ
I
R
B
+
γ
4
I
R
S
-
μ
3
+
δ
3
p
I
R
P
.Fleas
(10a)
d
S
F
d
t
=
ψ
2
s
-
β
G
3
t
S
F
-
μ
2
S
F
,
(10b)
d
I
F
d
t
=
ψ
2
i
+
β
G
3
t
S
F
-
μ
2
+
δ
2
I
F
.Pathogens
(11)
d
A
d
t
=
λ
4
t
+
η
1
t
I
H
P
N
1
+
η
3
t
I
R
P
N
3
-
μ
4
t
A
.


## 3. Basic Properties of the Model

In this section we discuss the feasible region and positivity of the plague disease model. For convenience purpose and easy presentation of the result we letC denote all continuous functions on the real line. If f is a periodic function in C then we use f¯ for the average value of f on time interval [0,T] defined by(12)f¯=1T∫0Tftdt,for a continuous T-periodic function f(t).


### 3.1. Invariant Region

Plague disease affects human, rodent, flea, and pathogens in the environment populations. For the possible modeling process all state variables and parameters of the model must be nonnegative for∀t≥0. We thus need to verify whether the solutions of the model system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) are in suitable feasible region where all state variables are positive. Inspired by Dumont et al. [43] and Mpeshe et al. [44] we first write system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) in the following compact form:(13)dXdt=AxX+F,where X=(SH,EH,IHB,IHS,IHP,RH,SR,ER,IRB, IRS,IRP,SF,IF,A)T, A(x) is a 14×14 matrix, and F is a column vector.We then have(14)Ax=A11A12A21A22,where(15)A11=-g10000ϖ0α1G1t-α2+μ1000000α2ν2-a100000α2ν3ρ3α3-a20000α2ν1ρ1α3α4ξ-a30000ρ2α3α41-ξα5-ϖ+μ10000000-g2,A12=0000000000000000000000000000000000000000000000000,A22=-γ2+μ3000000γ2τ3-a400000γ2τ2γ31-ϕ-a50000γ2τ1γ3ϕγ4-μ3+δ3p0000000-g3000000βG3t-μ2+δ20000η2tN300-μ4,A21=000000γ1G2t0000000000000000000000000000000000000η1tN10000,F=σ1ψ1,1-σ1ψ1,0,0,0,0,σ2ψ3,1-σ2ψ3,0,0,0,ψ2s,ψ2i,λ4T,where a1=(α3+μ1+δ1b),a2=(α4+μ1+δ1s),a3=(α5+μ1+δ1p),a4=(γ3+μ3+δ3b),a5=(γ4+μ3+δ3s),g1=(α1G1(t)+μ1),g2=(γ1G2(t)+μ3), and g3=(βG3(t)+μ2).Now from submatricesA11, A12, A21, and A22 we can deduce that matrix A(x) is a Metzler matrix such that all of its off-diagonal elements are nonnegative, ∀x∈R+14, and F≥0 is Lipschitz continuous. Thus the feasible region for the model system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) is the set(16)Φ=SH,EH,IHB,IHS,IHP,RH,SR,ER,IRB,IRS,IRP,SF,IF,A≥0∈R+14.This means that any trajectory of the system starting from an initial state in the positive orthant ofR+14 remains forever in Φ.


### 3.2. Positivity of the Solution

We need to show that all variables and parameters of the model are nonnegative,∀t≥0. We now solve the equations of the system in their patches for testing the positivity. We found that, by letting the initial values of the systems ((8a), (8b), (8c), (8d), (8e), (8f)), ((9a), (9b), (9c), (9d), (9e)), ((10a), (10b)), and (11) be SH(0)>0, SR(0)>0, SF(0)>0 and A0≥0EH(0)≥0, IHB(0)≥0, IHS(0)≥0, IHP(0)≥0, RH(0)≥0, ER(0)≥0, IRB(0)≥0, IRS(0)≥0, IRP(0)≥0, and IF(0)≥0, in the solution set SH(t), SR(t), SF(t), A(t), EH(t), IHB(t), IHS(t), IHP(t), RH(t), ER(t), IRB(t), IRS(t), IRP(t), and IF(t) are nonnegative, ∀t≥0.


## 3.1. Invariant Region

Plague disease affects human, rodent, flea, and pathogens in the environment populations. For the possible modeling process all state variables and parameters of the model must be nonnegative for∀t≥0. We thus need to verify whether the solutions of the model system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) are in suitable feasible region where all state variables are positive. Inspired by Dumont et al. [43] and Mpeshe et al. [44] we first write system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) in the following compact form:(13)dXdt=AxX+F,where X=(SH,EH,IHB,IHS,IHP,RH,SR,ER,IRB, IRS,IRP,SF,IF,A)T, A(x) is a 14×14 matrix, and F is a column vector.We then have(14)Ax=A11A12A21A22,where(15)A11=-g10000ϖ0α1G1t-α2+μ1000000α2ν2-a100000α2ν3ρ3α3-a20000α2ν1ρ1α3α4ξ-a30000ρ2α3α41-ξα5-ϖ+μ10000000-g2,A12=0000000000000000000000000000000000000000000000000,A22=-γ2+μ3000000γ2τ3-a400000γ2τ2γ31-ϕ-a50000γ2τ1γ3ϕγ4-μ3+δ3p0000000-g3000000βG3t-μ2+δ20000η2tN300-μ4,A21=000000γ1G2t0000000000000000000000000000000000000η1tN10000,F=σ1ψ1,1-σ1ψ1,0,0,0,0,σ2ψ3,1-σ2ψ3,0,0,0,ψ2s,ψ2i,λ4T,where a1=(α3+μ1+δ1b),a2=(α4+μ1+δ1s),a3=(α5+μ1+δ1p),a4=(γ3+μ3+δ3b),a5=(γ4+μ3+δ3s),g1=(α1G1(t)+μ1),g2=(γ1G2(t)+μ3), and g3=(βG3(t)+μ2).Now from submatricesA11, A12, A21, and A22 we can deduce that matrix A(x) is a Metzler matrix such that all of its off-diagonal elements are nonnegative, ∀x∈R+14, and F≥0 is Lipschitz continuous. Thus the feasible region for the model system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) is the set(16)Φ=SH,EH,IHB,IHS,IHP,RH,SR,ER,IRB,IRS,IRP,SF,IF,A≥0∈R+14.This means that any trajectory of the system starting from an initial state in the positive orthant ofR+14 remains forever in Φ.


## 3.2. Positivity of the Solution

We need to show that all variables and parameters of the model are nonnegative,∀t≥0. We now solve the equations of the system in their patches for testing the positivity. We found that, by letting the initial values of the systems ((8a), (8b), (8c), (8d), (8e), (8f)), ((9a), (9b), (9c), (9d), (9e)), ((10a), (10b)), and (11) be SH(0)>0, SR(0)>0, SF(0)>0 and A0≥0EH(0)≥0, IHB(0)≥0, IHS(0)≥0, IHP(0)≥0, RH(0)≥0, ER(0)≥0, IRB(0)≥0, IRS(0)≥0, IRP(0)≥0, and IF(0)≥0, in the solution set SH(t), SR(t), SF(t), A(t), EH(t), IHB(t), IHS(t), IHP(t), RH(t), ER(t), IRB(t), IRS(t), IRP(t), and IF(t) are nonnegative, ∀t≥0.


## 4. Model Analysis

### 4.1. Disease-Free Equilibrium Solution

The periodic model system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) with nonnegative, continuous periodic functions has disease-free equilibrium solution in which we consider the following equations:(17)dSHdt=σ1ψ1-μ1SH,(18)dSRdt=σ2ψ3-μ3SR,(19)dSFdt=ψ2s-μ2SF.Now given initial conditionsSH=SH0∈R+, SR=SR0∈R+, and SF=SF0∈R+ for (17), (18), and (19), respectively, we will have(20)SH=σ1ψ1μ1+SH0-σ1ψ1μ1e-μ1t,SR=σ2ψ3μ3+SR0-σ2ψ3μ3e-μ3t,SR=ψ2sμ2+SF0-ψ2sμ2e-μ2t.Ast→∞, (17), (18), and (19) admit unique solution SH≡σ1ψ1/μ1, SR≡σ2ψ3/μ3, and SF≡ψ2s/μ2, respectively, which is globally attractive in R+3.To find the disease-free equilibrium point we set the derivatives of system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) equal to zero. Then the model system has disease-free solution which is obtained by setting IHB=IHS=IHP=EH=RH=0, IRB=IRS=IRP=ER=0, IF=0, and A=0 for human, rodent, flea, and pathogen system, respectively. Hence system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) has a disease-free equilibrium point(21)E0SH0,EH0,IHB0,IHS0,IHP0,RH0,SR0,ER0,IRB0,IRS0,IRP0,SF0,IF0,A0=σ1ψ1μ1,0,0,0,0,0,σ2ψ3μ3,0,0,0,0,ψ2sμ2,0,0.


## 4.1. Disease-Free Equilibrium Solution

The periodic model system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) with nonnegative, continuous periodic functions has disease-free equilibrium solution in which we consider the following equations:(17)dSHdt=σ1ψ1-μ1SH,(18)dSRdt=σ2ψ3-μ3SR,(19)dSFdt=ψ2s-μ2SF.Now given initial conditionsSH=SH0∈R+, SR=SR0∈R+, and SF=SF0∈R+ for (17), (18), and (19), respectively, we will have(20)SH=σ1ψ1μ1+SH0-σ1ψ1μ1e-μ1t,SR=σ2ψ3μ3+SR0-σ2ψ3μ3e-μ3t,SR=ψ2sμ2+SF0-ψ2sμ2e-μ2t.Ast→∞, (17), (18), and (19) admit unique solution SH≡σ1ψ1/μ1, SR≡σ2ψ3/μ3, and SF≡ψ2s/μ2, respectively, which is globally attractive in R+3.To find the disease-free equilibrium point we set the derivatives of system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) equal to zero. Then the model system has disease-free solution which is obtained by setting IHB=IHS=IHP=EH=RH=0, IRB=IRS=IRP=ER=0, IF=0, and A=0 for human, rodent, flea, and pathogen system, respectively. Hence system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) has a disease-free equilibrium point(21)E0SH0,EH0,IHB0,IHS0,IHP0,RH0,SR0,ER0,IRB0,IRS0,IRP0,SF0,IF0,A0=σ1ψ1μ1,0,0,0,0,0,σ2ψ3μ3,0,0,0,0,ψ2sμ2,0,0.


## 5. Basic Reproduction Number

Let (Rk,Rk) be the standard ordered k-dimensional Euclidean space with a norm ·. For u,v∈Rk we write u≥v provided u-v∈R+k, u>v provided u-v∈R+k∖{0}, and u≫v if u-v∈int(R+k).Now letA(t) be the continuous, cooperative, irreducible, and T-periodic k×k matrix function with period T>0, ΦA(·)(t) be the fundamental solution matrix of the linear ordinary differential system(22)dxdt=Atx,and ρ(ΦA(·)(T)) be the spectral radius of ΦA(·)(T). By Aronsson and Kellogg [45] it follows that ΦA(·)(t) is a matrix with all elements positive for each t>0. By the Perron Frobenius theorem, ρ(ΦA(·)(T)) is the principal eigenvalue of ΦA(·)(t) in the sense that it is simple and admits an eigenvector v∗≫0. The following result is important for our subsequent comparison argument.Proposition 1.
letι=1/Tln⁡(ρ(ΦA(T))), and then there exists a positive, T-periodic function v(t) such that eιtv(t) is a solution of x′=A(t)x.Proof.
Letv∗≫0 be the eigenvector associated with the spectral radius ρΦA(·)(T).
By the change of variable(23)xt=eμtvt,system (22) becomes(24)dvdt=Atv-μv=At-μIv,where I is an identity matrix.
Thusv(t)=Φ(A(·)-μI)(t)v∗ is a positive solution of (24). We can easily see that(25)eμtΦA·-μIt=ΦA·t.
Moreover(26)vT=ΦA·-μItv∗=e-μTΦA·Tv∗=e-μTρΦA·Tv∗=v∗=v0.
Thus,v(t) is a positive T-periodic solution of (24) and hence, x(t)=eμtv(t) is a solution of (22).The plague disease model system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) has unique disease-free equilibrium point given in (21).We consider a heterogeneous population whose individuals are distinguishable by stage of the disease and hence identifiable and put into epidemiological compartments which areSH, EH, IHB, IHS, IHP, RH,SR, ER, IRB, IRS, IRP, SF, IF, and A. We sort the compartments so that the first m compartments correspond to infected individuals.We now letFi(x) be the rate of appearance of new infections in the ith compartments;Vi+(x) be the rate of transfer of individuals into compartment i by all other means, other than the epidemic ones;Vi-(x) be the rate of transfer of individuals out of compartment i.Then the plague disease transmission model in ((8a), (8b), (8c), (8d), (8e), (8f))–(11) is governed by a periodic ordinary differential system given in(27)dxidt=Fit,x-Vit,x≜fit,x,where Vi(x)=Vi-(x)-Vi+(x).We rearrange the system by sorting the infectious classes (EH, IHB, IHS, IHP, ER, IRB, IRS, IRP, IF, A) coming first. We then have(28)Fx=1-σ1ψ1+α1G1t¯SH0001-σ2ψ3+γ1G2t¯SR000ψ2i+βG3t¯SR0,(29)Vx=α2EH+μ1EHα3+μ1+δ1bIHB-α2ν2EHα4+μ1+δ1sIHS-α3ρ3IHB-α2ν3EHα5+μ1+δ1pIHP-α2ν1EH-α3ρ1IHB-α4ξIHSγ2ER+μ3ERγ3+μ3+δ3bIRB-γ2τ3ERγ4+μ3+δ3sIRS-γ2τ2ER-γ31-θIRBμ3+δ3pIRP-γ2τ1Er-γ3θIRB-γ4IRSμ2+δ2IFμ4A-η1tIHPN1-η2tIRPN3+λ4.Then we have(30)Ft=∂Fi∂xjx0,Vt=∂Vi∂xjx0,with 1≤i,j≤10.Now using (30) the matrices F and V are as given below:(31)Fx=00α1Γhsh¯α1Γhph¯00α1Γrsh¯SH0N3α1Γrph¯SH0N3α1Γfh¯SH0N2α1ω1¯SH000000000000000000000000000000000γ1Γhsr¯SR0N1γ1Γhpr¯SR0N100γ1Γrsr¯γ1Γrpr¯γ1Γfr¯SR0N2γ1ω2¯SR00000000000000000000000000000000βΓhbf¯SF0N1βΓhsf¯SF0N100βΓrbf¯SF0N3βΓrsf¯SF0N30000000000000,(32)Vx=v11v12v21v22,where(33)V11=α2+μ10000-α2ν2α3+μ1+δ1b000-α2ν3-α3ρ3α4+μ1+δ1s00-α2ν1-α3ρ1-α4ξα5+μ1+δ1p00000γ2+mu3,V22=γ3+μ3+δ3b0000-γ31-θγ4+μ3+δ3s000-γ3θ-γ4μ3+δ3p00000μ2+δ2000-η2N30μ4V21=0000-γ2τ30000-γ2τ20000-γ2τ100000000-η1N10,V12=0000000000000000000000000.Following the setting by Wang and Zhao [46] and van den Driessche and Watmough [47] for epidemic models we check conditions (A1)–(A7) for plague disease epidemic model. System ((8a), (8b), (8c), (8d), (8e), (8f))–(11) is equivalent to periodic ordinary differential system (27). Now considering this system we can easily see that conditions (A1)–(A5) stated below are satisfied.(A1)
Since each function represents a directed transfer of individuals (human, rodent, flea, and pathogens in the environment), they are all nonnegative. Thus, for each1≤i≤n the functions Fi(t,x), Vi+(t,x), and Vi-(t,x) are nonnegative and continuous on R×R+n and continuously differentiable with respect to x.(A2)
There is a real numberT>0 such that for each 1≤i≤n the functions Fi(t,x), Vi+(t,x), and Vi-(t,x) are T-periodic in t.(A3)
If a compartment is empty, there will be no transfer of individuals out of the compartment by any means. That is to say, ifxi=0 then Vi-=0. In particular if x∈Xs, then Vi-=0 for i=1,…,m.(A4)
The incidence of infection for uninfected compartments is zero. That is to say,Fi=0 for i>m.(A5)
If the population is disease-free then the population will remain free of disease. Thus ifx∈Xs, then Fi=0 and Vi+=0 for i=1,…,m.We know that system (27) has disease-free periodic solution given in (21). Now we define f(t,x(t))=F(t,x(t))-V(t,x(t)) and M(t)=∂fi(ti,E0)/∂xj, 11≤i,j≤14, where fi(t,x(t)) and xi are the ith components of f(t,x(t)) and x, respectively. Now from (28) and (29) we obtain a 4×4 matrix given in(34)At=-μ1ϖ000-ϖ+μ10000-μ30000-μ2.We then letΦA(·)(t) be the monodromy matrix of the linear T-periodic system dz/dt=A(t)z. Then ρΦA(·)(T)<1 which implies that E0 is linearly asymptotically stable in the disease-free subspace Xs=x≥0:xi=0,∀i=1⋯m, where i=1⋯m are the infected compartments. Thus condition (A6) stated below holds.(A6)
The disease-free periodic solution is asymptotically stable in a disease-free subspaceXs; that is, ρΦA(·)(T)<1, where ρΦA(·)(T) is the principal eigenvalue of ΦA(·)(t).Next we setF(t) and V(t) as two 10×10 matrices defined by (30); then using (28) and (29) we get matrices F(t) and V(t) given in (31) and (32), respectively. We can further see that matrix F(t) is nonnegative, and -V(t) is cooperative in the sense that the off-diagonal elements are nonnegative. Let Y(t,s),t≥s, be the evolution operator of our T-periodic system(35)dydt=-Vty.That is, for eachs∈R the 10×10 matrix Y(t,s) satisfies(36)dYt,sdt=-VtYt,s,∀t≥s,Ys,s=Id,whereId is a 10×10 identity matrix. Thus the monodromy matrix ΦV(t) of (35) equals Y(t,0),t≥0. Therefore condition (A7) stated below holds.(A7)
The evolution of individuals in the infectious compartments decays exponentially due to natural and disease induced mortalities. ThusρΦV(T)<1.Now using the standard theory of linear periodic system by Hale [48], there exist K>0 and ℘>0 such that(37)Yt,s≤Ke-℘t-s,∀t≥s,s∈R.We then have(38)Yt,t-aFt-a≤KFt-ae-℘a,∀t∈R,a∈0,∞.Considering the periodic environment we suppose thatΦ(s), T-periodic in s, is the distribution of the new infection at a rate F(s)Φ(s) produced by the infected individuals who were introduced at time s. Given t≥s then Y(t,s)F(s)Φ(s) yields the distribution of those infected individuals who were newly infected at time s and remain in the infected class at t. We then have(39)Ψt=∫-∞tYt,sFsΦsds=∫0∞Yt,t-aFt-aΦt-ada,which is the distribution of accumulative new infections at time t produced by all those infected individual Φ(s) introduced at previous time s to t(s≤t).LetCT be the ordered Banach space of all T-periodic function from R to Rn, which is equipped with the maximum norm ·∞ and the positive cone CT+={Φ∈CTΦ(t)≥0,t∈R}. Define a linear operator L:CT→CT by(40)LΦt=∫0∞Yt,t-aFt-aΦt-ada,∀t∈R,Φ∈CT.Now by Wang and Zhao [46], Diekmann et al. [49], and van den Driessche and Watmough [47] we name L as the next infection operator; then the basic reproduction number RT of the periodic system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) is given:(41)RT=ρL,where ρ(L) is the spectral radius of L.


### 5.1. Characterization ofRT

In this subsection we investigate whether the basic reproduction number in our periodic system can characterize the threshold of the disease invasion. To do this we consider the following linearT-periodic equation:(42)dwdt=-Vt+Ftλw,∀t∈R,with parameter λ∈(0,∞). Let W(t,s,λ),t≥s,s∈R, be the evolution operator of system (42) on R10. We can clearly see that ΦF-V(t)=W(t,0,1), ∀t≥0. Considering matrices (31) and (32) we note that, for each λ∈(0,∞), all off-diagonal elements of matrix -V(t)+F(t)/λ are nonnegative (cooperative matrix). It follows that the linear operator W(t,s,λ) is positive in R10 for each t≥s, s∈R. Now using Perron-Frobenius theorem by Smith and Waltman [50] it entails that ρ(W(T,0,λ)) is an eigenvalue of W(T,0,λ) with a nonnegative eigenvector. Also using matrix similarity concept by Shores [51] we can easily verify that matrix W(s+T,0,λ) is similar to the matrix W(T,0,λ) and hence σ(W(s+T,0,λ))=σ(W(T,0,λ)) for any s∈R, where σ(D) is a spectrum of the matrix D.Proposition 2 (see [46]).
We let (A1)–(A7) hold for system ((8a), (8b), (8c), (8d), (8e), (8f))–(11); then (i)
ifρ(W(T,0,λ))=1 has a positive solution λ0, then λ0 is an eigenvalue of L, and hence RT>0;(ii)
ifRT>0, then λ=RT is the unique solution of ρ(W(T,0,λ))=1;(iii)
RT=0 if and only if ρ(W(T,0,λ))<1∀λ>0.This result shows that, in order to find the basic reproduction number, we need to find the monodromy matrixΦF-V(t) of system (42) and evaluate it. We then find the spectral radius of ΦF-V(T) and solve the equation ρ(ΦF-V(T))=1 for λ which is the basic reproduction number.


### 5.2. Computation of the Basic Reproduction Number

We compute a time-averaged basic reproduction numberR0 using the next-generation matrix as outlined by Wesley and Allen [52], Heesterbeek [53], and Diekmann et al. [49]. The method has the advantage over the usual next-generation method in that the steps to reach an estimate of R0 and the matrix elements of the next-generation matrix have a clear biological basis. It is easy to handle complex diseases like plague disease which has multiple transmission roots from different infectious agents.To do this we first categorize individuals by their state at the moment they become infected (type at infection). These types at infection refer specifically to the birth of the infection in the individual. These categories (types at infection) differ in the way they transmit plague disease which in turn differentiates their ability to produce secondary cases.In our case we categorize the individuals into eight states and label them as follows: human infected with bubonic plague (type 1), human infected with septicemic plague (type 2), human infected with pneumonic plague (type 3), rodent infected with bubonic plague (type 4), rodent infected with septicemic plague (type 5), rodent infected with pneumonic plague (type 6), flea infested with pathogens (type 7), and the pathogens in the environment (type 8).We assume and label individual with bubonic plague as stage one of the disease, septicemic plague as stage two, and pneumonic plague as stage three. We also assume that when an individual in stage one graduates to stage two we only consider the current stage and ignore the latter. We assume that the infection only goes in ascending direction that is from stages one to two, or two to three, but not in the reverse direction.Since the system has eight types at infection, the next-generation matrix,K, will be an 8×8 matrix with elements kij’s. Each of the elements kij’s stands for expected number of new cases of i caused by one infected individual of j. For example, k11 is the expected number of new cases of humans infected with bubonic plague caused by one infected human with bubonic plague.We now define a matrixK whose entries are kij. The resulting next-generation matrix is as given in(43)K=k11k12k13k14k15k16k17k18k21k22k23k24k25k26k27k28k31k32k33k34k35k36k37k38k41k42k43k44k45k46k47k48k51k52k53k54k55k56k57k58k61k62k63k64k65k66k67k68k71k72k73k74k75k76k77k78k81k82k83k84k85k86k87k88.Then,R0=ρ(K), where ρ(K) is spectral radius of K.Some elements equal0 because not all types of infections cause all other types of infection. For example, humans with bubonic plague IHB (type at infection 1) do not produce type at infections 1 (human infected with bubonic plague), 4 (rodent infected with bubonic plague), 5 (rodent infected with septicemic plague), 6 (rodent infected with pneumonic plague), and 8 (pathogens in the environment). This means that k11, k14, k15, k16, and k18 are 0. The type at infection 2 (human infected with septicemic plague) also does not produce type at infections 1 (human infected with bubonic plague), 4 (rodent infected with bubonic plague), 6 (rodent infected with pneumonic plague), and 8 (pathogens in the environment). This also means that k21, k24, k26, and k28 are zero (0). The type at infection 3 does not produce type at infections 1 (human infected with bubonic plague), 2 (human infected with septicemic plague), 4 (rodent infected with bubonic plague), 5, and 7 which means that k31, k32, k34, k35, and k37 are zero. Type at infection 4 does not produce type at infection 1, 2, 3, 4, or 8 which means that k41, k42, k43, k44, and k48 are zero. Type at infection 5 does not produce type at infections 1, 3, 4, and 8; then k51, k53, k54, and k58 are zero. The type at infection 6 does not produce type at infections 1, 2, 4, 5, and 7; thus k61, k62, k64, k65, and k67 are zero. Type at infection 7 also does not produce type at infections 3, 6, 7, and 8; thus k73, k76, k77, and k78 are zero. And the type at infection 8 does not produce type at infections 1, 2, 4, 5, 7, and 8 which means that k81, k82, k84, k85, k87, and k88 are zero. Incorporating these, we modify the matrix K as shown in the following matrix:(44)K=000000k170k21k2200k250k270k31k32k3300k360k38000000k470000k54k550k570000k64k65k660k68k71k720k74k7500000k8300k8600.We will now explain the derivation of each matrix element in detail. We employ the derivation steps by Gail and Benichou [54] to drive the expressions for kij. We mainly base our derivation on the adequate contact rate between the infected individual type j and the susceptible individual type i, the expected duration of infection of individual type j, and the probability that the individual type j survives the duration between the latent stage and the time an individual experiences the onset of clinical disease as in(45)Kij=Effective contact Rate×Duration of infection×Probability that the individual survive the incubation period.The production ofIHB depends on the probability that the total number of fleas that become infectious at the rate of β and the infected immigrants survive the incubation period. We also consider the rate at which IF adequately bites the susceptible human and the bite results in a human infected with bubonic plague IHB at the average value of transmission rate Γfh¯. The total number of humans infected with bubonic plague caused by one flea infested with pathogens is as given in(46)k17=ββ+μ2+ψ2iψ2i+μ2ν2Γfh¯μ2+δ2.Septicemic plague in human may be produced in various ways: progression of untreated human with bubonic plague to human with septicemic plague, adequate contact (including sexual contact) between humans with septicemic plague, adequate contact between rodent and human with septicemic plague, and being acquired from the flea infested with pathogens. We consider the progression rate of infected human with bubonic to septicemicα3ρ3, the adequate contact (it may be sexual contact) rate between humans with septicemic plague, rodent infected with septicemic plague, and the infected flea to human with septicemic plague at the average rates γhsh¯, γrsh¯, and γfh¯. Then the number of humans infected with septicemic plague from all the mentioned infectious agents is as given in(47a)k21=α2α3ν2ρ3α2ν2+μ1μ1+α3+δ1b,(47b)k22=α3ρ3α3ρ3+μ1+α2ν3α2ν3+μ1Γhsh¯α4+μ1+δ1s,(47c)k25=γ2τ2γ2τ2+μ3+γ31-ϕγ31-ϕ+μ3Γrsh¯γ4+μ3+δ3s,(47d)k27=ββ+μ2+ψ2iψ2i+μ2ν1Γfh¯μ2+δ2.The proportionsρ1 and ξ of untreated IHB and IHS may progress and become IHP at the progression rates α3 and α4, respectively. We multiply the average period IHB remain infected by the rate at which they progress to IHP. IHP may also result from the airborne transmission from the human or rodent with pneumonic plague at the average rate γhph¯ or γrph¯, respectively, and through the direct interaction with the environment at the average rate ω1¯. Then the total number of humans infected with pneumonic plague from the stated five sources is given in(48a)k31=α2α3ν2ρ1α2ν2+μ1α3+μ1+δ1b,(48b)k32=α3ρ3α3ρ3+μ1+α2ν3α2ν3+μ1α4ξα4+μ1+δ1s,(48c)k33=α2ν1α2ν1+μ1+α3ρ1α3ρ1+μ1+α4ϕα4ϕ+μ1Γhph¯α5+μ1+δ1p,(48d)k36=γ2τ1γ2τ1+μ3+γ3ϕγ3ϕ+μ3+γ4γ4+μ3Γrph¯μ3+δ3p,(48e)k38=λ4¯λ4¯+μ4¯+η1¯η1¯+μ4¯+η2¯η2¯+μ4¯ω1¯μ4¯.Production of number of rodents with bubonic plagueIRB depends only on the flea infested with pathogens. The infection depends on the infection period of the flea that survives the incubation period and the proportion at which the adequate contact between infected flea and susceptible rodent causes bubonic plague at the average rate τ3Γfr¯ as given in(49)k47=ββ+μ2+ψ2iψ2i+μ2τ3Γfr¯μ2+δ2.The septicemic plague in rodent is produced in three ways; the first way is when infected rodent with bubonic plague progresses and becomes septicemic plague infective agent at the rateγ3(1-ϕ). The second way is after adequate contact (it may also be a rodent eating or biting an infected individual) between the susceptible rodent and a rodent infected with septicemic plague or human at the average rate Γrsr¯ or Γhsr¯, respectively. The third way is from the flea infested with pathogens with the proportion that the adequate contact between IF and the susceptible rodent results in IRS. The total number of IRS infected from these infectious agents is as given in(50a)k52=α3ρ3α3ρ3+μ1+α2ν3α2ν3+μ1Γhsr¯α4+μ1+δ1s,(50b)k54=γ2γ3τ31-ϕγ2τ3+μ3γ3+μ3+δ3b,(50c)k55=γ2τ2γ2τ2+μ3+γ31-ϕγ31-ϕ+μ3Γrsr¯γ4+μ3+δ3s,(50d)k57=ββ+μ2+ψ2iψ2i+μ2τ2Γfr¯μ2+δ2.I
R
P may be the result of airborne transmission between the susceptible rodent and the human and rodent with pneumonic plague at the average rates Γhpr¯ and Γrpr¯, respectively. It may also occur from the progression of untreated IRB and IRS at the rates γ3 and γ4, respectively. The pathogens in environment may also cause IRP after the adequate interaction at the average rate ω2¯. Now the total number of IRB resulting from these interactions is in(51a)k63=α2ν1α2ν1+μ1+α3ρ1α3ρ1+μ1+α4ϕα4ϕ+μ1Γhpr¯α5+μ1+δ1p,(51b)k64=γ2γ3τ3ϕγ2τ3+μ3γ3+μ3+δ3b,(51c)k65=γ2τ2γ2τ2+μ3+γ31-ϕγ31-ϕ+μ3γ4γ4+μ3+δ3s,(51d)k66=γ2τ1γ2τ1+μ3+γ3ϕγ3ϕ+μ3+γ4γ4+μ3Γrpr¯μ3+δ3p,(51e)k68=λ4λ4+μ4+η1¯η1¯+μ4+η2¯η2¯+μ4ω2¯μ4.Fleas are infested with pathogens from humans and rodents infected with bubonic and septicemic plague at the average ratesγhbf¯, γhsf¯, γrbf¯, and γrsf¯. The infection is dictated by the probability that humans and rodents with bubonic and septicemic plague survive the incubation period and the adequate rates of contact. From these interactions, we get the total number of infectious fleas, given in(52a)k71=α2ν2Γhbf¯α2ν2+μ1μ1+α3+δ1b,(52b)k72=α3ρ3α3ρ3+μ1+α2ν3α2ν3+μ1Γhsf¯α4+μ1+δ1s,(52c)k74=γ2τ3Γrbf¯γ2τ3+μ3γ3+μ3+δ3b,(52d)k75=γ2τ2γ2τ2+μ3+γ31-ϕγ31-ϕ+μ3Γrsf¯γ4+μ3+δ3s.The pathogens are released into the environment at the average ratesη1¯ and η1¯ from IHP and IRP, respectively. The released number of pathogens at a given time depends on the infectious period of the rodent and human infected with pneumonic plague and the probability that IHP and IRP survive the incubation period. The total pathogens in soil/environment is as given in(53a)k83=α2ν1α2ν1+μ1+α3ρ1α3ρ1+μ1+α4ξα4ϕ+μ1η1¯α5+μ1+δ1p,(53b)k86=γ2τ1γ2τ1+μ3+γ3ϕγ3ϕ+μ3+γ4γ4+μ3η2¯μ3+δ3p.Each element in the matrixK represents the expected number of secondary cases produced by infected individual j during the entire infectious period of that particular individual into a completely susceptible population i [55].


#### 5.2.1. Basic Reproduction NumberR0

We obtain the average basic reproduction numberR0 by computing the maximum modulus of the eigenvalues of the next-generation matrix K [49, 53]. Now using Maple computing software package, the basic reproduction number is(54)R0=1T∫0Tk22s+k55s4+12A1+1323A4+A53A4+12A2-1323A4-A53A4+A34A1+1/323A4+A5/3A4ds,in which (55)A1=3ϑ3+8ϑ112,A2=3ϑ3-8ϑ16,A3=4ϑ1ϑ3-ϑ33-8ϑ4,A4=13232ϑ13-72ϑ2ϑ1-9ϑ3ϑ4ϑ1+27ϑ42+27ϑ32ϑ2+2ϑ13-72ϑ2ϑ1-9ϑ3ϑ4ϑ1+27ϑ42+27ϑ32ϑ22-4ϑ12+12ϑ2-3ϑ3ϑ431/31/2,A5=23ϑ12+12ϑ2-3ϑ3ϑ4,where (56)ϑ1=k22sk55s-k17sk71s-k27sk72s-k57sk75s,ϑ2=k17sk55sk17sk71s+k21sk72s-k47sk25sk54sk72s+k22sk55sk74s+k54sk75s,ϑ3=-k22s-k55s,ϑ4=k22s+k55sk17sk71s+k47sk74s-k72sk17sk21s-k27sk55s+k25sk57s-k75sk47sk54s-k22sk57s.Since the system has multiple infectious types from multiple hosts, then the next-generation matrix produces the average value of the geometric mean of the number of infections per generation and the basic reproduction number is the average number of secondary infections [56]. It is shown that the basic reproduction number of plague disease depends on the expected number of new cases of humans infected with bubonic plague caused by one infected flea (k17), the expected number of new cases of humans infected with septicemic plague caused by one infected human with bubonic plague (k21), the expected number of new cases of humans infected with septicemic plague caused by one infected human with septicemic plague (k22), the expected number of new cases of rodents infected with bubonic plague caused by one infected flea (k47), the expected number of new cases of rodents infected with septicemic plague caused by one infected rodent with bubonic plague (k54), the expected number of new cases of rodents infected with septicemic plague caused by one infected rodent with septicemic plague (k55), the expected number of new cases of rodents infected with septicemic plague caused by one infected flea (k57), the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with bubonic plague (k71), the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemic plague (k72), the expected number of new cases of fleas infested withYersinia pestis caused by one infected rodent with bubonic plague (k74), and the expected number of new cases of fleas infested withYersinia pestis caused by one infected rodent with septicemic plague (k75). The result may also be interpreted as follows: among all elements of the matrix K, kij, which appear in RO, gives more significant involvement in the dynamics and spread of plague disease.


## 5.1. Characterization ofRT

In this subsection we investigate whether the basic reproduction number in our periodic system can characterize the threshold of the disease invasion. To do this we consider the following linearT-periodic equation:(42)dwdt=-Vt+Ftλw,∀t∈R,with parameter λ∈(0,∞). Let W(t,s,λ),t≥s,s∈R, be the evolution operator of system (42) on R10. We can clearly see that ΦF-V(t)=W(t,0,1), ∀t≥0. Considering matrices (31) and (32) we note that, for each λ∈(0,∞), all off-diagonal elements of matrix -V(t)+F(t)/λ are nonnegative (cooperative matrix). It follows that the linear operator W(t,s,λ) is positive in R10 for each t≥s, s∈R. Now using Perron-Frobenius theorem by Smith and Waltman [50] it entails that ρ(W(T,0,λ)) is an eigenvalue of W(T,0,λ) with a nonnegative eigenvector. Also using matrix similarity concept by Shores [51] we can easily verify that matrix W(s+T,0,λ) is similar to the matrix W(T,0,λ) and hence σ(W(s+T,0,λ))=σ(W(T,0,λ)) for any s∈R, where σ(D) is a spectrum of the matrix D.Proposition 2 (see [46]).
We let (A1)–(A7) hold for system ((8a), (8b), (8c), (8d), (8e), (8f))–(11); then (i)
ifρ(W(T,0,λ))=1 has a positive solution λ0, then λ0 is an eigenvalue of L, and hence RT>0;(ii)
ifRT>0, then λ=RT is the unique solution of ρ(W(T,0,λ))=1;(iii)
RT=0 if and only if ρ(W(T,0,λ))<1∀λ>0.This result shows that, in order to find the basic reproduction number, we need to find the monodromy matrixΦF-V(t) of system (42) and evaluate it. We then find the spectral radius of ΦF-V(T) and solve the equation ρ(ΦF-V(T))=1 for λ which is the basic reproduction number.


## 5.2. Computation of the Basic Reproduction Number

We compute a time-averaged basic reproduction numberR0 using the next-generation matrix as outlined by Wesley and Allen [52], Heesterbeek [53], and Diekmann et al. [49]. The method has the advantage over the usual next-generation method in that the steps to reach an estimate of R0 and the matrix elements of the next-generation matrix have a clear biological basis. It is easy to handle complex diseases like plague disease which has multiple transmission roots from different infectious agents.To do this we first categorize individuals by their state at the moment they become infected (type at infection). These types at infection refer specifically to the birth of the infection in the individual. These categories (types at infection) differ in the way they transmit plague disease which in turn differentiates their ability to produce secondary cases.In our case we categorize the individuals into eight states and label them as follows: human infected with bubonic plague (type 1), human infected with septicemic plague (type 2), human infected with pneumonic plague (type 3), rodent infected with bubonic plague (type 4), rodent infected with septicemic plague (type 5), rodent infected with pneumonic plague (type 6), flea infested with pathogens (type 7), and the pathogens in the environment (type 8).We assume and label individual with bubonic plague as stage one of the disease, septicemic plague as stage two, and pneumonic plague as stage three. We also assume that when an individual in stage one graduates to stage two we only consider the current stage and ignore the latter. We assume that the infection only goes in ascending direction that is from stages one to two, or two to three, but not in the reverse direction.Since the system has eight types at infection, the next-generation matrix,K, will be an 8×8 matrix with elements kij’s. Each of the elements kij’s stands for expected number of new cases of i caused by one infected individual of j. For example, k11 is the expected number of new cases of humans infected with bubonic plague caused by one infected human with bubonic plague.We now define a matrixK whose entries are kij. The resulting next-generation matrix is as given in(43)K=k11k12k13k14k15k16k17k18k21k22k23k24k25k26k27k28k31k32k33k34k35k36k37k38k41k42k43k44k45k46k47k48k51k52k53k54k55k56k57k58k61k62k63k64k65k66k67k68k71k72k73k74k75k76k77k78k81k82k83k84k85k86k87k88.Then,R0=ρ(K), where ρ(K) is spectral radius of K.Some elements equal0 because not all types of infections cause all other types of infection. For example, humans with bubonic plague IHB (type at infection 1) do not produce type at infections 1 (human infected with bubonic plague), 4 (rodent infected with bubonic plague), 5 (rodent infected with septicemic plague), 6 (rodent infected with pneumonic plague), and 8 (pathogens in the environment). This means that k11, k14, k15, k16, and k18 are 0. The type at infection 2 (human infected with septicemic plague) also does not produce type at infections 1 (human infected with bubonic plague), 4 (rodent infected with bubonic plague), 6 (rodent infected with pneumonic plague), and 8 (pathogens in the environment). This also means that k21, k24, k26, and k28 are zero (0). The type at infection 3 does not produce type at infections 1 (human infected with bubonic plague), 2 (human infected with septicemic plague), 4 (rodent infected with bubonic plague), 5, and 7 which means that k31, k32, k34, k35, and k37 are zero. Type at infection 4 does not produce type at infection 1, 2, 3, 4, or 8 which means that k41, k42, k43, k44, and k48 are zero. Type at infection 5 does not produce type at infections 1, 3, 4, and 8; then k51, k53, k54, and k58 are zero. The type at infection 6 does not produce type at infections 1, 2, 4, 5, and 7; thus k61, k62, k64, k65, and k67 are zero. Type at infection 7 also does not produce type at infections 3, 6, 7, and 8; thus k73, k76, k77, and k78 are zero. And the type at infection 8 does not produce type at infections 1, 2, 4, 5, 7, and 8 which means that k81, k82, k84, k85, k87, and k88 are zero. Incorporating these, we modify the matrix K as shown in the following matrix:(44)K=000000k170k21k2200k250k270k31k32k3300k360k38000000k470000k54k550k570000k64k65k660k68k71k720k74k7500000k8300k8600.We will now explain the derivation of each matrix element in detail. We employ the derivation steps by Gail and Benichou [54] to drive the expressions for kij. We mainly base our derivation on the adequate contact rate between the infected individual type j and the susceptible individual type i, the expected duration of infection of individual type j, and the probability that the individual type j survives the duration between the latent stage and the time an individual experiences the onset of clinical disease as in(45)Kij=Effective contact Rate×Duration of infection×Probability that the individual survive the incubation period.The production ofIHB depends on the probability that the total number of fleas that become infectious at the rate of β and the infected immigrants survive the incubation period. We also consider the rate at which IF adequately bites the susceptible human and the bite results in a human infected with bubonic plague IHB at the average value of transmission rate Γfh¯. The total number of humans infected with bubonic plague caused by one flea infested with pathogens is as given in(46)k17=ββ+μ2+ψ2iψ2i+μ2ν2Γfh¯μ2+δ2.Septicemic plague in human may be produced in various ways: progression of untreated human with bubonic plague to human with septicemic plague, adequate contact (including sexual contact) between humans with septicemic plague, adequate contact between rodent and human with septicemic plague, and being acquired from the flea infested with pathogens. We consider the progression rate of infected human with bubonic to septicemicα3ρ3, the adequate contact (it may be sexual contact) rate between humans with septicemic plague, rodent infected with septicemic plague, and the infected flea to human with septicemic plague at the average rates γhsh¯, γrsh¯, and γfh¯. Then the number of humans infected with septicemic plague from all the mentioned infectious agents is as given in(47a)k21=α2α3ν2ρ3α2ν2+μ1μ1+α3+δ1b,(47b)k22=α3ρ3α3ρ3+μ1+α2ν3α2ν3+μ1Γhsh¯α4+μ1+δ1s,(47c)k25=γ2τ2γ2τ2+μ3+γ31-ϕγ31-ϕ+μ3Γrsh¯γ4+μ3+δ3s,(47d)k27=ββ+μ2+ψ2iψ2i+μ2ν1Γfh¯μ2+δ2.The proportionsρ1 and ξ of untreated IHB and IHS may progress and become IHP at the progression rates α3 and α4, respectively. We multiply the average period IHB remain infected by the rate at which they progress to IHP. IHP may also result from the airborne transmission from the human or rodent with pneumonic plague at the average rate γhph¯ or γrph¯, respectively, and through the direct interaction with the environment at the average rate ω1¯. Then the total number of humans infected with pneumonic plague from the stated five sources is given in(48a)k31=α2α3ν2ρ1α2ν2+μ1α3+μ1+δ1b,(48b)k32=α3ρ3α3ρ3+μ1+α2ν3α2ν3+μ1α4ξα4+μ1+δ1s,(48c)k33=α2ν1α2ν1+μ1+α3ρ1α3ρ1+μ1+α4ϕα4ϕ+μ1Γhph¯α5+μ1+δ1p,(48d)k36=γ2τ1γ2τ1+μ3+γ3ϕγ3ϕ+μ3+γ4γ4+μ3Γrph¯μ3+δ3p,(48e)k38=λ4¯λ4¯+μ4¯+η1¯η1¯+μ4¯+η2¯η2¯+μ4¯ω1¯μ4¯.Production of number of rodents with bubonic plagueIRB depends only on the flea infested with pathogens. The infection depends on the infection period of the flea that survives the incubation period and the proportion at which the adequate contact between infected flea and susceptible rodent causes bubonic plague at the average rate τ3Γfr¯ as given in(49)k47=ββ+μ2+ψ2iψ2i+μ2τ3Γfr¯μ2+δ2.The septicemic plague in rodent is produced in three ways; the first way is when infected rodent with bubonic plague progresses and becomes septicemic plague infective agent at the rateγ3(1-ϕ). The second way is after adequate contact (it may also be a rodent eating or biting an infected individual) between the susceptible rodent and a rodent infected with septicemic plague or human at the average rate Γrsr¯ or Γhsr¯, respectively. The third way is from the flea infested with pathogens with the proportion that the adequate contact between IF and the susceptible rodent results in IRS. The total number of IRS infected from these infectious agents is as given in(50a)k52=α3ρ3α3ρ3+μ1+α2ν3α2ν3+μ1Γhsr¯α4+μ1+δ1s,(50b)k54=γ2γ3τ31-ϕγ2τ3+μ3γ3+μ3+δ3b,(50c)k55=γ2τ2γ2τ2+μ3+γ31-ϕγ31-ϕ+μ3Γrsr¯γ4+μ3+δ3s,(50d)k57=ββ+μ2+ψ2iψ2i+μ2τ2Γfr¯μ2+δ2.I
R
P may be the result of airborne transmission between the susceptible rodent and the human and rodent with pneumonic plague at the average rates Γhpr¯ and Γrpr¯, respectively. It may also occur from the progression of untreated IRB and IRS at the rates γ3 and γ4, respectively. The pathogens in environment may also cause IRP after the adequate interaction at the average rate ω2¯. Now the total number of IRB resulting from these interactions is in(51a)k63=α2ν1α2ν1+μ1+α3ρ1α3ρ1+μ1+α4ϕα4ϕ+μ1Γhpr¯α5+μ1+δ1p,(51b)k64=γ2γ3τ3ϕγ2τ3+μ3γ3+μ3+δ3b,(51c)k65=γ2τ2γ2τ2+μ3+γ31-ϕγ31-ϕ+μ3γ4γ4+μ3+δ3s,(51d)k66=γ2τ1γ2τ1+μ3+γ3ϕγ3ϕ+μ3+γ4γ4+μ3Γrpr¯μ3+δ3p,(51e)k68=λ4λ4+μ4+η1¯η1¯+μ4+η2¯η2¯+μ4ω2¯μ4.Fleas are infested with pathogens from humans and rodents infected with bubonic and septicemic plague at the average ratesγhbf¯, γhsf¯, γrbf¯, and γrsf¯. The infection is dictated by the probability that humans and rodents with bubonic and septicemic plague survive the incubation period and the adequate rates of contact. From these interactions, we get the total number of infectious fleas, given in(52a)k71=α2ν2Γhbf¯α2ν2+μ1μ1+α3+δ1b,(52b)k72=α3ρ3α3ρ3+μ1+α2ν3α2ν3+μ1Γhsf¯α4+μ1+δ1s,(52c)k74=γ2τ3Γrbf¯γ2τ3+μ3γ3+μ3+δ3b,(52d)k75=γ2τ2γ2τ2+μ3+γ31-ϕγ31-ϕ+μ3Γrsf¯γ4+μ3+δ3s.The pathogens are released into the environment at the average ratesη1¯ and η1¯ from IHP and IRP, respectively. The released number of pathogens at a given time depends on the infectious period of the rodent and human infected with pneumonic plague and the probability that IHP and IRP survive the incubation period. The total pathogens in soil/environment is as given in(53a)k83=α2ν1α2ν1+μ1+α3ρ1α3ρ1+μ1+α4ξα4ϕ+μ1η1¯α5+μ1+δ1p,(53b)k86=γ2τ1γ2τ1+μ3+γ3ϕγ3ϕ+μ3+γ4γ4+μ3η2¯μ3+δ3p.Each element in the matrixK represents the expected number of secondary cases produced by infected individual j during the entire infectious period of that particular individual into a completely susceptible population i [55].


### 5.2.1. Basic Reproduction NumberR0

We obtain the average basic reproduction numberR0 by computing the maximum modulus of the eigenvalues of the next-generation matrix K [49, 53]. Now using Maple computing software package, the basic reproduction number is(54)R0=1T∫0Tk22s+k55s4+12A1+1323A4+A53A4+12A2-1323A4-A53A4+A34A1+1/323A4+A5/3A4ds,in which (55)A1=3ϑ3+8ϑ112,A2=3ϑ3-8ϑ16,A3=4ϑ1ϑ3-ϑ33-8ϑ4,A4=13232ϑ13-72ϑ2ϑ1-9ϑ3ϑ4ϑ1+27ϑ42+27ϑ32ϑ2+2ϑ13-72ϑ2ϑ1-9ϑ3ϑ4ϑ1+27ϑ42+27ϑ32ϑ22-4ϑ12+12ϑ2-3ϑ3ϑ431/31/2,A5=23ϑ12+12ϑ2-3ϑ3ϑ4,where (56)ϑ1=k22sk55s-k17sk71s-k27sk72s-k57sk75s,ϑ2=k17sk55sk17sk71s+k21sk72s-k47sk25sk54sk72s+k22sk55sk74s+k54sk75s,ϑ3=-k22s-k55s,ϑ4=k22s+k55sk17sk71s+k47sk74s-k72sk17sk21s-k27sk55s+k25sk57s-k75sk47sk54s-k22sk57s.Since the system has multiple infectious types from multiple hosts, then the next-generation matrix produces the average value of the geometric mean of the number of infections per generation and the basic reproduction number is the average number of secondary infections [56]. It is shown that the basic reproduction number of plague disease depends on the expected number of new cases of humans infected with bubonic plague caused by one infected flea (k17), the expected number of new cases of humans infected with septicemic plague caused by one infected human with bubonic plague (k21), the expected number of new cases of humans infected with septicemic plague caused by one infected human with septicemic plague (k22), the expected number of new cases of rodents infected with bubonic plague caused by one infected flea (k47), the expected number of new cases of rodents infected with septicemic plague caused by one infected rodent with bubonic plague (k54), the expected number of new cases of rodents infected with septicemic plague caused by one infected rodent with septicemic plague (k55), the expected number of new cases of rodents infected with septicemic plague caused by one infected flea (k57), the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with bubonic plague (k71), the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemic plague (k72), the expected number of new cases of fleas infested withYersinia pestis caused by one infected rodent with bubonic plague (k74), and the expected number of new cases of fleas infested withYersinia pestis caused by one infected rodent with septicemic plague (k75). The result may also be interpreted as follows: among all elements of the matrix K, kij, which appear in RO, gives more significant involvement in the dynamics and spread of plague disease.


## 5.2.1. Basic Reproduction NumberR0

We obtain the average basic reproduction numberR0 by computing the maximum modulus of the eigenvalues of the next-generation matrix K [49, 53]. Now using Maple computing software package, the basic reproduction number is(54)R0=1T∫0Tk22s+k55s4+12A1+1323A4+A53A4+12A2-1323A4-A53A4+A34A1+1/323A4+A5/3A4ds,in which (55)A1=3ϑ3+8ϑ112,A2=3ϑ3-8ϑ16,A3=4ϑ1ϑ3-ϑ33-8ϑ4,A4=13232ϑ13-72ϑ2ϑ1-9ϑ3ϑ4ϑ1+27ϑ42+27ϑ32ϑ2+2ϑ13-72ϑ2ϑ1-9ϑ3ϑ4ϑ1+27ϑ42+27ϑ32ϑ22-4ϑ12+12ϑ2-3ϑ3ϑ431/31/2,A5=23ϑ12+12ϑ2-3ϑ3ϑ4,where (56)ϑ1=k22sk55s-k17sk71s-k27sk72s-k57sk75s,ϑ2=k17sk55sk17sk71s+k21sk72s-k47sk25sk54sk72s+k22sk55sk74s+k54sk75s,ϑ3=-k22s-k55s,ϑ4=k22s+k55sk17sk71s+k47sk74s-k72sk17sk21s-k27sk55s+k25sk57s-k75sk47sk54s-k22sk57s.Since the system has multiple infectious types from multiple hosts, then the next-generation matrix produces the average value of the geometric mean of the number of infections per generation and the basic reproduction number is the average number of secondary infections [56]. It is shown that the basic reproduction number of plague disease depends on the expected number of new cases of humans infected with bubonic plague caused by one infected flea (k17), the expected number of new cases of humans infected with septicemic plague caused by one infected human with bubonic plague (k21), the expected number of new cases of humans infected with septicemic plague caused by one infected human with septicemic plague (k22), the expected number of new cases of rodents infected with bubonic plague caused by one infected flea (k47), the expected number of new cases of rodents infected with septicemic plague caused by one infected rodent with bubonic plague (k54), the expected number of new cases of rodents infected with septicemic plague caused by one infected rodent with septicemic plague (k55), the expected number of new cases of rodents infected with septicemic plague caused by one infected flea (k57), the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with bubonic plague (k71), the expected number of new cases of fleas infested withYersinia pestis caused by one infected human with septicemic plague (k72), the expected number of new cases of fleas infested withYersinia pestis caused by one infected rodent with bubonic plague (k74), and the expected number of new cases of fleas infested withYersinia pestis caused by one infected rodent with septicemic plague (k75). The result may also be interpreted as follows: among all elements of the matrix K, kij, which appear in RO, gives more significant involvement in the dynamics and spread of plague disease.


## 6. Numerical Results and Discussion

Here we use the parameters values of model system ((8a), (8b), (8c), (8d), (8e), (8f))–(11) given in Table 1 to study the transmission trend of plague disease. Simulation results are given to show the effect of different parameters on the periodic reproduction number. We have also chosen temperature data obtained from Tanga region from January to December 2013 to show the seasonal distribution in the number of secondary cases of plague infections. It is observed that simulation results from time-averaged seasonal parameters and those seasonal parameters treated mathematically as sinusoidal functions match the real seasonal fluctuation data (temperature).Results shows that the increase in number of individuals infected with bubonic plague, to a large extent, affects the increases in number of individuals with septicemic and pneumonic plague disease. This is due to the fact that individuals with bubonic plague progress and become either septicemic or pneumonic plague infective agents. This in turn leads to the significant increase of plague disease transmission rate and the average number of secondary infections. Figures2 and 1 show how the progression rates from individuals with bubonic plague to individuals with septicemic plague affect the average number of secondary infections in human beings and rodents, respectively. It is illustrated that the increase of human beings and rodents progressing to become septicemic plague infective agents affects the disease dynamics by increasing the average number of secondary infections. We see similar result when we evaluate the periodic reproduction number based on the temperature data from Tanga region (Figures 1(c) and 2(c)) and time-averaged parameters (Figures 1(b) and 2(b)) for human beings and rodents, respectively. These findings necessitate the need for early treatment of plague disease infective agents especially the primary forms (bubonic and septicemic plague) before they progress to highly fatal and fast spreading plague forms like pneumonic plague disease. It is thus important for the government and other health stakeholders to ensure the availability of effective plague disease treatment especially in high risk areas.Figure 1
Effect of progression rates fromIHB to IHS on the periodic reproduction number.
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(c)Figure 2
Effect of progression rates fromIRB to IRS on the periodic reproduction number.
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(c)Increase of plague disease transmission through flea bite in human beings and rodents populations alters the whole dynamics of plague disease. Results in Figures3 and 4 show the effect of infection from infectious flea to human beings with bubonic and septicemic plague on the average number of secondary infections. The infection from flea to rodents with bubonic and septicemic plague disease also shows the significant effect on RT as illustrated in Figures 5 and 6, respectively. The results generally show that when the periodic infection rates from flea increase, those of the infectious human beings and rodent increase as well; this in turn affects the general plague disease periodic transmission and spread. These results are in conjuncture with what is observed when RT is evaluated using the temperature data and time-averaged seasonal parameter as in Figures 3(c), 3(b), 5(c), and 5(b) for human beings and rodents, respectivelyFigure 3
The effect of infection fromIF to human beings with bubonic plague on the periodic reproduction number.
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The effect of infection fromIF to human beings with septicemic plague on the periodic reproduction number.
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(c)Figure 5
The effect of infection fromIF to rodents with bubonic plague on the periodic reproduction number.
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(c)Figure 6
The effect of infection fromIF to rodents with septicemic plague on the periodic reproduction number.
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(c)The results call attention for the need to control the number of infectious fleas and flea population in general as the way of controlling the plague disease. The study recommends that for the appropriate and most effective way to control flea population we first need to study the flea’s ecology and its local patterns of disease transmission. One of the most important and cost-effective strategies of controlling the vector flea is environmental management strategies that can reduce or eliminate vector breeding grounds. For example, in residential areas people must be educated to improve their surrounding environment in a way that does not favor the survival and growth of vector flea. This may be through improving the design of water systems, improving waste disposal and water storage, discouraging deforestation and loss of biodiversity, and living in well ventilated housing that is not close to animals.There are also biological control tools like bacterial larvicides and larvivorous fish that may be used to control flea population [57]. These control methods aim at killing vector larvae without generating the ecological impacts as they do not use chemicals. Another strategy is using chemical methods, which mainly shorten the lifespan of vectors. These tools include indoor residual sprays, space spraying, and use of chemical larvicides and adulticides. Since most of these methods have side effects to the environmental ecology they are recommended to be used when other safe strategies fail. Moreover, even though these chemicals are not environmentally friendly, we advise the environmental stakeholders to recommend the use of chemical methods of vector control that reinforces linkages between health and environment.Reducing the number of flea population will reduce the infection rates to human beings and rodents and ultimately reduce the number of secondary infections. Figure7(a) shows how reducing the number of infectious fleas reduces the number of secondary infections. This is also true when the parameters that are affected by seasonal weather variation are evaluated using the temperature data in Tanga (Figure 7(c)) and using the time-averaged seasonal parameters (Figure 7(b)). This result is in light of the fact that the reduction of flea population will reduce the number of individuals with bubonic and septicemic plague and as a result reduce the number of pneumonic plague infective agents that result from the progression of individual with bubonic and septicemic plague.Figure 7
Effect of increased number of fleas’ death rate on the periodic reproduction number.
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(c)The reduction of flea population will reduce not only the infection from flea to other individuals but also the rate at which flea gets the disease from other individuals (human beings and rodents). When the flea population is reduced it will as a result reduce the interaction between susceptible fleas and other infectious individuals and vice versa. The number of fleas getting the disease increases with the increase of the rate at which fleas acquire infection from infectious human beings with bubonic plague (see Figure8(a)) and those with septicemic plague (see Figure 9(a)). We further observe that the increase of infectious fleas may be contributed by the infectious rodents with bubonic plague (see Figure 10(a)) and those with septicemic plague (see Figure 11(a)). We can also observe the similar results when the parameters are evaluated based on the temperature data in Tanga region and when the parameters are timely averaged as in Figures 8(c), 8(b), 10(c), and 10(b) for human beings and rodents, respectively. Therefore, using these results, we settle to the point that increasing transmission rate in flea population from human beings and rodents with bubonic and septicemic plague raises the average number of secondary plague disease infections.Figure 8
The effect of increased infection rate to fleas from the infectious human beings (IHB) on the periodic reproduction number.
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(c)Figure 9
The effect of increased infection rate to fleas from the infectious human beings (IHS) on the periodic reproduction number.
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(c)Figure 10
The effect of increased infection rate to fleas from the infectious rodents (IRB) on the periodic reproduction number.
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(c)Figure 11
The effect of increased infection rate to fleas from the infectious rodents (IRS) on the periodic reproduction number.
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(c)Physical contact that includes sexual contact between two infectious individuals (human beings and rodents) may lead to septicemic plague. The increase in the number of individuals with septicemic plague affects the general dynamics of plague disease, particularly the average number of secondary infections. It is illustrated that increasing infection rate from a human being with septicemic plague to the other susceptible human (see Figure12(a)) and from rodent with septicemic plague to the susceptible rodents (see Figure 13(a)) increases the average number of secondary infections. The result again shows a clear correlation when parameters are evaluated based on the temperature data from Tanga region (see Figures 12(c) and 13(c)) and when they are averaged (see Figures 12(b) and 13(b)) for human beings and rodents, respectively. This shows the necessity to educate human beings to practice safe sex using protective gears and taking necessary precaution when handling people or animals with septicemic plague. This also tells us that there is a necessity to quarantine people and animals that immigrate from areas that are infected by septicemic plague so that they do not affect other human beings or animals and thus increasing the endemicity of the disease.Figure 12
Effect of infection rate (Γhsh) on the periodic reproduction number.
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(c)Figure 13
Effect of infection rate (Γrsr) on the periodic reproduction number.
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(c)The distribution of the basic reproduction number is based on the seasonal weather condition of a particular area at a particular time. This is what causes the unpredictability of the number of secondary cases of plague disease infection (bubonic, septicemic, and pneumonic plague) as it will change whenever the weather conditions change. We evaluate the distribution of the basic reproduction number based on the data we obtained on daily temperature (°C) and humidity (%) from five regions in Tanzania from January to December 2013. Figures14, 15, 16, 17, and 18 show the seasonal distribution of basic reproduction number when evaluated based on the temperature and humidity data from Kigoma, Mbeya, Mtwara, Singida, and Tanga regions, respectively.Figure 14
Distribution ofR0 based on fluctuation of temperature (a) and relative humidity (b) data in Kigoma.
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Distribution ofR0 based on fluctuation of temperature (a) and relative humidity (b) data in Mbeya.
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Distribution ofR0 based on fluctuation of temperature (a) and relative humidity (b) data in Mtwara.
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Distribution ofR0 based on fluctuation of temperature (a) and relative humidity (b) data in Singida.
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(b)Figure 18
Distribution ofR0 based on fluctuation of temperature (a) and relative humidity (b) data in Tanga.
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(b)The features displayed in these results clearly show how seasonal weather fluctuation can have significant effects on the number of secondary cases of plague disease. It can be noted that there is a seasonal pattern in new plague disease infection cases. We therefore vindicate the fact that the number of plague disease infective agents peaks whenever the weather condition is favorable for plague disease transmission and it drops when the weather condition does not favor plague disease transmission.


## 7. Conclusion

The transmission of plague disease occurs in several pathways which makes the modeling of the disease challenging and very complex. Moreover all ways that lead to plague disease transmission are directly or indirectly affected by seasonal weather variation which causes seasonality in plague disease epidemic. The effect of seasonal weather variation has been a glowing concern in different epidemiological studies. This in turn dictates that in order to study the dynamics and propose the effective control strategies of the plague disease we must incorporate the effect of seasonal weather variation. In this study we have analysed the plague disease model incorporated with the factors that are affected by the seasonal weather variation in order to study its effects on the dynamics of the plagues disease. We have computed basic reproduction number and depicted how it can be affected by seasonal weather variation through numerical simulation. We were able to deduce that progression rates from one primary form to secondary form of plague infection, flea’s infection rate, and the vector flea abundance pose the significant effect on the increase of the average number of secondary cases of plague infection. Therefore the effective control strategies must take into account these factors as they have been shown to have a significant contribution to the increase of the average number of secondary cases of plague infection.



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*Source: 5058085-2017-08-10.xml*