Mathematical Modeling and Stability Analyses of Lassa Fever Disease with the Introduction of the Carrier Compartment

In this paper, a new mathematical model which takes into account the human and vector populations together with their interactions during Lassa fever disease transmission was developed. This transmission process is denoted by a seven mutually exclusive compartments for the human and vector populations. The proposed model is used to introduce the incubation period of the disease, a period in which an infected individual is yet to be symptomatic but infectious however, as denoted by the carrier human compartment. This carrier compartment was critically examined for its short and long term effects on the spread and control of the disease. Local and global stability analyses of the equilibrium points of the model was carried out using the first generation matrix approach and the direct Lyapunov method respectively. These analyses showed that the disease free equilibrium point of the developed model is locally asymptotically stable but not globally asymptotically stable. It was also observed that, although, there exist a unique endemic equilibria for the disease, this equilibria however is not stable. Numerical simulations of the model were carried out by implementing the MATLAB ODE45 algorithm for solving non-stiff ordinary differential equations. The results of these simulations are the effects of the various model parameters on each compartment of the developed model. Based on the findings of this research, necessary recommendations were made for the applications of the model to an endemic area.

The most recent Lassa fever outbreak in Nigeria was between January 28 and February 3rd, 2019 with a record of 68 new confirmed cases and 14 new deaths, making it a total of 275 confirmed cases and 57 confirmed deaths in the first quarter of 2019 alone [1]. Consequently, the federal government of Nigeria declared a LF outbreak in the country as stated in a signed statement by the Director-General of the NCDC, Dr Chikwe Ihekweazu [25]. He further added that this outbreak usually occur during the Nigerian dry season, ranging from January to April of each year. The mode of transmission of LASV is through direct contact with an infected rodent or via ingestion of food which has been contaminated by the droppings, such as excreta, urine, or saliva, of an infected rodent. Most people contract LF via anything contaminated with rat urine, faeces, blood, saliva or through eating, drinking or simply handling contaminated objects in the home. LF can also be transmitted via direct contact with the blood, urine, faeces, pharyngeal secretions or other body secretions of an infected person.
The incubation period of Lassa fever, a period in which an infected person do not exhibit any symptom of the infectiousness, is a duration of 6 to 21 days from the onset of the infection. Fever, general weakness and malaise are usually the first symptoms to appear in symptomatic patients. Moreover, Lassa fever disease, at its symptomatic stage, is quite difficult to distinguish from many other diseases which cause fever, including malaria, shigellosis, typhoid fever, yellow fever and other viral hemorrhagic fevers. Moreover, in contrast with some other viral diseases, such as the Human Immune-deficiency virus (HIV), of which the Center for Disease Control and Prevention, [9], states specifically that it cannot be transmitted via water, saliva, tears, or sweat, due to its very short lifespan outside the host, LASV, however, according to the Pathogen Safety Data Sheet, [22], is stable as an aerosol and has a biological half-life, between 24 0 and 32 0 , of 10.1 to 54.6 minutes outside host. This implies that contact with the secretions of an infected rodent or human over this period of time could still lead to the virus been transmitted.
Although, Mastomys atalensis species complex are widely distributed in sub-Sahara Africa, the ones harboring Lassa virus are the autochthonous of West Africa and that is why Lassa virus still spreads only in West Africa Countries [16]. However, there has been a report of some Lassa fever cases being "imported" into the U.S. and U.K. via travelers who acquired the disease elsewhere [13]. Presently, there is no licensed vaccine or Food and Drug Administration approved treatment against LASV [12]. However, replication-competent attenuated vaccine, one of which is Ribavirin, remains the most effective treatment against the disease especially if administered during the early stage of the infectiousness. In addition, LF disease can be prevented by practicing good hygiene, proper sanitation of the environment and the use of rodent-proof containers in storing food. [15] investigated the prevalence of LF disease in Northern part of Edo State, Nigeria with a high rate of infection on contact persons. The results of their investigations showed that, to control the spread of the virus, the average number of new secondary infection(s) generated by a single infected individual/rodent during their infectious period, 0 , must be brought below one. Consequently, [2] developed a mathematical model for the transmission of Lassa fever with isolation of infected individuals and obtained the basic reproduction number, 0 . The analysis showed that the disease free equilibrium (DFE) is locally and globally asymptotically stable whenever the threshold quantity, 0 , is less than unity ( 0 < 1). They also concluded that the endemic equilibrium point, a positive steady state solution when the disease persists in the population, of the model exists under certain condition. Similarly, [18] developed a mathematical model for the transmission of LF. This model was used to analyze the existence and stability of the DFE of LF disease. They, however, concluded that though the DFE is globally asymptotically stable (with 0 < 1), the disease will still continue to spread. Subsequently, [17] modeled the transmission of Lassa fever virus between humans and rodents with control strategies as a six-dimensional ordinary differential equation. Stability analysis of the DFE was performed and the basic reproduction number obtained using the next generation operator approach. The existence of endemic equilibrium was further determined. The study, then, concluded that more awareness should be conducted in the affected areas so as to prevent more outbreaks of the disease. [3] modeled the control of LF disease using an SITR model. The results of their investigation showed that, though there exist an endemic equilibria for the disease, the disease can still be controlled by using appropriate control strategies. As an extension of this model, we introduced the Carrier human compartment together with new model parameters and control strategies into the previously existing model, as discussed below, in order to have a more realistic model which is closer to what is obtainable in the real life situation.

Model Description and Formulation
Mathematical modeling is one of the most important tools used in understanding the dynamics of disease transmission. In the proposed model, in order to indicate individuals with unique mutually exclusive natures, we considered five mutually exclusive compartments for the human population in relation to the disease status. These compartments are: The Susceptible ℎ ( ), the Carriers ℎ ( ), the Infected ℎ ( ), the Treated ℎ ( ) and the Recovered ℎ ( ) human populations. For the rodent population, we considered two compartments namely: The Susceptible ( ) and the Infected ( ) vector populations.
The susceptible human compartment is made up of members who are not yet infected but stand a chance of contracting the infection if exposed to an infected individual. This is mostly because they live within a community in which the virus exists or has been previously reported. The carrier compartment consists of individuals who have the infection but do not show any clinical/noticeable symptoms even though they are infectious. The infected human compartment is made up of individuals who are with the fully blown infection with symptomatic evidence i.e they have survived the 6-21 days incubation period of the disease. The treated compartment is made up of individuals who are not only being treated but have been isolated from the other members of the community, hence are not infectious. The recovered compartment is made up of individuals who have either undergone treatment and have now fully recovered from the infection or have recovered by their own natural immunity.
We shall incorporate new model parameters such as: the rate of progression from the carrier class of the human population to the infected class to be denoted by ℎ , contracting rate for susceptible human population via interaction with the carriers to be denoted by and the rate of recovery of the carrier class by natural immunity to be denoted by 1 into the existing model of [3].

Rate of Recovery of Infected Human by Natural Immunity Treatment Factor
Model Assumptions: • The treated class are isolated from the remaining classes and are thus not infectious.
• Infected Rodents do not recover throughout their entire life-time.
• Treatment is given only to infected individuals.
• Infected Rodents are not given any form of Rodenticide.
• Each compartment in the model is made up of individuals or vectors with homogeneous characteristic (disease status). Model Equations:
From equations 2.1 to 2.7; Similarly, solving (3.2), we obtain; Hence, the feasible set of the solution of the model equations enter and remain in the invariant region:
Accordingly, since the threshold parameter 0 = 0.46252994 < 1, then the DFE is LAS and the EEP is unstable, hence the disease cannot invade the population. However, the existence of the Endemic equilibria implies that the disease may persist in the population.

Global Asymptotic Stability Analysis
In order to examine the DFE for global stability, we shall employ the procedure implemented by [8]. We shall denote the Lassa fever model by: where = ( ℎ , ℎ , ) denotes the uninfected population and = ( ℎ , ℎ , ℎ , ) denotes the Infected population.

Numerical Simulations and Discussion of Results
In order to solve the model equations numerically, we implemented the MATLAB ODE45 algorithm for the developed model, and plotted the graphs of each model compartment against time, with time ranging from 0 to 150 days, we obtained the following results: Figure 4.1 indicates that the susceptible human population decreases rapidly within the first few days due to the awareness/sensitivity of the affected population towards the infection and also because of the progression from the susceptible class to the carrier class. However, after these few days, an inflexion point is reached and then the population begins to increase steadily for the remaining 100 days due to the loss of immunity of the recovered human population. This result is in agreement with the results of [20] and [6]. It can be observed that an increase in the contracting rate of the susceptible class via contact with the carrier class, , leads to a decrease in the population of this compartment.

Carrier Human Population
In figure 4.2 to 4.4, the introduced carrier compartment reduces drastically in size from 40 to 19 during the first 5 days. This is unsurprisingly due to the progression of the carriers to the infected compartment since the incubation period of the disease is relatively short (6 to 21 days) and also due to the disease-induced death of the class. Thereafter, the compartment begins to increase in its population from 19 to 25 within a duration of 20 days due to the inflow from the susceptible compartment. Subsequently, the population was maintained at 25 due to the balance achieved in the inflow and outflow of humans in this compartment. It can be observed that an increase in the rate of recovery of the Carrier class, 1 and the progression rate from the carrier class to the infected class, ℎ , lead to a decrease in the population of the carrier class. Meanwhile an increase in the contracting rate of the susceptible class via contact with the carrier class, , leads to an increase in the population of the carriers.   literatures. According to [3] and [20], the infected human compartment experiences a rapid increase between time 0 and 5. This increase was ascribed to the progression from the susceptible human population to the infected human population. This, however, is not true in our case as there do not exist a progression into the infected human compartment directly from the susceptible human compartment. It can be observed that an increase in the progression rate from the carrier class to the infected class, ℎ , lead to an increase in the population of the infected class.  .6 shows that the treated human population increases rapidly within the first 0 days due to the progression from the infected class into this class. Thereafter, there exist little decrease in the population due to the recovery of the treated humans. 45 member of the treated human population was then maintained for the remaining period of the experiment due to the steady inflow and outflow within the population ascribed to the progression from the infected class and the progression into the recovered class respectively. This result is in sharp agreement with that of [3].  59 of the experiment due to its population by the treated humans who have recovered. Thereafter, there is a little decrease in this compartment ascribed to the progression of the recovered human population into the susceptible class due to loss of immunity. This decrease, however, does not last too long as the treated humans continues to move into the recovered class due to the efficacy of the treatment given. Hence, the recovered human population remains maintained at a stable state after the first 50 days. This result is in agreement with that of [3] but not with that of [4] since they did not consider immunity loss in their model construction. It can be observed that an increase in the rate of recovery of the Carrier class, 1 leads to an increase in the population of the recovered class.

.6 Susceptible Vector Population
From figure 4.8, the susceptible vector population decreases steadily and continues to decrease for a long period of time due to the progression into the infected vector compartment. After this period, however, the susceptible compartment is maintained at a steady state since it was continually populated either by new births or by recruitments from outside the studied population. However, since there do not exist the use of Rodenticide, decrease in this compartment was only due to natural death or the progression into the infected compartment. This result is in slight disagreement with that of [3] and [4] since they considered Rodenticide factor in their model. The use of Rodenticide, however, shall be introduced as a control parameter during the formulation of the optimal control problem.

Infected Vector Population
From figure 4.9, the infected vector population increases rapidly during the first 40 days due to its population by the susceptible vector population and thereafter decreases steadily due to the disease induced death rate and the natural death rate of the vectors. This decrease, however was not progressive since there was no use of Rodenticide nor was their recovery from the infectiousness. This is in contrast with the result of [3], in which there was a rapid decrease in this population due to the application of Rodenticide.

Conclusion and Recommendations
The results of the analyses of the developed model indicate that the disease free equilibrium of Lassa fever is stable while the endemic equilibrium point is unstable. These results imply that it is possible to stop the growth and spread of this disease within a studied population provided the assumptions and parameters of the developed model are implemented within such population including the isolation and treatment of the infected class. Similarly, from the simulation of the model, we observe that the carriers and infected class are maintained at a relatively low value by the end of the 150 observed days. However, the infected class is maintained at 17 which is relatively lower than the carrier class which was maintained at 29. This result is because of the availability of treatment for the infected class but not for the carrier class.
Hence, an OCP model which takes into consideration early diagnosis/early treatment of the carrier class alongside other control parameters should be developed and an optimal control application should be made on such model. Also, sensitivity analyses should be performed on the developed model in order to see the specific effect of each parameter of the model on the spread and control of the disease.