key: cord-0756562-hokrmcjq
authors: Ssebuliba, J.; Nakakawa, J. N.; Ssematimba, A.; Mugisha, J.Y.T.
title: Mathematical modelling of COVID-19 transmission dynamics in a partially comorbid community
date: 2021-11-24
journal: Partial Differential Equations in Applied Mathematics
DOI: 10.1016/j.padiff.2021.100212
sha: 0252da2dd9102613d481f19d492e767a98c1cb71
doc_id: 756562
cord_uid: hokrmcjq

A deterministic S , E m , E c , I m , I c , H , R epidemic model that describes the spreading of SARS-COV-2 within a community with comorbidities is formulated. Size dependent area is incorporated into the model to quantify the effect of social distancing and the results indicate that the risk of community transmission is optimumly minimised when the occupancy area is increased. The reproduction number is shown to have a positive relationship with the infection rate, the proportion of individuals with comorbidities and the proportion of susceptible individuals adhering to standard operating procedures. The model exhibits a unique endemic equilibrium whose stability largely depends on the rate of hospitalisation of individuals with underlying health conditions ( ω m ) as compared to those without these conditions ( ω c ), such that stability is guaranteed if ω m < ω c . Furthermore, if individuals with comorbidities effectively report for treatment and hospitalisation at a rate of 0.5 per day, the epidemic curve peaks 3-fold higher among people with comorbidities. The infection peaks are delayed if the area occupied by community is increased. In conclusion, we observed that community infections increase significantly with decreasing detection rates for both individuals with or without comorbidities.

Since the reporting of its first case in December 2019, Corona virus Disease (COVID-19) has had a significant impact on global public health with over 246 million reported cases and more than 4.9 million deaths as of October 2021 coupled with an imaginable economic devastation. 1 In some countries, as hospitals run out of patient admission facilities, disease severity is being considered among the factors to decide on whether or not J o u r n a l P r e -p r o o f Journal Pre-proof to admit a patient. This criteria calls for the evaluation of intervention scenarios with special consideration for the vulnerable people such as the elderly and the comorbid individuals.

For COVID-19, it can be concluded based on clinical observations and studies that among infected individuals, adverse clinical outcomes such as case severity and mortality seem to vary regionally and also vary within a 10 given population. The elderly and comorbid individuals seem to be the most vulnerable to severe outcomes of COVID-19. This variation is likely due to differences in immune response capacity related to age and the presence of medical comorbidities and pre-existing conditions that may exert pressure on the immune system. 2 There are reports of a strong link between severe and/or fatal COVID-19 and other communicable or noncommunicable diseases and such as age and risk factors such as smoking, exposure to polluted air. 15 The relationship between comorbidities and disease severity seems to have been observed globally. For example, Ye et al., 3 studied the hospitalisation data for Zhejiang Province in China and reported that COVID-19 patients with comorbidities had worse clinical outcomes. The study also revealed that extreme and serious manifestations of adverse outcomes were positively correlated with the number of comorbidities the patient 20 suffered from. Older obese patients were reported to experience more severe clinical outcomes. 4 Comorbidities such as diabetes, 5, 6 hypertension, cardiovascular disease, 7 chronic lung disease, tuberculosis 8 and the malnourished and those with HIV 2 were all reported to be more affected. For example, the incidences of hypertension, cardio-cerebrovascular diseases, and diabetes mellitus were 2-3-fold higher in intensive care unit (ICU)/severe cases than in their non-ICU/severe counterparts. 7

In a French study of 124 consecutive hospitalised COVID-19 patients, obesity and severe obesity were present in 47.6% and 28.2% of cases, respectively 9 and for the United States, Thomas et al. 10 reported approximately 42% as being obese of whom 85% had type 2 diabetes. Another modelling study with comorbidities parameterised using data from Nigeria 11, 12 showed that the top ranked parameters that drive the dynamics of the 30 co-infection model were the effective contact rate for COVID-19 transmission, the parameter accounting for increased susceptibility to COVID-19 by comorbid susceptibles, the comorbidity development rate, the detection rate for singly infected and co-infected individuals, as well as the recovery rate from COVID-19 for co-infected individuals. 35 Enhancing the overall efficiency of the public health responses, requires a thorough understanding of the impact J o u r n a l P r e -p r o o f Journal Pre-proof of COVID-19 on these comorbidities and mathematical modelling permits evaluation of various comorbidity and age-dependence scenarios on disease dynamics. Modelling comorbidity is not only seen from the perspective of the dynamics of re-infection and co-infection with comorbidities, but may be more to do with the impacts of the comorbidities on the severity of COVID-19. It is well observed that individuals with underlying health 40 conditions are more likely to report to health facilities than those without comorbidities. There is therefore high chance of persons without comorbidities likely to remain in communities spreading the infection as compared to persons with comorbidities.

Many aspects of the dynamics of COVID-19 are built on pre-existing health conditions and adherence to 45 standard operating procedures (SoPs). Models that can provide insights on how individuals with compromised health conditions are affected by COVID-19, need to be developed to help in designing effective mitigation and intervention strategies. Furthermore, size dependent area has been incorporated into the model in order to measure the effectiveness of social distancing. Therefore, a mathematical model has been developed to ascertain how COVID-19 dynamics impacts underlying comorbidities, and then used to investigate how early detection 50 and reporting for treatment benefits the overwhelmed health facilities, consequently minimising community transmissions.

In this section, we present a model for the transmission dynamics of COVID-19 amongst a population with a 55 proportion of individuals having underlying comorbidities. We consider the entire population to be susceptible S(t) and when infected with COVID-19, they are classified as exposed individuals with comorbidities (E m (t)) or without comorbidities (E c (t)). Individuals will then progress to the infectious classes (I m (t)) and I c (t) respectively depending on their comorbidity status. Individuals in the I m (t) and I c (t) will either by contact tracing or self reporting join the the hospitalised class (H(t)) . Recovered individuals from hospitals are grouped in the 60 R(t) class.

The transition process is as follows: Susceptible individuals get infected through contact with an infected person.

Once infected, depending on their co-morbidity status, a proportion m will join the E m (t) while the remaining proportion will be grouped in the E c (t) class. After a given latent period, latently infected individuals will J o u r n a l P r e -p r o o f Journal Pre-proof progress to the I m (t) and I c (t) classes at rates ρ m and ρ c respectively. Due to the underlying conditions of individuals in the E m (t) compartment, it is assumed that ρ c < ρ m . To explicitly capture and highlight the impact of comorbidities, we assume similar characteristics for asymptomatic and symptomatic and divide infected class into I c (t) and I m (t). Individual in the I m (t) and I c (t) classes may be traced and hospitalised or do self reporting at rates ω m and ω c respectively. It is assumed that those in the I m (t) class are likely to do self reporting much 70 easier given their status as compared to those in the I c (t) class. It is assumed that the disease related mortality rates will vary depending on whether the infected individual has a co-morbidity. The disease related mortality rate for individuals in I m (t) class is considered to be δ m while that of individuals in the I c (t) is given as δ c .

Hospitalised individuals may recover from COVID-19 at a rate α or die at a rate δ h . The recovered individuals have temporary disease induced immunity 11 which wanes off and they become susceptible again at a rate τ .

The model captures entry of individuals at a constant rate π of which a proportion θ being latently infected with a co-morbidity, e latently infected without co-morbidity, c being a proportion of those that are confirmed to be infected at the entry points and immediately hospitalised and the rest being susceptible. Susceptible and recovered individual exit the community at a per-capita rate µ.

To model COVID-19 transmission, we consider an area-dependent force of infection term given as βbS(I c + I m + gH) /A where β is the transmission rate and b is proportion of susceptible individuals that do not adhere to standard operating procedures, g is the percent reduction of hospital acquired infections and A is the total area occupied by the given community. 

Taking into account the above description, along with the dynamics as in Figure 1 , gives the model system as;

J o u r n a l P r e -p r o o f Journal Pre-proof

We ascertain epidemiological feasibility of the model system (2.1), by guaranteeing that starting with non negative initial conditions, the solutions will remain non-negative for all t ≥ 0. Consider the given initial

, H(t), R(t)), will remain non-negative for all t ≥ 0.

To prove this, it is sufficient to show that all the trajectories of system (2.1) are non-negative for time t > 0.

From the first equation of system (2.1), the evolution of susceptible individuals over time will be given by the inequality;

By solving the inequality and taking the limit as t → ∞, we have 

We observe that N > π µ whenever dN dt < 0. Since the right hand side of dN dt is always bounded, then by standard comparison Theorem, 13 it can be shown that

The domain D is positively invariant under the flow of system (2.1). Therefore, system (2.1) is biologically feasible and mathematically well posed in D.

Journal Pre-proof 3.1. Local stability of the disease free equilibrium and computation of R 0

The model system (2.1) has a disease free equilibrium (ξ 0 ) given by ξ 0 = (π/µ, 0, 0, 0, 0, 0, 0). Using the next generation matrix described by van den Driessche and Watmough, 14 the local stability of ξ 0 is investigated. The 100 basic reproduction number, R 0 (defined as the average number of secondary infections generated by COVID-19

infectious individuals through out their infectious period if introduced into a population with a proportion of individuals having a comorbidity) is also determined by the same method. Let F denote the rate of appearance of new infections into the infected compartments and V the transfer in and out of the infected compartments.

We obtain the derivatives of the matrix and F and V evaluated at the ξ 0 . Thus,

The spectral radius, ρ, of matrix F V −1 gives the reproduction number as,

The threshold quantity R 0 is the basic reproduction of the model system ( in Theorem 2, the following lemma is established.

community where a fraction of individuals are having comorbidities, then the resulting number of secondary cases will not lead to an outbreak whenever R 0 < 1. But if R 0 > 1, then the number of cases in the subsequent generation will be greater than the former and as a result the disease will spread and become endemic in the 125 community. In this case the disease will continue to spread until the proportion of susceptible individuals is too small such that the probability of infecting a new person is very low. To ensure that elimination of the virus from the population is independent of the initial population size, we investigate the global stability of the disease free equilibrium.

130 Theorem 3.1. The DFE, ξ 0 , of the model system (2.1), is globally asymptotically stable in the invariant region

Proof. Consider the positively definite Lyapunov function,

The time-derivative of function (3.1) is given by, 

In the absence of infected recruits, the parameters θ, e and c (the proportions of latently infected undetected individuals with or without comorbidity and detected infected individuals respectively recruited into a closed community) are zero. Therefore equation (3.2 ) reduces to;

Clearly, λ * = 0 is a solution and in this case, corresponding to ξ 0 and when λ * = 0, the coefficient C 1 summarises as

In this case, a J o u r n a l P r e -p r o o f Journal Pre-proof unique endemic equilibrium point (ξ 1 ) is obtained as;

Thus, the following lemma is established. Re-defining the state variables (S, E m , E c , I m , I c , H, R) as (x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 ), the associated system (2.1) 155 is given as;

The bifurcation parameter φ obtained by equating R 0 to one is given by

.

By Linearising system (3.3) at disease free equilibrium (ξ 0 ) and with φ the bifurcation parameter, we obtain

The Jacobian matrix J ξ0 has zero eigenvalue and the rest are negative.

The left eigenvector associated with the zero eigenvalue of (3.4) is given by

Similarly, the right eigenvector of (3.4) associated with the zero eigenvalue is given by w = (w 1 , w 2 , w 3 , w 4 , w 6 , w 7 ) T with v.w = 1 where,

Next we compute the non-zero partial derivatives of system (3.4) with respect to the state variables that are used in the computation of coefficients a and b defined as,

Thus, we obtain,

(3.8)

The non-zero partial derivatives of system (3.3) with respect to state variables and the bifurcation parameter, φ are obtained as;

Aµ , 

According to the Center Manifold Theorem, 16 if B > 0 (ω c > ω m ) and R Ic < 1, then the coefficients a > 0 and 160 b > 0. In this case, a backward bifurcation would occur but it is not possible because the absence of infected entrants into the community yields a unique endemic equilibrium. If B < 0 (ω c < ω m ) and R Ic < 1, then the coefficients a < 0 and b > 0. In this case, the model exhibits a transcritical bifurcation at R 0 = 1 and the "No-imported-case" equilibrium is locally asymptotically stable. On the contrary, if R Ic > 1, then a < 0 and b < 0 implying that the "No-imported-case" equilibrium is unstable. Therefore the following theorem is 165 established.

Theorem 3.2. The unique endemic equilibrium point ξ 1 (i) is locally asymptotically stable only if ω m < ω c , R Ic < 1 and R 0 > 1.

(ii) is unstable if ω m > ω c and R Ic > 1.

According to Theorem 3.2, it is important for COVID-19 comorbid patients to report early for treatment and 170 to also strictly observe SOPs since the reaped associated benefits of a significantly reduced infectious period would consequently lead to halting the spreading of the disease within the community. It is worth noting that in such a community, the disease dynamics is driven by individuals without comorbidities. Since some of these individuals are asymptomatic, they continue transmitting the disease and make its control very complicated. Proof. Suppose θ, e, c = 0, then within a community, the endemic equilibrium point will be given by roots of polynomial (3.2) . In this case, the coefficient C 0 < 0 and C 2 > 0. Therefore the only positive root of polynomial

which gives the endemic equilibrium point as;

,

Lemma 3.3 guarantees the possibility of an endemic equilibrium point ξ 2 and its existence reveals that the disease will persist as long as the infected individuals are continuously recruited into the community.

In this Section, numerical simulations are presented to gain more insights on the model properties under various scenarios. Simulations are performed using MATLAB, version 2020a (Math Works, Inc.) software. The 185 parameter values given in Table 1 are obtained from existing literature and some are as estimated basing on the Ugandan data. 

This study develops and analyses a habitat-size dependent deterministic mathematical model that is then used to gain insights into the impact of early detection and treatment on the COVID-19 epidemic curve in a community with some co-morbid individuals. The model formulated by subdividing the population into susceptible, 220 exposed with and without comorbidities, infectious with and without comorbidities, hospitalised and recovered classes. It considers a habitat area-size dependent force of infection, whereby the transmission is based on the possibility of individuals to social distance within the community.

The solutions derived from Lyapunov stability analysis show that the model has a globally asymptomatically 225 stable disease free equilibrium (obtained when infected individuals are denied entry to the community) whenever the basic reproduction number is less than unity. This implies that, COVID-19 can be eliminated irrespective of the initial number of infected individuals introduced in the community with some individuals having underlying health conditions. The basic reproduction number was found to be directly proportional to the infection rate, the proportion of individuals with comorbidities and those adhering to standard operating procedures and 230 inversely proportional to the recovery rates, COVID-19 detection rates and the area size occupied by the community. Our results indicate that without infected entrants, the stability of the endemic equilibrium point when R 0 > 1 also depends on the basic reproduction number associated with the individuals without comorbidities.

It is shown that this endemic equilibrium is only locally asymptotically stable if the basic reproduction number associated with individuals without comorbidities is less than unity else it is unstable. These results reveal 235 that as long as the individuals without comorbidities fail to observe the standard operating procedures, then COVID-19 cases especially amongst individuals with underlying conditions will increase which will overwhelm the health system.

Simulation result indicate that the number of detected and undetected cases increases with increasing propor-240 tions of individuals with comorbidity (see Figure 2 ). This has been observed in several countries where there is significantly high prevalence and number of deaths as a result of comorbidities 23 

In conclusion, our study emphasises adherence to standard operating procedures, early detection and treatment of COVID-19 patients especially those that do not have underlying health conditions, so as to lessen community transmission of the disease.

The authors declare that they have no known competing financial interests or personal relationships that could 255 have appeared to influence the work reported in this paper.

COVID-19 Coronavirus pandemic updates

COVID-19 Across Africa: Epidemiologic Heterogeneity and Necessity of Contextually Relevant Transmission Models and Intervention Strategies

Impact of comorbidities on patients with COVID-19: A large retrospective study in Zhejiang

COVID-19 and comorbidities: A role for dipeptidyl peptidase 4 (DPP4) in disease severity?

Presenting Characteristics, Comorbidities, and Outcomes Among 5700 Patients Hospitalized With COVID-19 in the New York City Area

Mathematical modeling of COVID-19 transmission dynamics in Uganda: Implications of complacency and early easing of lockdown

Prevalence and impact of cardiovascular metabolic diseases on COVID-19 in China

COVID-19 and tuberculosis: A mathematical model based forecasting in Delhi, India

High Prevalence of Obesity in Severe Acute Respiratory Syn-280

drome Coronavirus-2 (SARS-CoV-2) Requiring Invasive Mechanical Ventilation. Obesity (Silver Spring)

A Primer on COVID-19 Mathematical Models

Analysis of COVID-19 and comorbidity co-infection model with optimal 285 control

COVID-19 and postinfection immunity: Limited evidence, many remaining questions

Stability analysis of nonlinear systems

Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission

The stability of dynamical systems

Dynamical models of tuberculosis and their applications

Government of Uganda/UN High Commissioner for Refugees

Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study

Mathematical model for COVID-19 management in crowded settlements and high-activity areas

A conceptual model for the coronavirus disease 2019 (COVID-19) outbreak in

China with individual reaction and governmental action

Prevalence of Underlying Diseases in Hospitalized Patients 305 with COVID-19: a Systematic Review and Meta-Analysis

Prevalence of comorbidities among individuals with COVID-19: A rapid review of current literature

Mathematical modelling of the dynamics and containment of COVID-19 in Ukraine

The authors thank the Government of Uganda and Makerere University Research and Innovation Fund for the support to carry out this study, and the Ministry of Health for the data-sets on COVID-19. We also appreciate the suggested useful comments of the anonymous reviewers towards the improvement of this paper.